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Measurement Scales That Are Used in Collecting Essay

… ¶ … measurement scales that are used in collecting and organizing research data. The scales discussed in this text are: nominal; ordinal' interval' and ratio. The differences between the measurement scales determine how data can be manipulated and studied and which kinds of conclusions can be reached using the data.

Nominal scales are the simplest of the measuring devices. These scales only measure the classification of items. Either the data does or does not fit the scheme. An example of this is gender. When you ask for a respondent's gender on a survey form, this is a nominal question. There are only two answers, and the only arithmetic you can use on the data is to count how many of each kind are in each classification.

Ordinal scales include the classification involved in nominal scales, but they can also measure preferential order. In addition to classifying popcorn, for example, as air-popped or oil-popped, the respondent can also rank the different flavors of popcorn in order of best to worst. The limitation of using ordinal scales is that there is no uniform interval between the rankings. The difference between "highest" and "high" isn't necessarily the same in the respondent's mind between "lowest" and "low."

Interval scales take care of the problem with ordinal scales, as they allow for classification, order and an equality of difference between rankings. The difference between 1 and 2 is the same as the difference between 4 and 5, for example. The data is relatively symmetrical and researchers can do operations like plotting a standard deviation and finding the mean. Intervals can be used to find averages, which is not possible with ordinal rankings.

Ratio scales are the most complicated and can be the most useful. They include all the qualities of nominal, ordinal, and interval scales, plus the provision for an absolute zero. Measurements like distance, volume, length, money, population counts, etc. are all measured on a scale that begins at zero. Ratios are most useful with data that can be definitively measured and they are more useful in the hard sciences than behavioral sciences.

Choosing a measurement scale inherently involves looking at white…… [read more]

Early Reading or Fluency and Word Identification Essay

… Jeffery Case Study

Background for Jeffery

Jeffery has problem understanding algebraic concepts such as polynomials and factoring. Jeffery doesn't require any special aids and his learning capabilities are normal for his age.

Ability to solve polynomials and factorization with the help of Mnemonic device such as FOIL

develop the ability to understand order in which polynomials are solved

Become familiar with complex algebraic terms with the help of vocabulary teaching.

Supporting information for set goals

John Steele (2003) explains that children having difficulty with mathematical concepts can be helped with the use of mnemonic devices because "Mnemonics are useful for memorizing rules, steps, and procedures." (p. 624)

Vocabulary teaching is also critical for developing good understanding of algebraic concepts. This has been proven by Richard Drake's research of 1940 and was later supported by studied carried out by others. Myszczak found that vocabulary learning is significant because it teaches students the ability to focus on the question and what is being asked as "students often search the problem for numbers rather than attempting to comprehend what is truly being asked in the problem" (p. 28)

Schoenberger and Liming (2001) focused on the teaching of a specific list of words that could help students learn mathematical concepts. They believed that "Students should be able to use and understand vocabulary in order to think about and discuss mathematical situations" (p. 27).

Apart from teaching vocabulary, it was also found that simply communicating with students while teaching mathematical concepts could help facilitate better understanding of the concepts. In other words, students could communicate the problem and seek the solution by talking to their teachers. "Language is a major medium of teaching and learning mathematics; we serve students well when we support them in learning mathematical language with meaning and fluency" (Rubenstein, 2007, p. 206).


First Jeffery needs to understand that any two sets in parenthesis without any sign in the middle means multiplication. Once he understands that he needs to understand the…… [read more]

Dot Maps of Three Datasets Term Paper

… In the same figure, pine tree data points are randomly distributed in grids. The distribution density changes by changing the size of quadrat area (see Figure 5). For example, 8th and 13th quadrats of redwood data exhibit the highest density of data points whereas 1st, 6th and 14th quadrats do not contain any data point.

In order to test the CRS hypothesis, the appropriate quadrat counts were visually investigated (see Figure 2, 3, and 6. Eventually, the grid size is chosen as 10x10. The mean value, variance, standard deviation and VMR for each dataset were calculated by using MATLAB scripts (i.e., mean, var, std). The results are shown in table 3.

Table 3. Mean, variance, standard deviation and VMR values calculated for 10x10 quadrats.




















The mean values of Redwood and pine datasets are similar; however, it was already analyzed that these two datasets show different distribution characteristics. Variance (square of standard deviation) is another value used to describe the distribution of datasets. Since the variance provides the information about the relative distance between mean and data point, one could say that the cell data is the most uniform data within analyzed datasets. The last value calculated in table 3 is the variance mean ratio (VMR). This ratio of the variance value to the mean value is used to identify whether the dataset is dispersed or clustered compared to CSR hypothesis. Finally, VMR value would quantify the distribution model of any dataset. The VMR value for Poisson distribution is defined as 1.The negative binomial distribution requires a VMR value larger than 1 while binomial distribution has a VMR value smaller than 1. In this regard, the cell and the pine datasets are under-dispersed. In other words, the data points are distributed uniformly in the spatial domain. Therefore, these two datasets obey the definition of binomial distribution. The redwood data is over-dispersed meaning the data points are clustered together. Thus, the underlying distribution model for redwood is negative binomial distribution.

Taken all together, one can define the distribution models of the datasets; however, the relationship between individual data points cannot be discussed only by quadrat count method.

Figure 6. 25x25 Grid.

To validate a null hypothesis the z-scores of three datasets is calculated using CrimeStat-III. The criteria for z scores (95% confident) are as follows.

(1) Region of significance (rejection of the hypothesis) if the set of z scores is outside of the range -1.96 to 1.96

(2) Region of non-significance (acceptance of the hypothesis )if the set…… [read more]

Digital Audio Book Report

… Digital Audio

Over the last several years, digital audio has been continually innovating the way that people listen to and write various forms of music. This is because, how sound waves travel can be challenging, as the different instruments will reflect: the pitch, tone, bass and timber in varying degrees. When trying to record these sounds this can be challenging, with the various waves reflecting certain amounts of frequency. The problem begins with trying to reflect these various tones in real life, as different analogue and acoustic devices are only focused on the continuous flow of the music. When this is replayed, the various sounds will often be distorted, because they cannot reflect the actual extremes of these reverberations in real life. In the last few years, digital audio has become an increasingly popular solution in addressing these different challenges. With this form of recording focused on the specific mathematical values that the music will represent. As various mathematical formulas using decimals will be used to more accurately reflect, the different sounds that are being recorded. This is important, because it shows how digital recordings are working to accurately reflect the various sounds that are recorded. To fully understand how this new technology is able to accurately, capture the overall collection of different sounds requires: comparing these forms of technology with one another. Once this takes place, it will provide the greatest insights, as to the how digital audio technology is improving the way everyone listens to and records music. (Pohlmann, n.d. )

Digital Recording vs. Analogue and Acoustic Recording

The big difference between digital recording and analogue / acoustic recording is the way the various sounds are represented in a binary number. These are the different mathematical calculations that are used to convert the various sounds being recorded, into actual resonance on the recording device. What makes digital audio more accurate is the way that the binary code is represented. Where, it is more concerned about the actual decimals of the numbers vs. The underlying wave. In mathematics,…… [read more]

Spiritual Principle: So Teach Us to Number Term Paper

… Spiritual Principle:

So teach us to number our days, that we may apply our hearts unto wisdom. (KJV Psalm 90:12)

The school year consists of two semesters. Within each semester are three units. During unit one of the first semester which is four weeks, students will learn about functions. During the second unit of the first semester which is five weeks long, students will learn about algebra investigations. During the third unit of the first semester which is seven weeks long, students will learn the geometry gallery. The second semester of the school year also consists of three units. During the fourth unit which is six weeks in duration, the chance of winning will be covered. The fifth unit of the second semester is also six weeks long and students will learn algebra in context. The objective of the sixth and last unit is for students to learn coordinate geometry. The last unit is four weeks long.

Suggested Activities and Experiences:

1. To learn functions, students will spend lab time exploring the National Library of Virtual Manipulatives ( By clicking on the functions button, students can play the game that allows them to drag an input number into a machine which then gives the output. Based on the pattern of inputs and outputs, students can figure out what the remaining inputs and outputs will be based on the pattern established by playing the game. This game will allow the students to learn the basic concepts of function in order to move on to other more challenging concepts.

2. To learn algebra investigations, students will download the Mathematics I Frameworks: Student Edition document ( On page seven (7) of this document is an exercise called the…… [read more]

Geometry of Design Elam, Kimberly. ) Book Review

… Geometry of Design

Elam, Kimberly. (2001). The Geometry of Design. New York: Princeton Architectural Press.

The Geometry of Design is not a book about nature, physics, or even of design. Instead, it is a relatively short and simple overview of the role of geometry within nature -- whether it is the analysis after the fact from a human perspective or the way nature works that we find pleasant, the book explains the prevalence of the Golden Mean and other geometrical thermos within nature's design.

Proportion in Man and Nature - Proportion is all around us, it is in everything designed within the sphere of nature; a leaf, a shell, a flower. And these proportions are instinctively pleasurable for us, which is likely the reason why much of design and architecture is based on the very same principles of ratio, proportion, and structure. The basis for this design structure is the Golden Ratio, or 1:1.618. Since the Renaissance, this is the proportion that has been used by artists and architects to proportion their works for mass appeal. Fascinating, however, is just how many objects in nature follow this exact proportion.

Talking Points-

Nature is typically proportionate in design, showing smaller objects to be part of a greater whole.

Even animals show this same proportion, a fish for example, when split into individual rectangles, retains the 1:1.618 ratio.

Similarly, the human body in classical drawing (Leonardo, the Greeks, etc.) form similar ratios.

Preferred facial proportions also follow the ratio; faces that do not are often considered less pleasing.

Chapter 2 -- Architectural Proportions - Through a series of dynamic rectangles, humans have developed their entire building system off this ratio. The harmony of space, e.g. windows, doors, arches, etc., especially in public buildings (governmental locations, arenas, religious buildings), all serves both to inspire and make one comfortable.

Talking Points

Ancient architects were very concerned with the way a building was shaped, laid-out, and built. It had to conform to strict proportions in order to be appropriate from a symboligist viewpoint to its function.

Each architectural discovery and innovation resulted in a reestablishment of the principles of appropriate design (e.g. circular stained glass windows in cathedrals, etc.)

This tradition remained in effect for several centuries; progressing through styles like the Baroque, Gothic, Romantic, etc.

In 1931, a French architect, Le Corbusier, expanded this into a more complex merging of mathematics and geometry -- regulating lines. He believed "with regulating lines, you make God a recipe."

In a way, this invigorated the reemphasis on proportion and meaning to form a more 20-21st century way of applying the Golden ratio to modern construction and design.

Chapter 3- Golden Section- the Golden section of any rectangle is a ratio of the Divine Proportion. The Divine Proportion is derived from the division of a line segment into two segments such that the ratio of the whole segment is the same, as 1: 1.61803. This ratio can be found in any portion or sub-portion of a triangle, rectangle,… [read more]

Group Will Behave, We Make a Hypothesis Term Paper

… ¶ … group will behave, we make a hypothesis, a testable proposition (or set of propositions) that are believed to be true which seeks to explain the occurrence of some specified group of phenomena, (Random House, 2010). For example, let's say that the widget making department is producing fewer widgets per hour this year than last year despite the fact that the number of employees has remained constant. You hypothesize that their decreased productivity is because of low morale but how do you know if your hypothesis is correct?

Hypothesis testing is a statistical way of testing the validity of a hypothesis. In business and the social sciences, hypothesis testing allows us to generalize about a population based on sample information by using methods that allow the research to separate the effects of systemic variation of a variable from mere chance effects (Sarich, 2010). This is particularly important in business because we often cannot isolate or control for phenomena in a laboratory -- type setting the way a physicist or biologist can (Sarich, 2010).

A 1999 study on the automobile insurance industry appearing in the Journal of Economics and Business illustrates the real world applicability of hypothesis testing. The study entitled Modeling Market Shares of the Leading Personal Automobile Insurance Companies, looks to identify the advantages that give one firm more market share over another. The author uses several hypothesis tests to analyze the market share of the leading personal auto liability insurers from 1980 to 1994, discovering in the process that automation and advertising are significant sources of competitive advantage, and that price-cutting, reductions in commission rates and concentration in the private passenger line of insurance are not - useful information in helping an insurer to decide where to…… [read more]

History and Present Day Applications of Logarithms Essay

… Logarithm

History and Modern Applications of Logarithms

The first time a publication contained a mention of logarithms, their method of derivation, and a table of logarithms was in 1614 with the publication of Mirifici Logarithmorum Canonis Descriptio by the Scottish nobleman John Napier (ST 2005). Napeir's book did not describe or list logarithms as they are known today, but rather the logarithms contained in this work were meant to replace the trigonometric multiplication functions needed in astronomy and other branches of science with a simplified addition from other established figures (Capmbell-Kelly 2003). Henry Briggs, a professor of geometry at Oxford, was very inspired by Napier's work, developing his own ideas based on those in the Descriptio and meeting with Napier to discuss developments and recalibrations of the logarithms contained in Napier's original and pioneering work on the subject (Campbell-Kelly 2003).

Briggs would go on to publish his own table of logarithms for common numbers (as opposed to the logarithms for sines contained in Napier's column); Briggs' tables showed the logarithms for every whole number below 1000 carried out to eight decimal places, providing a very useful tool to the navigators, astronomers, and mathematicians working in this day centuries before the advent of computer and calculators (ST 2005). This was published in 1617, the year of Napier's death, and by 1624 Briggs had expanded his tables to include all integers from 1 to 20,000 and from 90,000 to 100,000, carried out to fourteen decimal points (ST 2005). These tables led to a massive increase in use of logarithms in certain fields where their usefulness was already established, and this subsequently led to expansions in the applications for logarithms generally (Campbell-Kelly 2003).

There are many different modern applications for logarithms that have nothing to do with the distances of navigation and astronomy -- or any physical measurements at all -- proving that logarithms are indeed an incredibly useful mathematic tool on a scale that Napier himself did not really envision. Anything that involves exponential growth can most easily and accurately be calculated using logarithms; studies of population growth, nuclear reactions, and any other scientific inquiries depend on the use of logarithms to develop real and usable data and projections (Tom 2002). Logarithmic scales also exist in electrical engineering, as a means for testing for signal decay, and there are many bodily functions and reactions that are logarithmic in nature, leading to many other biological and medical uses and needs for an understanding and utilization of logarithms (Tom 2002).

Another common use for logarithms is in the world of banking, specifically in the calculation of interest and periods of repayment on…… [read more]

Digital Audio Broadcasting System Case Study

… Analogue and Digital Converter

This is an electronic device that helps in the conversion of continuous signals to discrete or isolated digital numbers. When an analogue voltage or current is fed into the device as an input, it converts it into a digital number relative to the voltage or current magnitude. There are a number of terminologies related to ADC, which include resolution, accuracy, response type, sampling rate, aliasing, dither, oversampling, relative speed and precision, and the sliding scale principle, however, just a few of them will be considered in this study.

The resolution of an ADC refers to an indication of the isolated values that in can generate over the range of analogue values. Since the electronic storage of these values is in binary form the resolution is normally expressed in bits and the available discrete values being a power of two. For instance, an ADC whose resolution is 6 bits has the capability of encoding an analog input to one in 64 varied levels, given that 26=64. It is also possible to define resolution electrically and give the expression in volts. The voltage resolution of an ADC is found by dividing the overall voltage measurement range by the number of isolated intervals. The formula is written as:


Q= resolution in volts/step i.e. (volts/output codes-1)

EFSR= full scale voltage range which is given by VRefHi -- VRefLow

M= ADC's resolution in bits

N= number of intervals= 2M -- 1 (Knoll 1989)

Consider an example given by Knoll (1989) where the Full scale measurement range is 0 to 7 volts, the ADC resolution will be 3 bits which means 8 quantization level sie. 23. When this is given in terms of ADC voltage resolution it equals 7V/7 steps which give 1V/step.

The ADC is not exempted from errors that are encountered by other instruments and has errors that have a number of sources which brings about the question of accuracy. These errors are categorized as quantization error, non-linearity error and aperture error. Quantization error is caused by the finite resolution of the ADC and cannot be avoided in any ADC while non-linearity error occurs due to the physical imperfections of the ADC which leads to a deviation between the output and the input from a linear function. The third error is caused by a clock jitter and is usually exposed when digitizing a signal that is time variant. The non-linearity error can be toned down by calibration or eve averted by testing. In most ADCs the range of input values that map to every output value are linearly related to that output value and are referred to as linear ADCs.

The speed and precision of an ADC varies depending on the type of the ADC with the Wilkinson ADCs being considered the best since they exhibit the best differential non-linearity. ADCs are usually represented using a symbol; the conventional electrical symbol used is as below (schematic).

Demodulator (Band pass filter)

A band pass filter is a device that helps… [read more]

Educational Standards Thesis

… Communicating No Child Left Behind Daily Standards

Grade-Appropriate Restatement of New York State 2nd Grade Math Standards

Original Statement of 2.PS.1:

"Explore, examine, and make observations about a social problem or mathematical situation"


of 2.PS.1:

We're going to look at the kinds of problems people have and the kinds of problems that mathematics can help us solve.

Original Statement of 2.PS.2:

"Interpret information correctly, identify the problem, and generate possible solutions"

Restatement of 2.PS.2:

We're going to learn how to understand what kinds of problems we have to solve and how we can use mathematics to do that.

Original Statement of 2.PS.4:

"Formulate problems and solutions from everyday situations (e.g., counting the number of children in the class, using the calendar to teach counting)"

Restatement of 2.PS.4:

Some of the problems we're going to look at are the kinds of things that people need to figure out all the time, like how to count how many students are in a big room without counting on our fingers and toes.

4. Original Statement of 2.RP.3:

"Investigate the use of knowledgeable guessing as a mathematical tool"

4. Restatement of 2.RP.3:

We're going to learn what an "educated guess" is, how that is different from regular guessing, and how to use educated guesses in mathematics.

5. Original…… [read more]

Operational Definitions Essay

… ¶ … Operational Definitions of Each of These

It states clearly the expected relationship between the variables

It states the nature of the relationship

It states the direction of the relationship

It implies that the predicted relationship can be tested… [read more]

Teaching Calculus to Young Children Thesis

… Mamikon's Approach To Teaching Calculus

Mamikon's A. Mnatsakanian, often along with his colleague Tom M. Apostol, has published many papers detailing new instructional methods for explaining otherwise complex concepts in the realm of calculus, as well as new ways of understanding these concepts. His emphasis is on a visual understanding of calculus, which is more easily observed and intuited by students -- and at a younger age, it seems increasingly evident -- than traditional textual and purely mathematic explanations and understandings. For years, a website has been available with several puzzles and games that help to visually express many of the mathematical measurements and principles of calculus. Several brief examples of Mamikon's teaching style make it clear how the principles of calculus build on lower mathematic understanding, and are in fact easily understood themselves.

Measuring the area of a curved space is essential for many applications of calculus, yet can be one of the more difficult among the basic principles and practices of the average calculus student. Mamikon's illustration of the curving bicycle, and subsequent related illustrations, show quickly and easily how the area described by such a curve is the same as the area -- or partial area -- of a circle (CalTech). The preceding sentence is proof of how difficult such concepts can be to clearly and efficiently explain, but accompanied by Mamikon's illustrations the principle is instantly observed and far more easily remembered and recognized.

A more thorough and elegant explanation of the same concept is provided on Mamikon;s paper (with Apostol) entitled "Subtangents -- An Aid to Visual Calculus." Again, Mamikon starts with a visual explanation of the principle, but goes on to detail this principles work in calculus (Apostol & Mamikon 2002). Thus, his method of teaching calculus visually creates at least a rudimentary understanding of a principle or practice before any theorem or even a simple equation is introduced. This is the opposite…… [read more]

Errors Type I/Type II Errors Statistical Analysis Thesis

… ¶ … Errors

Type I/Type II Errors

Statistical analysis can lead to many different errors of many different types, both in the gathering of data and the manipulation of it to produce results in a practical and relevant manner. Often, errors arise as a result of the complex mathematical manipulations that must occur in order to make useful sense of data. These mathematical errors can compound and lead to wildly incorrect interpretations of data, producing results that cannot be trusted or validly used. Other errors can occur in the interpretive phase of data analysis; these can often be far more egregious, and at the same time they are often more difficult to catch. Errors made in that actual mathematic manipulation of data often delivers results that -- for obvious reasons -- simply do not make sense. Interpretive errors, however, are more difficult to catch almost by definition. The data itself may be entirely sound, and therefore the results are more likely to be trusted, but an error in interpretation can still cause the data to be incorrectly applied.

There are two rather basic and fairly straightforward errors, known as Type I and Type II errors, that are commonly made in data analysis. Both refer to a basic mistake regarding the status quo from which the analysis is meant to measure change. This status quo is called the null hypothesis, the idea/belief that there was no change in the phenomenon measured during the test. When there is no change in the situation or phenomenon, the null hypothesis is said to be true (that is, nothing happened). If a change in the situation/phenomenon has in fact occurred, then the null hypothesis (the idea that nothing has happened) is quite clearly false. A Type I error occurs when there is a false positive -- when the data analysis suggests a change has occurred, when in fact there has been no change. Thus, in a Type I error the null hypothesis is true…… [read more]

Fractal, in Its Completed and Perhaps Complex Essay

… Fractal, in its completed and perhaps complex form, resembles a fracture or a series of complicated and uncoordinated breaks. Indeed, the word can trace its origins to the Latin fractus, which means fractured or broken. A fractal, as is mathematically… [read more]

Pi Is Interwoven With the History Essay

… ¶ … pi is interwoven with the history of humanity. Remarkably, "By 2000 BC, men had grasped the significance of the constant that is today denoted by pi, and…had found a rough approximation of its value," (Beckmann 9). Math historians assume that the study of pi began as an analysis of magnitude: that circles remained circles no matter how big or small. Beckmann suggests that early humans contemplated "the peculiarly regular shape of the circle," which was visible everywhere in nature in "its infinite symmetry," (9). Pi remains a mystery in spite of thousands of years of scholarship and investigation. The number is both irrational (it cannot be represented as a ratio of two integers) and transcendental (it is never the solution of a polynomial equation that involves rational numbers). Pi is remarkable in its scope. Professor Yasumasa Kanada of the University of Tokyo writes computer programs that are designed to calculate pi, and has continually broken his own world records. Kanada has computed pi to well over one trillion decimal places and remains "intent on achieving new world records" (Arndt, Haenel, Lischka, and Lischka 1). Because of Kanada's work, pi is now the mathematical constant "which has been calculated to the greatest number of decimal places," (Arndt et al. 1). In addition to performing the calculations for pure pleasure, Kanada and other mathematicians study pi in search of patterns. Understanding pi would be a significant epiphany, a major evolution in human history.

So far pi has yet to reveal itself fully and the number remains a major mathematical mystery. Pi can be understood easily on its most basic level: that of Euclidian geometry. The fundamental realization that the wider a circle is "across," the longer it is "around" is what led to the discovery of pi in the first place (Beckmann 11). That discovery seems to have occurred in multiple cultures, as pi was studied among the ancient Mesopotamians, Egyptians, and Chinese. The ancient Greeks delved deeply into the study of pi, especially pi's relationship to geometry. Pi was revealed as a constant ratio not just of circumference to diameter but also of radius to area. The existence of both constants was well-known, but the fact that both constants were in fact one and the same number represented a major breakthrough. Arndt et al. note that the ancient Greeks first drew the connection between both ratios as they related to the circle. In 414 BCE, Aristophanes presented the problem known as "squaring the circle," which has become the quintessential problem of pi.

Pi has numerous applications, and not just in the world of geometry. Number theorists hope to discover meaning in the endless stream of digits represented by pi, and pi could in fact be meaningful to the study of theoretical physics. Arndt et al. point out that calculating pi sometimes depends on time as well as space. Pi is also meaningful for probability theories, such as the Wallis product (Arndt et al. 9). Moreover, pi may be related… [read more]

Greek Numeration Systems Thesis

… Greek numeration system is one of the oldest in the world and still in use in many parts of Greece, especially for the ordinal numbers. The Greek numeration system was based both on its internal invention, as well as the constant interaction with some of the neighboring people, most notably the Phoenicians, the Egyptians and the Babylonians, all who had developed their own numeration systems and who thus influenced the Greek one.

There are two types of Greek numerations systems, depending on the moment they came into existence. The first type, predominantly referred to as Herodianic was used as early as 500 BC. As most of the old numeration systems, this was primarily an additive one, with the letters being allocated to the numbers based primarily on the first letter of the way the number was said. For example, penta was five, so the letter pi was designated to be the one representing the number five. In a similar manner, the letter symbol for 10 was the letter delta, which was because the number was referred to as deka and thus started with that respective letter.

This system was pretty much replaced with the Ionic system later on. The Ionic system was primarily based on the Greek alphabet, because it implied that for each unit 1 through 9, a letter of the alphabet would be allocated, a mechanism which was also applied for the tens (10 through to 90) and the hundreds (100 through to 900). However, the Greek alphabet only had 24 letters, which meant that three new ones were added for this purpose alone. These letters were digamma (an almost double gamma), qoppa and sampi. These letters were allocated for 9, 90 and 900 respectively.

At the same time, after each number thus written, a small sign would be added in the form of…… [read more]

Oxford Murders Matinez, Guillermo. Book Report

… Oxford Murders

Matinez, Guillermo. The Oxford Murders. MacAdam/Cage, 2005.

The Oxford Murders is the story of an unnamed Argentinean mathematician studying at Oxford. One day, while accompanied by his landlady's friend, the don and professor of mathematics Arthur Seldom, the two find Mrs. Eagleton murdered on her sofa. The only clue, other than the fact that the old woman worked on the Enigma Code during World War II, is a circle left by the killer in a mysterious note sent to Seldom, along with the lines, "the first in a series." Soon it becomes clear there is apparently 'serial' killing occurring, on a very literal level. Seldom receives a note, accompanied by a symbol, every time a murder takes place. Seldom fears that the killer is effectively parodying his mathematical work on theories of patterns or series in mathematics. One of Seldom's areas of expertise is Wittgenstein's theories about series and the possibilities for deviation in numerical series.

The…… [read more]

How Does My Calculus Class Help or Relate to a Business Management Major? Essay

… Calculus

The world of business is comprised of many unique disciplines. The manager can expect to synthesize all of them as part of their work. Therefore, a strong multidisciplinary background is essential in the pursuit of a major in Business Management. The subjects learned in this major will require a wide range of basic knowledge including economics, sociology, psychology and statistics. Calculus also plays a role, providing both a functional and a theoretical backdrop.

It is often considered that the basic functions of calculus are not used in the acquisition of a business education. However, understanding the fundamentals of calculus allows the student to derive formulae in financial courses. Students of Business Management should have a strong knowledge of corporate finance, and this requires some basic calculus. The models used to price derivatives are based on calculus. A stock option, for example, comprises an underlying asset with an intrinsic value and a fluid time value. This concept can be extended to any asset. Business Management is, at its core, managing assets. But to manage those assets requires the ability to understand how the value of those assets is derived.

Differential calculus forms a key component of the business world. Many important concepts in business management relate directly back to calculus. For example the yield curves on bonds, or the demand curve of a product relative to macroeconomic variables. There are many instances where a manager must interpret complex, interrelated and fluid variables in order to predict the future.

Integral calculus is useful when examining concepts involving fluidity. The business world is constantly changing. The numbers used to interpret the world in order to make managerial decisions are also constantly changing. The relationship between those numbers is also subject to constant flux. It is impossible to understand the business environment without understanding these relationships. A sound knowledge of calculus fundamentals allows for that.

The concept of limits also proves useful to the Business Management student. Management is a subject based on finding ways to achieve objectives. In many cases those objectives are quantifiable, thus to derive the best way to achieve those objectives requires calculus. The concepts, however, can also be applied to non-quantifiable objectives once the student understands the basic principles. The manager can then understand how to bring a variable such the organizational culture closer to a limit such as having a strong emphasis on integrity. Even without numbers, the principles of analyzing the relationships between variables remain the same.

A…… [read more]

Eudoxus of Cnidus Essay

… Eudoxus of Cnidus

Boyer, in his "A History of Mathematics" gives a quote from Eudoxus that is quite self-descriptive of this genius, "Willingly would I burn to death like Phaeton, were this the price for reaching the sun and learning its shape, its size and its substance."

It is descriptive of the man from Cnidus because it shows us the mind of this genius, the curiosity he displayed during his lifetime and why he contributed so much, in particular, to the fields of mathematics and astronomy.

Unfortunately, all of his works are lost to history. We have snippets, pieces, basic facts about Eudoxus' life and work, and some words from others through the ages who have dug up what could be found and put it together in biographies and descriptions of his work.


We know that Eudoxus was born in Cnidus, Asia Minor (now Turkey). Actually, historical documents claim a birth sometime between 408 B.C. To 390 B.C. And his death at the age of 50 to 53 years old. Best guess is 408-355 B.C.

He is known for his revolutionary work as a mathematician, astronomer, and philosopher. However, at some point in his life he was also a theologian, meteorologist, doctor, and "http: geographer. He studied mathematics in Italy under the tutelage of Archytas, the Greek mathematician and philosopher. Many historians claim that Eudoxus worked with Plato in Athens, but others dispute whether there is enough data to support that and are unclear about this relationship between the two great intellectuals. (O'Connor & Robertson, 1999) Archytas and Plato were close friends, so it is possible that Eudoxus met Plato, and, perhaps, this too, could explain part of the confusion whether or not Plato and Eudoxus actually worked with each other.

It is somewhat clear from historical records that Eudoxus had little respect for Plato's analytic ability, but since Plato was not the mathematician that Eudoxus was, that is to be expected. It does not appear as if either had much influence on the other's work. (O'Connor & Robertson, 1999)

Diogenes Laertius, the Roman biographer of Greek philosophers, claims that Eudoxus did, indeed, study in Athens under Plato. However, some of Laertius' usually solid work has come under question by other scholars, and, since Laertius' lived in the third century A.D., we can't be certain he was correct, since, again, all of Eudoxus' work is lost. (Soylent Communications, 2008)

He traveled to Sicily where he studied medicine with Philiston. After that, we surmise, with the help of financial aid from friends, he went to Egypt to learn astronomy with the priests at Heliopolis, and made astronomical observations from an observatory located between Heliopolis and Cercesura. From there Eudoxus travelled to Cyzicus, in northwestern Asia Minor on the south shore of the Sea of Marmara. There he established his own school which proved to be quite popular. As a matter of historical record, it appears that Plato became somewhat jealous of Eudoxus' success with his school. Not much more is… [read more]

Math Webliography Term Paper

… Math Webliography (

By Karen," this site is self-described as an "amusement park of math and more." colorful icons against a black background constitute child-friendly visuals, and menu items like "Number Monster" and "The Geometry of Crop Circles" are also guaranteed to please curious young and adult minds alike. (

Not as visually appealing as it could be, Kidsnumbers is still a valuable resource tool for teachers and parents. Several "Let's Practice" sections encourage children to play and interact. ( chalkboard cat icon welcomes children and their parents to the Web site, which several separate sections include "Math Cats Explore the World." However, the activities contained on the site are geared toward children older than the cute drawings would suggest. Mathcats is not for youngsters but is lacking in the sophistication that might draw a more mature audience.

TeachRKidsMath (

Seemingly geared toward teachers instead of students, TeachRKidsMath is not as child-friendly as it could be. The exercises are, however, good resources for math teachers needing some activities for their students.

Wolfram MathWorld (

Wolfram might indeed live up to its self-proclaimed subtitle, "web's most extensive mathematics resource." Containing a wealth of information on every mathematics topic of interest to advanced students from…… [read more]

PHI Golden Ratio Term Paper

… History Of Phi, Mathematical Connections, And Fibonacci Numbers: Nature's Golden Ratio

Throughout history, humans have been seeking to define beauty in quantifiable and meaningful ways. For many observers, the connection between beauty and the rhythmic patterns evinced in the Fibonacci… [read more]

Memory Ronald T. Kellogg's Working Memory Components Term Paper

… Memory

Ronald T. Kellogg's "Working Memory Components in Written Sentence Generation": A Review and Further Research Inspired by the Study

In his article, "Working Memory Components in Written Sentence Generation," Ronald T. Kellogg used quantitative research in order to how working memory is impeded during distraction. Kellogg begins his article with a literature review detailing the types of research that psychologists have completed regarding memory in the past. He cites Baddely's 1986 work, which created the Baddeley model, a "phonological loop for storing and rehearsing verbal representations, a visuospatial sketchpad for visual object representation and their locations, and a central executive for attentional and supervisory functions" (341). In addition to this model, Kellogg also more modern research by Jonides and Smith that suggests the "visual, spatial, and possibly semantic stores are dissociable from the verbal store" (341). By considering both this old and new research concerning working memory, Kellogg designed his own research in order to increase psychologists' understanding of working memory.


Kellogg designed his study in order to increase others' understanding of working memory. Specifically, Kellogg was interested in understanding if "planning conceptual representations" and "linguistically encoding these into words and sentences" depends on working memory (341). According to Kellogg, putting together a sentence, or sentence generation, requires "planning conceptual content," or deciding what one wants to say in writing, and "linguistically encoding it into a grammatical string of words," or placing those ideas and concepts into a well-formed, grammatically correct sentence (341). If one or either depended on working memory, a second purpose of the study was to review the strength of this dependence.


College students in a General Psychology class were chosen as the subjects to be tested. The students were assembled, given keyboards, and then given a visual prompt of two words, nouns. Subjects were then to write a "meaningful sentence" using the two nouns (344). At the same time, students were told to complete a "memory task," such as remembering certain digits. First, students' typing speed was assessed via trials. Next, students were given directions, followed by the two visual prompts, and then time to type their sentences and submit them to a computer. Finally, students were asked to complete the memory task they had been assigned correctly and were given feedback concerning whether or not they had made the correct response. Some students were asked to write sentences using nouns that were related, while others were asked to write sentences using nouns that were unrelated. In one group, students were asked to produce complex sentences; while in the other students were told to write simple sentences (344).

IV. Findings

In accordance with the methods above, the researchers derived results concerning, initiation time, sentence length and typing time, grammatical and spelling errors, and concurrent task performance. Only those students who were asked to remember six digits produced shorter sentences, while the memory tasks did not affect the length of any other students' sentences. Similarly, spelling and grammar were not affected by the memory… [read more]

Mathematician Biography and Works: The Mathematician Blaise Term Paper

… ¶ … Mathematician

Biography and Works: The Mathematician Blaise Pascal

The Life of Pascal

Blaise Pascal, along with Rene Descartes, is the rare case of a mathematician equally famous for his religious devotion and contributions to theology as he is for his work with numbers. In fact, Pascal would likely prefer to be remembered as a philosopher of religion rather than a mathematician, theorist, and scientist, as he is today. One biographer of famous mathematician tartly observed that Pascal's "mathematical reputation rests more on what he might have done than on what he actually effected, as during a considerable part of his life he deemed it his duty to devote his whole time to religious exercises" (Ball 1908). However, other biographers have seen Pascal's religion and mathematical gifts as complementary.

For example, "Pascal's Wager" is based upon the proposition that a person should believe in God because as a bet, the idea makes sense. There is so much to be gained potentially by believing in God if one is 'correct.' Likewise, there is so much to lose if one is an unbeliever, and one is 'incorrect.' Conversely, there is little to be lost by a person whose belief is in error. A proof of Pascal's Wager might look like this: 1) the probability of God's existence is 50/50. (2) Wagering for God brings infinite reward if God exists (Hajek 2005). If God does not, there is no net loss. Wagering against God brings no gain, and a great loss.

Despite the modernity, even humor, inherent in such moral calculations, Pascal was largely a man of his time, and a devout Christian. Blaise Pascal was born during the 17th century at Clermont on June 19, 1623, and died in Paris on August 19, 1662. Although the Frenchman's early education was confined to modern languages, when his father noted that the boy had unusual mathematical aptitude in geometry (Pascal intuited as a child why the sum of the angles of a triangle is equal to two right angles), his father gave his son a copy of Euclid's Elements. It would not be an understatement to call the young Pascal a prodigy. At the age of fourteen Pascal was admitted to the weekly meetings of French geometricians, at sixteen he wrote an essay on conic sections and at the age of eighteen, he constructed the first arithmetical machine, a kind of prototypical adding machine or calculator (Ball 1908).

However, Pascal suddenly abandoned mathematics in 1647, "after being advised to seek diversions from study and attempted for a time to live in Paris in a deliberately frivolous manner," because of his health ("Blaise Pascal," Island of Freedom, 2008). Pascal's interest in probability theory "has been attributed to his interest in calculating the odds involved in the various gambling games he played during this period" ("Blaise Pascal," Island of Freedom, 2008). However, Pascal's account in his Pensees is different. He says wished to "contemplate the greatness and the misery of man" in a purely religious… [read more]

Math Curriculum Development Term Paper

… Math Curriculum

Science and its modes of studies are very much reliant on the mathematical techniques which have been heavily evaluated over the past few years. Numerous studies like the ones conducted by Mike Cass et al. (2003) and Lynn… [read more]

Pythagoras, the Pythagorean Theorem and Its Relationship Term Paper

… ¶ … Pythagoras, the Pythagorean theorem and its relationship to the area of a circle.

Biography of Pythagoras:

Pythagoras was a Greek sage of the 6th century B.C.. He was born on the Greek island of Samos, off the coast of Asia Minor. Pythagoras was introduced to mathematics by Thales of Miletus and his pupil Anaximander, according Iamblichus, the Syrian historian. He traveled to Egypt, around 535 B.C., to continue his studies, but was captured by Cambyses II of Persia, in 535 B.C., and was taken to Babylon ("Pythagorean," 2007).

Eventually, Pythagoras emigrated to the Greek colonial city-state of Croton, in Southern Italy (Mourelatos, 2007; "Pythagoras," 2007).

Pythagoras was a "teacher and leader of extraordinary charisma. Pythagoras founded in Croton a society or brotherhood of religious-ethical orientation. The society fostered strong bonds of friendship and a sense of elitism among its initiates through ritual, esoteric symbolism and a code of rigorous self-control, including lists of taboos" (Mourelatos, 2007). This was known as Pythagoreanism. Pythagoreanism became politically influential in Pythagoras' home town of Croton, and eventually spread to other cities in the region ("Pythagoras," 2007).

Pythagoras' teachings were basically ethical, mystical, and religious. He believed in the transmigration of souls from one body to another, known as metempsychosis, either human or animal.

It's unclear whether Pythagoras believed that this led to the immortality of the soul; however, it did lay the foundations for some of the practices of the Pythagorean society he founded. These included vegetarianism and the rituals of purification, in an effort to promote the chances of superior reincarnation (Mourelatos, 2007).

A legend grew around Pythagoras, according to Mourelatos (2007), involving superhuman abilities and feats. However, he believes that this legend was based on the historical reality that Pythagoras was a Greek shaman. Some modern scholars theorize that the religious movement of Orphism, as well as Indian and Persian religious beliefs, influenced Pythagoras.

Although Pythagoras' contemporaries honored him as a polymath, modern scholars question this. Today, many "discount the tradition that he was the founder of Greek mathematics, or even that he proved the geometric theorem named for him" (Mourelatos, 2007).

Pythagoras died in Metapontum, near modern-day Metaponto, in approximately 500 B.C. ("Pythagorean," 2007).

History of the Pythagorean Theorem:

The Pythagorean theorem holds that "the square of the hypotenuse of a right triangle is equal to the sum of the squares of its other two sides" (Meserve, 2007). During Pythagoras' lifetime, the square of a number was represented by the area of a square with the side of a length of that number. With this representation, the Pythagorean theorem can then be stated as "the area of the square…… [read more]

Finding the Diameter Term Paper

… Diameter Problem

In this experiment, measuring of the diameter of the sun relative to the average radius of the earth's orbit requires some basic geometric knowledge, particularly in the properties of angles. Understanding these properties will provide us the mathematical relationship between the data provided in the problem and the data to be taken throughout the experiment. We must consider, however, that the mathematical aspect is just half of the story. Reckless experimentation often results to errors which are intolerable.

The choosing of favorable conditions under which an experiment is to be done, the careful set up of the materials to be used, and the precise measurements of data all contribute to finding the best result to the experiment. These three factors, however, are subjects to a certain degree of errors no matter how careful the person conducting the experiment may be. Human error is always present; and materials have limitations that affect the result of the experiment. Precision of the materials used and of the measurements taken are determining factors of how precise the result is.

As I conducted this experiment, however, I carefully set up the materials needed. I used a flat mirror covered by a white paper with a hole in the middle. I measured the hole to be approximately seven (7) millimeters. Finding a suitable place for the experiment was relatively easy; the hard part was putting the mirror in the right angle such that the image of the sun on the beige-colored wall was not oblong or ellipse, but rather very close to a circle; and what's even harder was setting the distance of the mirror to the reflected image at almost exactly six (6) meters. I finished setting up at 3:07 in the afternoon. Afterwards, starting at 3:10, I took the required measurements with ten-minute intervals.

Treating then the problem mathematically and scientifically, I had to identify what is required in the problem, what data are given, what variables are needed to be derived, and what formulas are involved in the solution. This step-by-step procedure is the key to getting the desired result. Hence, the required in the problem is D: the diameter of the sun; the given datum is L: the average radius of the earth's orbit which is equal to 150,000,000 kilometers; the derived measurements are: the distance "l" from the mirror to the reflected image on the wall, and the diameter "d" of the image itself; and the formula to be used is simple ratio and proportion, which is: D:L = d:l or D/L = d/l. From this formula, we can derive the formula for D. where we can supply our data. Hence, we get D = (d/l)

L. The…… [read more]

8th Grade Math Introduction to Fractals Lesson Term Paper

… 8th Grade Math

Introduction to Fractals

Lesson Title:

Why Study Fractals and What Are They?

Learning Objectives/EALRs:


Investigate situations and search for patterns


Extend mathematical patterns & ideas to other disciplines

Describe examples of contributions to the development of mathematics


Recognize use of mathematics outside the classroom



Quotes by Mandelbrot - "Fractals represent a new geometry that mirrors the universe."

Quote from Fractals, the Patterns of Chaos, p. 70:."..whether the fractal is..."

Definition of a fractal


Summary of lives of Sierpinski,

Mandelbrot & Koch [

Page 1 & Page 2]

Internet sites loaded on computers ahead of time:

Fractal of the Day:

African Fractals:

Photo posters:

Sierpinski, Mandelbrot, Koch, Mandelbrot Set, Sierpinski Gasket, Snowflake

Butcher paper for chart

Set up:

Schedule computer lab

Overhead projector

Collect overheads & visuals

Display visuals on white board for reference and interest

Draw chart on butcher paper for recording student ideas


We're exploring and collecting ideas and perceptions about fractals because:

They're something fairly new in math.

We, as 8th graders, can understand lots about them.

Fractals often look like objects in nature.

Point out the photos displayed of Sierpinski, Mandelbrot, and Koch and the fractals they are associated with.

Distribute the mathematician background handout and briefly talk about Koch, Sierpinski, and Mandelbrot. Ask students to look for mention of these names as they browse the fractal internet sites marked on their computers.

Search the following internet sites. Go to Fractal of the Day:

Number a piece of paper 0-15. View the fractals for today and…… [read more]

Hart, B. And Risely, T ) Term Paper

… Hart, B. And Risely, T (2003). The early catastrophe: The 30 million word gap by age 3. American Educator, Spring

Fairly commenting on an investigator's research endeavor is a task that must be taken seriously. Although it is quite easy to have an opinion of another's research, it is something quite different to be able to evaluate the research activity in terms of topic specificity and soundness, intent or purpose, data analysis, and informational importance. The focus of this paper was on whether or not the research investigators of the above cited research publication were prudent in stating a research question and a testable hypothesis along with informing the reader of the chosen research design, statistical data analysis and reporting the results, limitations, limitations and implications for future practice - all of which must lead to a best fit research decision.

The authors of this particular research report not only failed to state a research question and testable null hypothesis but selected a sample (N=42 families) on a non-random basis. In fact the sample selection was reported as being "pre-selected." As such any results garnered from a statistical data analysis can only be inferred back to the selected population and not to a wider universe of language growth deficient children. In fact, the authors set out to examine language deficiency of lower income children yet, included in their analysis a disproportionate number of upper income (13), middle income (10), lower income families (13), and welfare (6). Not only was there disparity among family selection the authors failed to report how many children were included in each of the four socio-economic status categories, thus producing error contamination of the results.

In addition to the errors associated with failure to state a research question and testable null hypothesis the authors of the article failed to alert the readers that a cross-sectional research investigation is point in time…… [read more]

Archimedes Many Experts Consider Term Paper

… Archimedes

Many experts consider Archimedes to have been the greatest mathematician of his era. The contributions that he made to the field of math, including geometry are considered phenomenal. In addition he is often credited with understanding and anticipating the advent of calculus 2,000 years before it happened. When he was not busy cracking the code to mathematical equations he spent his time inventing machines that included the pulley. Today, many commonly used mathematical concepts are directly related to the mind and development of Archimedes (Archimedes of Syracuse (


He began his life in 287 BC in a city called Syracuse by the sea in Sicily, Italy. His birth date was determined when he died and those who knew him claimed that he was 75 years old at the time (Biography (

He was born to a father who spent his life as an astronomer.

He spent his life building the foundation for many of today's mathematical calculations, formulas and concepts as well as providing the world with valuable inventions like the pulley. He died in 212 BC in the middle of the Second Punic War.

According to the popular account, Archimedes was busy contemplating a mathematical drawing in the sand. He was interrupted by a Roman soldier and replied impatiently: "Do not disturb my circles." The soldier was enraged by this, and killed Archimedes with his sword (Biography ("

Discoveries and Achievements

Many experts refer to Archimedes as the first math physicist. He contributed the foundation for the later works of Newton and Galileo. One of the things he is most well-known for discovering is the principal behind buoyancy. Legend has it that a crown was prepared for a king and Archimedes was asked to verify its gold qualities and to determine whether gold had been placed in it as well.

He was asked to make these determinations without destroying the crown so he figured out that the density of the crown would determine how fast it would sink in liquid.

Another achievement of his was the Archimedes screw. This invention is a machine that has a revolving screw shaped end that was often used to transfer water from low lying bodies to irrigation canals.

Archimedes cannot be credited with inventing the level however, he was the one who developed the principles that explained how a lever works.

His Law of the Lever states: Magnitudes…… [read more]

Proof David Auburn's "Proof" -- Catherine Term Paper

… Proof

David Auburn's "Proof" -- Catherine vs. Claire

How can two sisters from the same family be so different?

Both fulfill different functions and roles in the family dynamic. Catherine is the caretaker, and the mathematical problem-solver. Claire is the problem-solver, in a practical sense.

Temperamentally, Claire takes risks in finance, Catherine hides from the world, locked in her father's reality, and does not embrace risk, only intellectual risks on the Both represent two different aspects of mathematics. Catherine represents mathematics' theoretical side, while Claire's career in finance and concern with money demonstrate the worldly value of numbers

Catherine's characteristics

Compassionate, as manifested by her role in the caretaker of her mathematician father until recently freed by his death

Confined socially, limited professionally, and kept in a childlike state personally by her father's mental illness

Depressive, reluctant to move forward even after a change, pessimistic

Allied with her father, even after he dies -- experiences his presence in a ghostly fashion

Still young in terms of her sexual experience, even though she is in her 20s

Over the course of the play, says she has discovered revolutionary mathematical theory amongst her dead father's papers -- Claire expresses surprise

Protective of father's memory as a great man

Protective also of father emotionally, did not, when he was living, allow him to be fully cognizant of the extend of his illness, while Claire believes she sees her father's illness (and her sister's) clearly However, in doing so, may also have been protecting herself from moving out into the world, which her older sister Claire was able to do much more successfully

Loves father without reservation -- finds her identity in caring for her father, unlike Claire who finds her identity by breaking away from the family and becoming engaged in professional, normal life

Moody, withdrawn, emotionally unstable

Might be author of the supposedly new proof --…… [read more]

Euclid -- 323-285 Term Paper

… The cause that such a proof was in great demand for such a long time was due to the fact that Euclid's other postulates seem to be easy, self evident, and naturally clear, the fifth postulate basically described the intersection of lines at potentially infinite distances -- the notion of infinity being at the time, was mathematically, problematic. (Euclid:

Therefore, the fifth postulate seemed as a sort of blotch in the otherwise apparently perfect logical edifice which is Euclid's Elements. Whereas Elements was referred and used into the 20th century as the original geometry textbook and has been regarded a fine example of the formally precise axiomatic method, Euclid's study fails to meet the modern standards and several logically necessary axioms are not there. The first correct axiomatic treatment of geometry was given by Hilbert during the year 1899. Nearly nothing is available regarding Euclid, apart from what is presented in Elements and some of his other surviving books and whatever scanty biographical information the world has comes mostly from explanations by Proclus and Pappus of Alexandria. (Euclid:


Dietz, Elizabeth. "Euclid 323-285 B.C. Biography" Retrieved from

Accessed on 8 August, 2005

Dietz, Elizabeth. "Euclid 323-285 B.C: Discoveries" Retrieved from

Accessed on 8 August, 2005

"Euclid's Elements" Wikipedia, the free encyclopedia. Retrieved from

Accessed on 8 August, 2005

"Euclidean Geometry" Wikipedia, the free encyclopedia. Retrieved from

Accessed on 8 August, 2005

"Euclid" Retrieved from

Accessed on 8 August, 2005

"Euclid" Wikipedia, the free encyclopedia. Retrieved from

Accessed on 8 August, 2005… [read more]

Hypatia of Alexandria Term Paper

… Historian Coffin continues, "She wanted to help her students understand the mathematics she was teaching, so she wrote books that gave explanations that were easier to understand than the original books" (Coffin, 1998, p. 95). She taught many well-known young men of the time, both Christians and pagans, and most found her an excellent teacher, scientist, and philosopher. She had many friends and supporters in Alexandria, but she also had many detractors.

It is important to note that at the time, it was not especially prestigious or even socially wise to be a mathematician. At the time, astrology and mathematics were closely linked, there were many astrologists and numerologists who were considered mathematicians too, and so, the profession came to be seen in a bad light. Hypatia, "The bad currency drove out the good. Reputable astronomers and geometers like Theon and Hypatia got confused in the popular and in the ecclesiastical mind with these fly-by-nights. All were lumped together as 'mathematicians'" (Deakin, 1997). This was a dangerous position at the time, because Christianity was becoming the dominant religion in Egypt and beyond, and many Christians distrusted science and education. Coffin states, "The Romans did not appreciate Greek mathematics and in fact thought it was subversive. The Romans had not been greatly involved in the development of mathematics; hence their mathematicians did not compare favorably to the Greeks" (Coffin, 1998, p. 96). Thus, as Christianity spread throughout the area, Hypatia found herself in a dangerous situation. Many pagans left the city to save themselves, but Hypatia did not.

In addition to her notoriety as a female scholar and philosopher, Hypatia made no secret of the fact she was a pagan who did not believe in Christianity, and this further alienated her from the Christian majority. She made a strong political and religious enemy when she angered Cyril, the Roman Catholic bishop and leader of Alexandria. Cyril was adamantly against pagans and Jews, and issued many sanctions against them during his forty-year reign. Cyril considered Hypatia an enemy, and where there are not actual texts tying him to her death, most historians regard Cyril as the force behind the mob that eventually attacked and killed Hypatia. Some say Cyril made it known there would be a reward for those who killed the woman.

Eventually, Cyril and his supporters did manage to murder Hypatia, and her death was especially grisly. As she rode through the city in a carriage, her enemies overtook her. Another historian states, "She was seized from her carriage and dragged into the Caesareum, the former temple of the imperial cult, which was now the cathedral. There she was stripped and stoned to death with broken roof tiles. Her body was then hacked to pieces and burned" (Russell, 2000, p. 9). Later, legend has it that her remains were scattered all throughout the city as a warning to other pagans and scientists. The year was 415, and Hypatia was anywhere between 40 and 60 years old.

Unfortunately, none of the… [read more]

Mathematician Nassar, Sylvia. A Beautiful Mind Term Paper

… ¶ … Mathematician

Nassar, Sylvia. A Beautiful Mind. New York: Touchstone Books, 1998.

The story of the 1994 Nobel-Prize winning mathematician and economist John Nash has proved to be an inspiration to all individuals who have heard and read about the great Princeton genius, and not simply because of Nash's ground breaking insights about the mathematics of game theory. Nash is equally famous for his return from his prison of mental illness. He is now once again dwelling in the lucid world of his previous, brilliant mind, in a state of sanity. Author Sylvia Nassar's biography A Beautiful Mind tells the story of how Nash was born in the Ozarks, one of the poorest regions of America. As a young man, he was arrogant yet intellectually talented. After becoming a graduate student at Princeton, he even challenged Albert Einstein face-to face about the older man's theory of relativity. Einstein dismissed the young Nash's concerns from a mathematical point-of-view, but was impressed by Nash's bravado.

Nash became famous as a young man because of his unique insights into game theory, a theory that attempted to rationally predict how human beings made decisions with imperfect information, as people must in certain kinds of games, worldwide diplomacy, and economics. Ironically, although Nash's mathematical theories are used to predict human behavior with numbers and equations, even before he began to lose his reason, Nash had a great deal of difficulty relating to other people, even his fellow mathematicians. One of the reasons Nash loved mathematics was that he did not need to deal with other people's emotions in a world of numbers. According to Nassar's book, even before he developed the symptoms of schizophrenia, his contemporaries, "found him immensely strange," and "aloof." (Nassar, p. 13) Even so, by the 1950s, Nash had carved a brilliant career "at the apex of the mathematics profession, traveled, lectured, taught," and met "the most famous mathematicians of…… [read more]

Why Algebra Term Paper

… Algebra

How Algebra can be used in real Life

Algebra can easily be used in real life, in many of the calculations which we make every day. The most common example I can think of is using algebra in some way to calculate a rate of change. Let's say for example the hypothetical person who has passed his algebra classes with flying colors and has won a job working for the college has been asked to calculate the rate of change in the number of students who will be attending the school and need to take classes in the next ten years. In this matter, one has to take the formula, which just uses x's and y's and change them into something which has meaning, in this case, the people that will be attending the school. Since we know that in this case, the independent variable we are discussing is the number of students coming to school, we can also make the subject a little bit more complex. We know that the variable can increase, increase exponentially, decrease, remain constant, or any of a number of different things.

So in this case let us make the number of students the function x. By just looking at x we cannot tell what it is going to do, it is too unpredictable. We can only give meaning to x by placing it in context to our study. By this we…… [read more]

Calculus? Calculus Is a Vast Topic Term Paper

… ¶ … Calculus? Calculus is a vast topic and it also forms the basic foundation of any calculations that are based on math. Calculus is divided into two branches, one being differential, and the other being integral. Differential calculus deals with the study of the rates of changes that may happen in functions. For example, it teaches how to find the derivative of a particular function when the function is to determine the angle of the slope of a graph, which deals with that function, at a particular point. Integral calculus generally helps the beginner to deals with certain primitive functions, like for example, indefinite integrals, and also for finding, for example, the area under a curve, known as a definite integral. (Quick math, Automatic Math Solution)

In other words, calculus can be defined as a branch of mathematics that pertains to the rates of change. Although its basic roots can be traced back to Ancient Greece and to Ancient China, its actual origins are indicated in the time of Newton and Leibnitz in the seventeenth century. In the modern world, calculus has proved its usefulness time and again, and it is used extensively in many different areas of science. The there ideas of 'limit', 'derivative', and 'integral' are interwoven into the principles of calculus, wherein while derivative indicates the instantaneous rate of change, in comparison to something else, the integral generally indicates the area under its graph, or a sort of total 'over time'. For example, when the derivative of 'height' in respect to its position, is 'slope', then the derivative of 'position' with respect to 'time', is velocity', and when 'velocity' is taken with respect to 'time', then the derivative would be 'acceleration', and so on. As far as integrals are concerned, the area under the graph is taken into consideration, and this means that when the integral of 'slope', in respect to a constant, is 'height', then the integral of 'velocity', up to a constant, 'position', and thereafter, the integral of 'acceleration', with respect to 'time', is 'velocity'. It is quite evident therefore, that derivatives and integrals are inter-related and are also at times, complete opposites. (Preparing for University Calculus: At Atlantic Canadian Universities)

Why is 'calculus' important in today's world? Science today studies many processes that involve change, and since calculus deals with changes, it is very important. (Preparing for University Calculus: At Atlantic Canadian Universities) One particular High School teacher gave her students concrete examples of how calculus can be applied in real life to solve real problems. The links between the physics concepts of position and velocity and acceleration and the totally calculus concepts of function, derivative and anti-derivative were found. (Dosemagen; Schwalbach, 54) Another teacher toot found that when physics is applied to calculus, the problems would be solved easily. When compared to the way in which these students found it difficult to use mathematics for their calculations, and how they preferred to use physics as a better option, it can be seen… [read more]

Development and Application of These Concepts in Real Life Term Paper

… ¶ … Real Life

Golden Ratio:

The golden ratio is acknowledged as the divine proportion golden mean, or golden section is represented as a number mostly confronted while considering the ratios of distances in simple geometric diagrams like pentagram, decagon… [read more]

Pre-Post Test Knowledge of Ultrasound Content Term Paper

… Pre-Post Test Knowledge of Ultrasound Content Knowledge for Sonographers

When an investigative research study ins set up on the basis of a pre and post testing situation the investigator is afforded the opportunity not only to determine differences in mean test scores for the before and after criterion but also establish whether or not there exists a relationship between the pre and post test situation. The latter statistical process is known as a correlation study. In addition the investigator is also able, on an individual participant basis, able to determine wherein each individual places with respect to the average scores on a pre-test basis and on a post-test basis in respect to the group average. This calculation is known as a Z Score.

Descriptive Statistics Results

Pre-Test Ability to Detect Abnormalities







B. Post-Test Ability to Detect Abnormalities







C. Mean and Median Graphic Presentation of Differences

Between Pre-test and Post-Test Results

Conclusion: On the basis of the raw data (i.e., non-statistical significant difference) there appears to be a slight increase in ultrasound content knowledge as result of individuals participating in the academic course. Whether or not the reported difference is statistically significant is not known from a review of the raw data. In addition, the conclusion can be drawn that the variability of ultrasound knowledge is more variable before the course vs. after the course.

D. Statistical Analysis of Raw Data visa via "t" Test Calculation:

value and statistical significance:

The two-tailed P. value equals 0.0033

By…… [read more]

Java Application (Parking Simulator) Interface Description Form Term Paper

… Java Application (parking Simulator)

Interface description


When user launches the application the form1 appears. From form1 we can go to parking form by clicking on "Park." Actions of "Park" button will be viewed later. By the use of panel menu we can do the following:

File-exit - exits the application

Change statistics->Coins available- calls from2 that shows coins available in parking machine

Clicking on button "OK" returns user to Change statistics->Add funds calls form3 that allows to add funds to the machine


Clicking "OK" button adds funds inputted into the machine and return the user to the form1. In order to change the available funds user can view menu Change statistics->Coins available;

Change statistics->General statistics

About-> contains information about the program's


By clicking on park button on form1 user gets to the form4 which shows parking actions. Here the driver has to choose number of hours and pay for them. He can view the indicator of current time and ending time of parking for which he is supposed to pay. Also the form shows the amount that has to be paid and the amount the driver had already paid. If user makes a mistake in inputting data either for parking time or for amount paid he can cancel payment and will be returned to the form1. In the upper section of the window there are coins of different value for parking payment. By clicking on these coins user will pay corresponding amount of money for parking. When payment is complete buttons become unavailable- user can not add any more money. After the payment is complete and user had paid more than it is supposed, he will get a change menu message form5.


On the change menu he can choose the way he want to get change in different variants. He can also watch the change available. After he chooses the variant of payment the program checks for available funds and if the funds are unavailable program gives an error message-Form6


Then the application returns user to form1.

The data about parking payments can be viewed from report file or in the menu: Change statistics->General statistics Form7


Information about payments is saved into dynamically creating file Report.txt for future…… [read more]

Job Satisfaction Survey Research Studies Term Paper

… Job Satisfaction Survey

Research studies, whether they are clinical trial based, experimental, case study designed, or descriptive, must exhibit and command interest, enthusiasm, and passionate commitment. The research investigator must catch the essential quality of the excitement of discovery that comes from research well done if expected results are to be gained. To this end the researcher is bound by a very stringent protocol for the development of all research endeavors. Not only must the investigator clearly define the research problem but must also plainly state a research question, followed by a testable research null hypothesis. Contained within the format of the research purpose, question, and hypothesis are various inherent constraints that will alert the reader as to the investigator's knowledge of, and adherence to, those tenets that make for sound, credible, and purposeful research (Ohlson, 1998). Included in the aforementioned three research requirements are statements of, and a rational for the use of, specifically chosen variables (independent and dependent), measurement or data assessment tools, statistical data analysis techniques, and potential error sources. Wherein most research fails to deliver scientific information for the advancement of content knowledge is in the area of study error. The remainder of this report will examine one source of possible error, namely that associated with sampling. Once sampling theory has been examined the garnered information will be applied to a specific study using a job satisfaction survey.

Sampling. The most succinct and effective way to view research sampling is to look at the process as being a part of a whole that represents a larger connection (Ohlson, 1998). Briefly defined sampling is the taking any portion of a population or universe, as representative of that population or universe. Sampling alone can skew testing results, infuse…… [read more]

Permutations and Combinations Term Paper

… (Dr. Math, 2004)

Beyond speculating about advertising, however, the use of permutations and combinations can be useful, for instance, when a business desires to allocate its advertising for several products along several potential sources, all from the same media such as television or magazines, but wishes to make sure that the combinations of the different publications did not overlap in terms of the combinations. To calculate, the various letters could be used to represent the publications, and the products also allocated various letters, to ensure that there was no overlap of particular products and sources.

Restaurant owners could use permutations and combinations to determine various pairings of toppings, to determine if this was economically feasible to achieve -- for instance, what is the maximum amount of combinations a pizza can have, given a fixed array of toppings, and also, what is the maximum amount of toppings that any singular pie can possess? Thus, the owner could calculate the maximum amount of funds the pizza could possibly allocate to the owner of the pizza business.

Permutations in combination could also be used to pair off different members of staff, when needed, on the floor of a store or a restaurant. For instance, it could be assumed that the establishment would likely have a certain number of tables filled on one particular night, and thus would require a certain number of members of the wait staff, busboys, and cooks, per table, to ensure that the table was adequately attended to over the course of the evening. From an inventory perspective as well, certain products must always be paired together in combination in terms of need, such as coffee cups and stirrers. Thus, permutations in combinations are not simply interesting and speculative statistical tools, but are used in the daily world of business mathematics.

Works Cited

Dr. Math. "Permutations." (2004) Math FAQ. Retrieved on October 7, 2004 at

Dr. Math. "Fast Food Combinations." (2004) Math FAQ. Retrieved on October 7, 2004 at… [read more]

Scientific Investigation Includes Both Independent Term Paper

… This particular type of ANOVA shows differences that are a result of "effect" and not differences between mean scores. Again, a wrong choice was made.

Assuming the "t" test were the appropriate tool the authors failed again in not pre-selecting their level of statistical acceptance. Whether or not this was a clever move on the part of the authors or not is not know. What is being implied here is that when there is no stated testable null hypothesis there is no reason to pre-set an alpha or probability level for accepting or rejecting the null hypothesis. In cases like this researchers are free to explain any result they choose to and not bound by controls pre-set in the beginning of the study. This, as most research pundits would express is sloppy research.

Results and Implications Because of all the errors in research design, measurement instrument selection, and statistical tool selection interpreting the results of this study cannot take place. The errors are so great that offering an interpretation of the results would only add error to error. At not time can there be faith placed in research findings that are fraught with error. To warrant the results of this study as being useful to the advancement and understanding or learning disabled student simply cannot take place. The bar of excellence must be raised significantly for this study to garner support, as the external error for this study is magnanimous. What is suggested is as follows:

Restate the research question in proper investigative form.

Specifically state all testable null hypotheses.

Pre-set a probability level for the acceptance or rejection of any and all null hypotheses

Select statistical tools that can effectively analysis data gathered in support of the specific research question.

Fully expound upon the measurement tools used to collect the test data to be analyzed.

Specifically state the limitations and delimitations of the study

Support the need for such a study through a review of literature

Explicitly define the terms used in the study including both independent and dependent variables.

Explain all testing procedures in detail.

List all possible intervening variables and internal and external error.

Make use of…… [read more]

Nursing Research Report the Structure Term Paper

… Other considerations that a researcher must account for within the design section of the study are directly related to the manner by which a sample has been selected, the fashion by which the assessment instrument is administered, possible limitations and… [read more]

Power of Statistical Analysis Term Paper

… The mean, median and mode are all equal in this type of measurement, and the scores at either end of the distribution, those which are extremely high, or extremely low, occur less often. For example, a curve representing the results… [read more]

Frege's Much-Discussed Book, the Foundations Book Review

… "

Importantly, Frege argued that analytic judgments govern the laws of arithmetic, and thus these laws exist a priori. This analysis was likely one of the most important contributions of The Foundations of Arithmetic.

As defined in The Foundations of Arithmetic, Frege's main analysis that the laws of arithmetic are a priori has important consequences. Frege notes that the definition of the laws of arithmetic as a priori results in the following: "Arithmetic thus becomes simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one. To apply arithmetic in the physical sciences is to bring logic to bear on observed facts; calculation becomes deduction."

Frege has a great number of influential and important admires. For example, the well-known philosopher Michael Dummet notes that Frege's work on the criterion of identity is "brilliant and philosophically fruitful." Further, Dummet gives Frege high praise for uniqueness, noting that The Foundations of Arithmetic is truly the first work of analytical philosophy (Frege Biography).

In spite of the great praise heaped upon the book, Frege's analysis is sometimes somewhat flawed. For example, Frege's analysis is often somewhat weak in terms of psychological references. He easily notes the difference between concept and object, and yet he does not clearly define sense and references. Barbosa notes that after The Foundations of Mathematics was written, Frege rejected his contextual preference, which argued that words refer to something based on context.

Frege has also been accused of distorting the ideas of other philosophers to his own benefit. Notes Barbosa, "Sometimes (Frege) distorts a little bit what others say about logic, so he argues against those thinkers more effectively." This is indeed a serious criticism, and implies that Frege was, at least in this case, not scrupulously rigorous or honest in his philosophical arguments.

Frege was always willing to correct inconsistencies in his works, and the issue of contextual preference was no exception. Barbosa notes that after Frege "wrote the book, he would reject this principle, because of his doctrine of sense and reference: the sense of the words determine the sense of the sentence; and the reference of the words determine the reference of the sentence."

In conclusion, Gottlob Frege's The Foundations of Arithmetic has made a lasting and influential contribution to the philosophy of mathematics. This book is essential reading for anyone interested in the philosophy of mathematics, and is also invaluable for anyone interested in the broader field of analytical psychology. The Foundations of Arithmetic provides a convincing argument that logic is the basis of arithmetic, rather than psychology, and also makes the important argument that analytic judgments govern the laws of arithmetic, and thus these laws exist a priori. It was in these discussions that Frege likely made his greatest contributions to the philosophy of mathematics. Despite his great critical success and long-lasting influence, Frege's works were not without their weaknesses. Importantly, many psychological terms within The Foundations of Arithmetic are not thoroughly defined, and critics have argued… [read more]

Multiplicative Number Theory Term Paper

… All integers are divisors of 0. All integers 'a' are divisible by and If a has any other divisors, then it is called a composite. Otherwise, it is called a prime, unless which are called units. Of the positive integers… [read more]

Algebra Lesson Plans and Curriculum Term Paper

… (p. 38).

A large category of algebra misconceptions has been documented in Algebra: Some Common Misconceptions under the category of "The Meaning of Letters." These relate to students' difficulty with the meaning of letter variables. They may completely ignore letters… [read more]

Structure the Research Term Paper

… Individual researchers must monitor a vast number of research results and activities that have a bearing on their own particular specialties.

By definition, measurement must be objective, quantitative and statistically valid. Simply put, it's about numbers, objective hard data. A scientifically calculated sample of people from a population is asked a set of questions on a survey to determine the frequency and percentage of their responses. For example: 240 people, 79%, of a sample population, said they are more confident of their personal future today than they were a year ago. Because the sample size is statistically valid, the 79% finding can be projected to the entire population from which the sample was selected. Simply put, this is quantitative research. Qualitative research, is much more subjective than quantitative research and uses very different methods of collecting information, mainly individual, in-depth interviews and focus groups. The nature of this type of research is exploratory and open-ended. Small numbers of people are interviewed in-depth and/or a relatively small number of focus groups are conducted. Participants are asked to respond to general questions, and the interviewer or group moderator probes and explores their responses to identify and define peoples' perceptions, opinions and feelings about the topic or idea being discussed and to determine the degree of agreement that exists in the group. The quality of the findings from qualitative research is directly dependent upon the skill, experience and sensitivity of the interviewer or group moderator.

The term "Epidemiology" comes from three separate words: "epi" - which means "upon," "demos" - which means people/population, and "ology" - which means the science. Hence it means the study of things that occur upon the population. Epidemiology is the study of how diseases and other health outcomes are distributed in the population and the factors that influence or determine this distribution.

Today, research is increasingly diverse in its objectives, scope and modes of organization. Research involves individual endeavors, small and large teams, major regional or national research networks, and international consortia. The need to concentrate intellectual and financial resources has spurred the creation of new research centers and institutes that transcend the traditional academic organization based on disciplines. Budgetary pressures and the benefits of collaboration encourage the pooling of expertise…… [read more]

Lives of Archimedes and Carl Term Paper

… The Latin translation influenced the work of the foremost mathematicians and physicists of the time, including Johannes Kepler and Galileo. A 1615 Latin translation of his complete works, was enormously influential in the work of Rene Descartes and Pierre de Fermat. The mathematical advances in Europe from 1550 to 1650 was build largely on the work of the ancient mathematicians, especially Archimedes ("Archimedes").

In contrast, Gauss's work had immediate impact on both theoretical and applied mathematics. This was partly because Gauss himself was interested in the applications of his work. Gauss also lived during the dawn of the Industrial Age, a time of unprecedented scientific advancement.

Gauss's interest in gravitation and magnetism led to a published paper in 1840 on real analysis. This paper became the starting point for the modern theory of potential. In the beginning of the 20th century, mathematicians re-developed potential theory, on the basis of Gauss's initial conclusions. In 1830, Gauss's mathematical investigations of fluids at rest contributed to the development of the Law of Conservation of Energy. His collaborative work with William Weber on electromagnetism paved the way for the invention of the telegraph. In addition to electromagnetism, Gauss's works in algebra and numbers theory continues to have significant impact on current mathematical research, including geometry and modern telecommunications ("Gauss").


In conclusion, the work of Archimedes and Gauss continues to make significant contributions to all fields of mathematics. Many mathematical disciplines would not even be possible without their work.

Both men faced significant obstacles to their mathematical research. Archimedes lived in a time of very limited mathematical knowledge. Even commonplace mathematical concepts today - the ideas of the infinitesimal, pi, and infinity - were unheard of during his time. The cumbersome Greek and Roman numeral systems were unwieldy for his computational needs so he had to devise his own.

Gauss, on the other hand, was born into poverty and did not have his family's support. He lived in or close to poverty for much of his life.

However, both men eventually overcame these obstacles to produce their research. Centuries after their deaths, the work of these two great minds continues to revolutionize the science of mathematics.

Works Cited

Archimedes," in Guide to the History of Calculus. Retrieved 30 November 2002 from

Bell, E.T. Men of Mathematics: The Lives and Achievements of the Great Mathematicians from Zeno to Poincare. New York and London: Simon and Schuster, 1965.

Boyer, Carl B. A History of Mathematics, 2nd ed. New York: John Wiley and Sons, 1991.

Gauss," in Guide to the History of Calculus. Retrieved 30 November 2002 from

Muir, Jane. Of Men and Numbers: The Story of the Great Mathematicians. New York: Dover Publications, 1996.

Riley, Mark T. "Archimedes"…… [read more]

Isaac Newton Was the Greatest Term Paper

… Newton's three laws of motion explained the hitherto inexplicable behavior of all physical bodies in motion. Still more astounding was Newton's discovery of gravity. All these four laws put together explained the mechanical motion of all earthly and heavenly bodies. Newton not only proposed these laws but also ratified them by using the integral calculus.

Newton published his laws of motion and gravity in his famed work 'Mathematical Principles of Natural Philosophy' (Principia) in 1687. [Microsoft Encarta] Newton's revolutionary discoveries were applied to a wide array of subjects. The field of Astronomy in particular got a tremendous impetus from his laws of motion. Using these laws Newton was able to precisely identify and predict the position of planets and other heavenly bodies. This was a significant milestone in the field of astronomy and for this reason Newton is rightly regarded by many as the greatest of all astronomers.

As for Newton's accomplishments we can do better by quoting some of the other great scientists. The great French scientist Laplace said, " The Principia is preeminent above any other production of human genius." Lagrange, another famous mathematician opined that Newton was the greatest genius who ever lived. Ernst Mach on his part said "All that has been accomplished in mathematics since his day has been a deductive, formal, and mathematical development of mechanics on the basis of Newton's laws." [Michael.H. Hart]. In conclusion we can say that in the history of science Newton's name is indelibly imprinted and perhaps his laws and discoveries are the most referenced by the scientific world. It is clear that Newton is not only one of the greatest scientists but also one of the most influential scientific personalities.


Michael.H. Hart, "The 100, A Ranking of the Most influential Persons in History"

Isaac Newton', 1999, Meeraa publications.

Microsoft Encarta, "Isaac Newton," Accessed on 2nd, December 2002,

D.R.Wilkins, "Sir Isaac Newton (1642-1727)," Accessed on 2nd, 2002… [read more]

Euclid's Fifth Postulate Philosophical Term Paper

… Whether the fifth postulate is true or false also affects the kind of geometry that is being used. For example, if the postulate is true, than Euclidian geometry works. If the postulate is false, then it creates a non-Euclidian type of geometry where similar triangles must always be congruent and the Pythagorean Theorem is no longer valid (Bennett, 2000).

Whether Euclid's fifth postulate is true, false, or both, it will continue to be debated by scholars and mathematicians for many years to come. Whether it will ever be 'solved' or proved remains to be seen. Either way, it is a fascinating and problematic piece of information that will carry on Euclid's legacy for centuries. If someday it is proven to be either true or false, and the decision is agreed upon, then it could change the way mathematics are done and the way geometry is looked at during the present time and also well into the future. Euclid would have likely enjoyed the attention his simple thoughts are receiving.

Works Cited

Bennett, Andrew G. The Axiomatic Method. 2000. Math 572 Home. 2 December 2002.

Bogomolny, Alexander. The fifth postulate: attempts to prove. 2002. Cut the Knot. 2 December 2002.

Parallel lines and planes. 2002. Connecting Geometry. 2 December 2002.… [read more]

Fractal Geometry Term Paper

… His mind was a visual one, a geometric mind, yet he was not taught this way.

Mandelbrot claims he could not do algebra well yet managed to receive the highest grades by translating the questions mentally into pictures. When he finished school, he came to the United States, where IBM gave him the freedom to pursue his mathematical interests, as he deemed worthy.

Mandelbrot's research led to a huge breakthrough summarized by a simple mathematical formula: z -> z^2 + c. This formula is now called the Mandelbrot set. It is important to understand that this formula, and the Law of Wisdom which it represents, could not have been discovered without computers. Many say that this mathematical breakthrough, which occurred in the research laboratories of I.B.M., is the greatest in twentieth century mathematics.

The Mandelbrot set is a dynamic calculation based on the iteration of complex numbers with zero at the beginning. The order behind the chaotic production of numbers created by the formula can only be seen by the computer calculation and graphic portrayal of these numbers on computers. Otherwise the formula appears to generate a totally random and meaningless set of numbers. It is only when millions of calculations are mechanically performed and plotted on a computer screen that the hidden geometric order of the Mandelbrot set is seen. The order is of a beautiful yet different kind, containing self-similar recursiveness over an infinite scale.

Euclidian geometry was unable to describe the shape of a cloud, coastline, a hill or a tree. As Mandelbrot says in his book the Fractal Geometry of Nature:

Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line."

Mandelbrot broke the long-term chain of thinking that held that most of nature's systems were too complex and strange to be described in mathematical terms. Fractal geometry opened a new world of mathematics that is capable of describing mathematically the most strange and complicated forms of the real world. In Mandelbrot words: "Fractal geometry is not just a chapter of mathematics, but one that helps Everyman to see the same world differently."

The Mandelbrot set has done much more than just produce beautiful pictures. When pictures are thoroughly examined, innumerable empirical observations are found that can be restated in the form of mathematical conjectures. Many of these have already led to brilliant theorems and proofs. Fractal geometry has inspired a new approach to mathematics, using a computer screen.

The discovery simply started a new way of thinking, which will undoubtedly change the way many things in the world work. Fractal geometry is opening doors to significant new insights into complex phenomena, including cells, hearts, brains, coastlines, mountains, earthquakes, economies, political systems, art, music, climate & life.


Mandelbrot, Benoit B. The Fractal Geometry of Nature W.H. Freeman and Company, 1977.

Crilly, R.A. Fractals and Chaos. Springer-Verlag, 1991.

Dictionary of Scientists, Oxford University Press, Market House Books Ltd., 1999… [read more]

Georg Cantor: A Genius Out Term Paper

… Cantor's most criticized concept presented in this paper is that the power of the continuum is independent from its number of dimensions. It was commonly believed that points in two dimensional space cannot be traced back to one dimensional space,… [read more]

Euclid of Alexandria: 325 Term Paper

… "; a clear insight into his apparent wisdom. The second story concerns a student who, after his first lesson, asked what he would gain in life from learning such things as he was in the school; Euclid called his slave and said: "Give him a coin since he must make gain by what he learns."

Arabian and Syrian writers have said Euclid's father was Naucrates, and his grandfather, Zenarchus. They also said he was a Greek who was born in Tyre and lived in Damascus. Unfortunately, most of this information has little evidence of validity. Added perplexity began around 14th century when the Byzantine writer Theodorus Metochita (d. 1332) wrote "Euclid of Megara, the Socratic philosopher, contemporary of Plato." This Euclid, Euclid of Megara, lived around 400 B.C.E. And was actually a pupil of Socrates who founded a philosophical school, which Plato did not like. Nothing is known of Euclid's death.

Other works of Euclid include: The Data, for use in the solution of problems by geometrical analysis, On Divisions (of figures), The Optics, and The Phenomena, a treatise on the geometry of the sphere for use in astronomy. His lost Elements of Music may have provided the basis for the extant Sectio Canonis on the Pythagorean theory of music.


Gillispie, Charles C. ed. The Dictionary of Scientific

Biography, 16 vols. 2 supps. New York: Charles Scribner's

Sons, 1970-1990. S.v. "Euclid: Life and Works" by Ivor


Heath, Thomas L. The Thirteen Books of Euclid's Elements, 2

vols. Cambridge: Cambridge University Press, 1926.

Frankland, William Barrett. The first book of Euclid's Elements…… [read more]

Stigmatization and Therapy Counseling of Gay Men Research Paper

… Unnatural sexual orientation is not an independent, standalone issue for those professing inclination towards same sex individuals for their sexual needs. Such individuals invariably face difficulties, humiliation, and consternation in many other areas of their lives, thereby being isolated, stigmatized,… [read more]

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