Monthly Archives: April 2015

Proofs Perceived

I hope to impart some extremely awesome facts on the intrinsic elements of being a mathematician and knowing mathematics by explaining, broadly, the fundamentals of proofs. I will do this by first giving you the basics and then move to more interesting theorems/proofs. For simplicities’ sake all information contained in this blog post is from the book “The Nuts and Bolts of Proofs” by Antonella Cupillari and I have included the page numbers at the end of the sentences.

Let’s start with if/then statements. If/then statements are like statements of a scientist who happens to states his hypothesis when he first starts talking only to end on some conclusion that was dependent upon his hypothesis (11). The tricky part of proving if/then statements is that some of the information might not be explicitly stated (12). For instance, sometimes people will ask you how long it takes to get a college degree. The amount of fluctuation in the length of time it takes differs greatly because of different requirements and different people, but it is implied that the person is looking for average length of time. This implied part can slip you up when trying to do these proofs because you might need to use information that is implied. For example, let’s say you were given an assignment to prove that two is prime. The original statement if two satisfies the condition of being prime then it is prime, will very likely not state that you can use all the properties of real numbers. You could accidentally see things being implied that aren’t being implied. If/then statements are a basic statement in mathematics and fundamental to many theorems and proofs.

All proofs fall into one of two categories, or a mixture of these categories (12-29). The categories are indirect proof and direct proof. Direct proof is like performing a calculation or computing a problem in most math classes. You just carry out a series of logically valid calculations to come to a final answer. You are given a beginning claim that you want to show leads to some particular conclusion. Essentially, you want to move from point “A” to point “B”, and in order to do so in a direct proof you carry out a series of logical steps. Direct proofs are used when there is enough information in the hypothesis to come to the conclusion (12). Indirect proofs are a bit more complicated than direct proofs. Indirect proofs involve altering the hypothesis in order to prove a different statement that is logically equivalent to the original hypothesis (22). A specific example of indirect proof is proof by contradiction. Proof by contradiction is assuming the hypothesis to not be true, and if this leads to a contradiction, then the hypothesis must be true.

Negation of quantifiers is also fundamental to proofs. I will identify the quantifiers in bold followed by an example and another example of its negation. All of this information on quantifiers can be found on page twenty-five from the book “The Nuts and Bolts of Proofs” by Antonella Cupillari as previously mentioned earlier. There are six quantifier negations that you just have to know. The first is at least one, as in there is at least one mathematician in our class (Hint: Lamb). The negation of this statement would be none of our classmates are mathematicians. The second one you have to know is some. Today some new myths will be made. The number of new myths made today was none. If you assert that all birds in America drink water, the negation of this statement is there is at least one object in the collection that does not have that property. Specific to the example given, the negation would be there is at least one bird in America that doesn’t drink water.

The fourth quantifier that you need to know is none. None of the boys in Salt Lake City like to eat cake. There is at least one boy in Salt Lake City that likes to eat cake. Fifth is there is no unicorn. There is at least one unicorn. The sixth is every object in a collection has a certain property. Every book at the University of Utah is worth reading. There is at least one book in the University that is not worth reading.

Moving on to the theorems and special proofs. By special proofs I mean specific versions of direct and indirect proofs. Theorems are general truths that are proven to be true by a series of logical steps, hence their connection to proofs (50). Existence proofs involve either an algorithm as a proof or an argument that proves the existence of the object in question and are often in the category of theorem (58). Just in case anyone doesn’t know what an algorithm is, it is a series of steps that are carried out sequentially. Also, an argument is a claim that is usually supported by reason and evidence, but as religion has shown, neither reason nor evidence is needed to argue—just some claim that causes controversy is necessary.

Another special type of proofs is the uniqueness proof, which shows that the steps/algorithms to show that the object exists were unique (61). Or by assuming that the object isn’t unique and then showing that the other object with the same properties isn’t a different object, and by constructing a valid argument you are able to prove uniqueness (61).

If and only if proofs involve the proof of two separate if/then proofs in order to prove the if and only if statement (35). An example of this is necessary because it is difficult to explain in words. “I will checkout more books if and only if I can read the books I have” would require the proof of “if I will checkout more books then I can read the books I have” and the proof of “if I can read the books I have then I can checkout more books.” Once you have done this then you have proven the statement “I will checkout more books if and only if I can read the books I have.” This makes more sense when you think about the fact that both checking-out more books and reading the books I already have must both be true in order to get a true statement. If either is false then the statement is false but if both are false then the statement is true (35). In this relationship checking-out books and reading the books I already have must follow each other in their truth value in order to get a true statement (35).

Proof by induction is also another cool, interesting tool in the world of mathematics and it is of great utility. All information on mathematical inductive proofs can be found on pages 48-58.  The first step in a proof by induction is to prove that the hypothesis is true for the smallest base case. Then you assume that it is true for some arbitrary value. After that you deductively show that it is true for an arbitrary value plus an additional defined magnitude. If you said for any natural number there is always a number greater in magnitude, the first thing you would do is show that it is true for the smallest case. In this instance the smallest case is one with two being greater. Then you would assume that an arbitrary number represented by some variable, usually “n”, has the same property. After that you would show that your hypothesis is true for the “n” plus one case. The steps taken in a proof by induction are awesome because it gives you the ability to conquer the infinite with a logical domino effect.

You now know the basics of proofs—which are if/then statements, negations, direct proofs, indirect proofs, and quantifiers. You also know some more interesting aspects of proofs such as proof by induction, if and only if proofs, existence proofs, and uniqueness proofs. How is this information important to your life?

Henri Poincaré: A twentieth century polymath

Many of the first scientists would now be considered madly interdisciplinary. Aristotle’s fields of study ranged from mechanics and optics to medicine and the classification of animals, not to mention philosophy and other fields outside the natural sciences. Archimedes not only was fascinated by proving mathematical principles, he also applied them to physics, astronomy, and engineering. Newton invented principles which now are part of calculus while developing his theory of motion. Leonardo da Vinchi and other known Renaissance men were notoriously broad in their fields of knowledge and investigation. Gradually, mathematicians and scientists became more specialized. Darwin focused on biology, Cauchy on mathematics, Einstein on physics, and so on. Now, we recognize some academics as experts in such fields as number theory, particle physics, or Lie groups.

Henri Poincaré was one of the last of the generation of Renaissance men. While he was principally a mathematician, some of his work extended firmly into the world of physics. On the side he was a mining engineer and a philosopher. To see how varied and numerous his contributions were, see this list of things named after him, most of which are mathematical or physical topics.

Henri Poincaré
Image: Connormah via Wikimedia Commons.
Public domain.

Classical physics works very well for large objects with low speeds. In the late 1800s, physicists simultaneously realized that their understanding of the universe utterly failed to explain the behavior of small objects or fast objects. Two theories forever revolutionized our understanding of the universe: relativity, which explains fast moving objects, and quantum mechanics, which explains the behavior of very small objects like electrons. Poincaré contributed mathematically to both of them. Hendrik Antoon Lorentz derived the famous Lorentz transforms which explain relativity on a simple level. Lorentz discovered the Lorentz transforms without collaborating with Poincaré. However, Poincaré did critique Lorentz’ papers and offer additional input, ideas, and encouragement. It was this relationship with Lorentz that would later lead Poincaré into quantum mechanics.

Out of quantum mechanics and relativity, quantum mechanics has by far influenced the world more. It contributed to several major developments, including the understanding of atoms, nuclear power, and semiconductors. Of course, to semiconductors we owe much of our modern society. The development of the transistor would not have been possible without quantum mechanics. Transistors enabled the building of modern computers, cell phones, and the Internet.

For these reasons, Poincaré’s contributions to quantum mechanics are among his most important contributions to math and science. Poincaré was invited to the first Solvay Conference in 1911 on quantum theory by Lorentz. This appears to be the first time Poincaré was exposed to this new theory. In spite of this, his energetic participation in the discussions at the conference were noted by the other participants. In that conference, Max Planck presented a new theory about black body radiation.

Participants in the First Solvay Conference, 1911.
Image: Fastfission via Wikimedia Commons.
Public domain.

Black body radiation simply refers to the light given off by all objects as they cool. By 1911, enough experiments had been done that the wavelengths of light emitted from black bodies of different temperatures were known. However, classical physics failed to explain these results. Plank attempted to explain them by introducing the idea of “resonators” which could produce electromagnetic radiation. Although Planck didn’t consider matter to be made up of these resonators, this is a natural extension of his theory. Poincaré thought of this and questioned how Planck’s theory could explain the transfer of heat within an object. He quickly got to work rederiving Planck’s result and putting it on a more solid theoretical ground. In keeping with quantum theory, his reasoning used probability rather than absolute knowledge about particles. He did arrive at the same result as Planck, although he was more rigorous in doing so:

Unfortunately, just eight months after the First Solvay conference, Henri Poincaré passed away without living to see the impact his research would have on math and physics.

References

McCormmach, Russell (Spring 1967), “Henri Poincaré and the Quantum Theory”, Isis 58 (1): 37-55, doi:10.1086/350182

Plank’s Law on Wikipedia

Henri Poincaré on Wikipedia

Poincaré’s original paper on Planck’s theory (in French) can be seen here.

“The Copernicus of Geometry”

A young Nikolai Lobachevksy. Image: Lev Kriukov (father), via Wikimedia Commons.

On December 1st, 1792 one man, who would create a revolution in geometry, was born. Actually, a lot of people were born on December 1st, 1792. I can’t name any others, but I’m 99% sure that more than one person was born on that day. I’m not a betting man, but if I was I’d even gamble that more than three were born that day. But I don’t really care about them (no offense to them of course, I’m sure they were fine people). I only care about Nikolai Lobachevsky, the man who would take geometry from the ideas of Euclid, throw those ideas away (he didn’t do that), and change the rules and our perception of shapes, angles, and all things geometric.

Since Euclid’s Elements, circa 300 BC, geometry had been looked at in Euclidean way. Euclid’s axioms and postulates were how it was, and mathematicians had to work within those confines. One similarity is how humanity thought that everything revolved around the Earth, the very human, egotistic geocentric model of the cosmos. In Elements, Euclid’s mathematical magnum opus, which may be the most influential treatise of all time, Euclid creates axioms, propositions, and proofs giving an overview of Euclid’s ideas on number theory and geometry. One of the most important axioms within Elements is Euclid’s parallel postulate.

The parallel postulate states that if a line segments intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles. For many, many years, around two thousand, various mathematicians attempted to prove this postulate. With no success, again and again mathematicians would never see the day the postulate was proven, nor were their names engraved in history as the person to prove such an important postulate. However, ideas of negating the postulate altogether came into fashion in the early 19th century. Lobachevsky took this task to hand and worked his way towards an entirely new view of geometry.

Lobachevsky decided to abandon the idea of the parallel postulate, negating its meaning trying to see if there was a possibility for geometry that did not follow the rules Euclid put into place. Lobachevsky worked around the idea that there exist two lines parallel to a given line through a given point not on the line. In 1829, Lobachevsky published a paper in the Kazan Messenger on his new, non-Euclidean geometry, doing this before anyone else had. Unfortunately due to the small nature of the paper, as well as the fact that it was Russian, his work went largely unnoticed. While others like the famous János Bolyai later discovered this new non-Euclidean geometry completely separate from Lobachevsky, they discovered it years after Lobachevsky did. This new geometry became known as hyperbolic geometry, a geometry that Pringles would sponsor if people didn’t hate math and for some reason math had sponsors. A new form of geometry was born, and Lobachevsky discovered his own personal heliocentric cosmos.

Lobachevsky had many other findings. He discovered the angle of parallelism in hyperbolic geometry, the computation for the roots of a polynomial, and the “Lobachevsky criterion for convergence of an infinite series.” When it comes to his life, it unfortunately wasn’t as great as his discoveries. The combination of his radical new theories, findings that were found same time others discovered them (this can be seen with the Graeffe’s method, which is the computation of the roots of a polynomial that I previously mentioned, and Peter Dirichlet’s definition of a function), and being Russian led him to quite the sad ending. Left without the ability to walk and blind with no job due to his quickly deteriorating health, his life ended in poverty. He had lost his son he loved the most to tuberculosis, came from a poor family, died a poor man, and worked hard all his life without much humor or relaxation. He is quite the Russian stereotype. If I were to make a movie about Russia, he would be the person who symbolizes the Russian winter.

Luckily for Lobachevsky, and moreover mathematics as a whole, his legacy and ideas in his works have lived on. Much work has been done in hyperbolic geometry since his time, as well as the extension of non-Euclidean geometry to Riemannian geometry. Taking what we consider as fact and not only negating it but also proving there is more to it, in this sense going from Euclidean to non-Euclidean geometry, is a revolutionary task that not many people in the history of, well, the universe, have done. It’s like that one Arcade Fire song, they just tell us lies.

On February 24th, 1856, a lot of people died. Like, a lot of people. I don’t know how many people, but I assume there were quite a few. When you think of how many people die each day, it’s slightly horrifying. On that day Nikolai Lobachevsky died, a poor man with no vision and not much left to live for. However on February 24th, 1856, many people were born. And even today, even more people were born. And who knows, maybe the next Nikolai Lobachevsky was born today.

Sources and Further Reading:

http://en.wikipedia.org/wiki/Nikolai_Lobachevsky

http://en.wikipedia.org/wiki/Euclid%27s_Elements

http://www.math.brown.edu/~rkenyon/papers/cannon.pdf

http://www.britannica.com/EBchecked/topic/345382/Nikolay-Ivanovich-Lobachevsky

http://www.regentsprep.org/regents/math/geometry/gg1/Euclidean.htm

http://www.encyclopedia.com/topic/Nikolai_Ivanovich_Lobachevsky.aspx

http://en.wikipedia.org/wiki/Hyperbolic_geometry

http://www.cut-the-knot.org/triangle/pythpar/Drama.shtml

http://www-history.mcs.st-and.ac.uk/HistTopics/Non-Euclidean_geometry.html

Archimedes’ principle

Background of Archimedes

Archimedes was born in 287 BC into a wealthy family of nobility and his hometown was a small village near to Greece. He also had a great father who was a great astronomer and mathematician. His father was very friendly to his children, thus Archimedes was greatly influenced by his father. This made him take a keen interest in mathematics, astronomy and ancient Greek geometry when he was a child. In 267 BC, Archimedes was 11 years old. At that time, his father sent him to Alexandria, Egypt and let him learn mathematics with Euclid’s student. Alexandria, located in the mouth of the Nile, was the knowledge and cultural center of the world at that time. There were also a lot of scholars and professionals in various fields. During his stay in this city, Archimedes met many mathematicians, and he learned a lot of knowledge and skills from them. This knowledge made a major impact for his scientific career and is also the basis of his science research in the future.

Achievements of Archimedes

Fig.1 Archimedes' principle

Fig.1 Archimedes’ principle. Image: Yupi666, via Wikimedia Commons.

Archimedes is considered by most great mathematicians as one of the greatest mathematicians of all time. And he had a lot of important achievements because of his early life of learning in Alexandria, Egypt. He was very good at learning, and this skill made him to find a way to solve areas, surface areas, volumes and other many geometrical objects. In addition to geometrical objects, he also had a important achievement in buoyant force. I think that Archimedes’ principle is his most important achievement. This principle told us the basic rule of buoyant force. According to Wikipedia, “the upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially submerged, is equal to the weight of the fluid that the body displaces.”[1] We also can express this principle by using a formula such that F = G (F is the buoyancy and G is the weight of the liquid that the object displaces ). Another different expression is  (ρ is the density of liquid, g is the acceleration of gravity, V is the volume of liquid).

Story of Archimedes’ principle

Fig.2 Archimedes run to palace

Fig.2 Archimedes runs to the palace. Image: Public domain, via Wikimedia Commons.

According to legend, one day, the king of Greece asked his craftsman to make a gold crown for him. However, the king suspected his crown was not made of real gold after his craftsman completed the crown. The king was afraid that his craftsman pocketed his gold. Although the weight of the crown is equal to the weight of gold that he give to the craftsman, but he could not destroy the crown and check. This question stumped the king and his chancellor. At that time, Archimedes had a very good relationship with the king of Greece and he was already very famous in Greece. After he listened to the suggestion of his minister, the king was going to invite Archimedes to test the crown. In the beginning, Archimedes also had no idea how to solve this problem. One day, he was about to bathe in his bath tub. When he got into the bath tub and saw the water spill, he suddenly had a good idea to solve the king’s problem. He thought that he could measure the displacement of a solid in the water and use this method to determine if the crown was made of real gold. And then, he excitedly jumped out from bath tub and ran to the king’s palace. He even forgot to wear clothes and he said “Eureka! Eureka!” (Eureka means “Found it!”) When he arrived at the palace, he immediately began to test the crown. He put the same weight of pure gold and crown into the two bowls that filled with full water and to compare the water of overflow. Then he found that the bowl with real crown overflowed more water than another bowl. It means that crown was made of other metals. This proves that the craftsmen deceived the king. The significance of the test is not that whether the goldsmith deceive the king, but Archimedes discovered the Archimedes’ principle.

Conclusion

Archimedes’ principle is a very important theory for the world. We can see that many modern inventions were made by using this theory, like big ships, submarine and so on. Thus Archimedes was really a great scientist, and he has made an indelible impact for social progress and human development.

Reference

[1] http://en.wikipedia.org/wiki/Archimedes’_principle

Using Math to make Change: The Hull-House Statistics

In the late 1800’s, thousands of immigrants came to the United State seeking for “The American Dream”. Factories boomed as they flooded into cities such as Chicago and New York. But with so many incoming desperate workers, factory owners realized that they could demand long hours for little pay and the foreign immigrants would comply because they had no other option. They couldn’t speak English, they could be easily replaced, they didn’t know of any other life and so they obeyed. This was the era that Jane Addams was born into.

Jane Addams was born in September of 1860 to a very affluent family in Northern Illinois. Jane grew up wanting to make a difference. She read stories by Dickens and watched her mother care for the poor of her neighborhood. In this environment, she decided that she wanted to become a doctor so that she could work among the poor. Unlike many women of her time, Jane went on to get a higher education at the Rockford Female Seminary and the Women’s Medical College of Philadelphia. She was unable to complete her degree due to health problems, but decided that there were other things she could do to help the poor. In 1889, she created Hull-House, a settlement house in Chicago, Illinois [1].

Now, you may be wondering what all of this has to do with math. Well, in that day and age, there was a prevalent idea that those who were poor lived in such conditions because of their physical inability to do better. They believed that you were poor because you were “stupid” and you couldn’t do anything to help that. As Jane lived at Hull-House among the poor of Chicago, she quickly realized that this idea was false. She grew to understand and love these people and saw that they could rise to higher positions in life if they were just given the opportunity. But to do so would require institutional change. And she, as a woman, could not even vote. She understood that for this change to come about, she would need support of the government and policies put into place to protect these immigrants. In order for her to gain the support needed for this change, she would need quantitative data.

In 1895, she co-authored The Hull-House Maps and Papers. This document provided a statistical analysis of the surrounding blocks of Hull-House. To gain this data, the women of Hull-House went door to door and asked a survey of the families who lived inside. These questions included questions such as: how many families lived in each apartment, how many were in each family, how long had they lived in the United States, did they speak English, where did they work, what was their wage, etc [2]. This data was then summarized and quantified into a series of color-coded maps. These maps divided the city blocks according to the wage earned by the family and their nationality. Each apartment was then color-coded so that the data could be easily seen and understood.

Wages Earned Map No. 1. Image: Thomas Y. Crowell & Co via The Newberry Library.

caption

After collecting this data and compiling it into The Hull-House Maps and Papers, the women of Hull-House presented the information and data they had gathered to Congress. The data they gathered would eventually lead to new laws instating maximum working hours, minimum wage, and child-labor protection. This was one of the first times quantitative data was used to prove theories and change the political structure of an institution in this way. It was “a new type of Science” [3]. Through this statistical approach, these women were able to redefine what the general populace believed was the cause of poverty.

I think that there are many times that we get so caught up in the mathematical equations and the proofs that we distance ourselves from the social differences that can be brought about by it. This methodology of quantitative use of data is still used to make differences. The research of a pharmacist student to discover a new medicine and cure a disease. The survey of a teacher amongst his/her students to figure out what more he/she can do to better their educational experience. The analysis of a city-wide opinion on what to do with an area so that their community can be bettered. Each of these examples uses statistical data to bring about change. Each of us have within us the opportunity to do so. We just need to find a problem, grasp at the answers needed and go for it.

Bibliography:

[1] http://en.wikipedia.org/wiki/Jane_Addams

[2] http://media.pfeiffer.edu/lridener/DSS/Addams/family.html

[3] http://xroads.virginia.edu/~hyper/INCORP/Hull-House/hullrd.html

Map Projections

The world is not flat (citation needed). This is a very important aspect of our planet Earth; indeed, were our world flat it would not rotate on its axis the same way, would have to have an edge, and would probably crumble into a non-flat ball of rubble from its own sheer weight. We should all, therefore, be grateful that the world we live on is the 3D almost-perfect-sphere that it is. Cartographers tend to be a little less happy about our world’s roundness than others because it presents them with an irksome problem: how do we model a non-flat 3D world on a flat 2D surface? Initially this may seem like a straightforward issue. We could, for example, just draw the earth by what it looks like from space! Well this doesn’t quite work because, among other things, you would only have a 2D projection of the side of the earth facing you. Angles, sizes, and shapes get distorted, especially as we approach the edges of our disc. Perhaps we could imagine we have a globe that we cut a slice through from North Pole to South Pole, which we can spread out flat on a table! This method is unfortunately flawed as well, since the sphere will never lie flat regardless of how many cuts are made. Clearly this problem isn’t quite as straightforward as we initially hoped! Well luckily for us, mathematicians and cartographers love these types of problems, and many have offered many possible solutions, of which we will discuss a few.
The first and probably most familiar solution (called a projection) to this problem is the Mercator projection. This is probably the map that you had hanging on the wall of your elementary school classroom. Gerardus Mercator developed it in 1569. The goal of this particular projection is to maintain direction of rhumb lines (aka paths of constant bearing), which are lines that meet each meridian (lines between the two poles) at the same angle. It was particularly useful for navigation because of these lines. The Mercator projection’s biggest failure is generally that it distorts sizes more and more as we venture away from the equator, causing the poles to have infinite size. For example, Greenland and Africa take up roughly the same size on the Mercator projection when in actuality Africa is nearly fourteen times larger!

The Mercator Projection with red dots showing size distortion. Image: Stefan Kühn, via Wikimedia Commons.

The Mercator can be created by projecting the Earth onto a vertical cylinder with circumference equal to the circumference of the Earth. The next projection, called the Transverse Mercator, is obtained using a horizontal cylinder instead. This projection does not maintain straight rhumb lines like its counterpart and distorts scale, distance, and direction away from the central meridian used.

Transverse Mercator projection. Image: Public domain, via Wikimedia Commons.

Next among the more famous projections is the Robinson. This map features a flat top and bulging sides, with meridians starting and ending equidistant to each other but spreading out as they approach the equator. This projection can be seen as a compromising projection: it loosely preserves size, shape, and distance by not being exact in any one of them in particular. The Robinson, like the Mercator, is frequently used in classroom maps due to providing good guesses for relative shapes and positions and being very easy to understand.

Robinson projection. Image: Strebe, via Wikimedia Commons.

Next up are the Stereographic and Orthographic projections. These projections have existed for thousands of years. They were even used by the ancient Greeks! These two methods are projections of the sphere onto a plane, resembling what it would look like if you were to view the earth from space. The Orthographic projection maps along straight lines perpendicular to the tangent plane of the sphere (think looking through a window from space) while the Stereographic projection maps each point by constructing a line through a predefined point (like the north pole) and drawing where it intersects the tangent plane (think the image in a mirror that the earth is placed on). Thus, the main difference is that Orthographic takes the projection from infinity while the Stereographic takes the projection from a point on the sphere. This means that the Orthographic projection only shows one hemisphere, where the Stereographic can show the entire sphere (except the pole) but in a more distorted way. For both of these projections, directions are true from the center point. With the Orthographic projection, any line going through the center is a great circle.

Orthographic projection. Image: Strebe, via Wikimedia Commons.

Stereographic projection. Image: Strebe, via Wikimedia Commons.

There are many more projections to be found online and in books. Indeed, Wikipedia has a stellar list of some of the many different types. These projections all have different uses and are able to convey information in their own clever way, but they share at least one thing in common: they take quite a bit of ingenuity and creativity to come up with and they reflect a deep love and understanding of math in their creators.

Ancient Egyptian mathematics

Fig.1 Ancient Egyptian mathematics. Image: Ricardo Liberato, via Wikimedia Commons.

Egypt, located around the Nile, is one of the earliest developed culture areas in the world, and established a unified country near 3200 BC. The great characteristic of the Nile is the regularity of its floods. The periodic floods would inundate the whole land, which had to be re-measured when the waters receded. Egyptian mathematics developed to make accurate measurements for the division of land. In addition, mathematics also played an important role in prediction and preparation of flood events.

Fig. 2: Rhind Mathematical Papyrus. Image: Public domain, via Wikimedia Commons.

Our main awareness of Egyptian mathematics is based on two papyri, named the Moscow Mathematical Papyrus and the Rhind Mathematical Papyrus, which are preserved in Moscow and London respectively. Why were so few documents from ancient Egypt preserved? It’s because any kind of paper doesn’t last as long as stone, clay, or some other materials. The Egyptian utilized a kind of grass as “paper”, named papyrus, which looks like reeds and grows broadly around Nile Delta. However, this paper made of grass would easily dry and crack, and the records on it were really hard to preserve. Instead, some records on stone, clay, and other materials were preserved well. A Frenchman, whose name is Bastien, spent a very long time studying this information, and he finally figured out the meaning of the words on papyrus. His findings help us to understand some applications of the old mathematics on managing civil and religious matters. Specifically, the mathematics can be used in the division of land and wages, and the calculation of amounts of bricks required to build a building. In summary, the realistic problems motivated ancient Egyptians to master arithmetic operations, a fraction method and so on. We can see the ancient Egyptians had a gift in mathematics.

Fractions played an important role in ancient Egyptian mathematics. When an Egyptian conducted a fraction calculation, they only used as numerator. One of the famous examples in Egyptian fraction is how to divide bread. When ancient Egyptians divided nine loaves between ten people, instead of saying that each person should get 9/10 of a loaf, they would say each person should get 1/2+1/4+1/5+1/12+1/30 of a loaf. So why do they express fractions like that? According to Wikipedia, “An Egyptian fraction is the sum of distinct unit fractions, such as 1/2+1/3+1/16. That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators differ from each other. The value of an expression of this type is a positive rational number a/b; for instance the Egyptian fraction above sums to 43/48.[2]” We can see another bread example of how to divide only two loaves between three people. Firstly, they would divide two loaves to four halves, and each person would get half, the remaining 1/2 would be divided into three equal parts, and those three parts are divided between the three people. In the end, each person would get 1/2 plus 1/3 of 1/2, which sum is 2/3. Thus I think that they express fraction like this way because it was easy to understand for them. The Rhind Mathematical Papyrus also recorded some information about this method[3].

The origin of Egyptian mathematics was not as theory. Instead, the mathematics was applied to real life and to solve realistic problems. The beginning of mathematics was just a method to solve problems instead of a subject. Ancient Egyptian used mathematical methods to pay the workers, and found a way to pay them evenly. As time went by, people used this method and to improved this method. Thus, Egyptians studied mathematics and opened up a road for further development of mathematics. The ancient Egyptians not only invented the fraction, but also found other stuff. For example, the method of calculating area of a circle, their own calendar and so on. Those achievements brought great change and convenience for future life.

Reference:

[1]http://en.wikipedia.org/wiki/Ancient_Egyptian_mathematics

[2]http://en.wikipedia.org/wiki/Egyptian_fraction

[3]Clagett, Marshall Ancient Egyptian Science, A Source Book. Volume Three: Ancient Egyptian Mathematics (Memoirs of the American Philosophical Society) American Philosophical Society. 1999 ISBN 978-0-87169-232-0