History of Mathematics: Pythagoras Theorem

Math has existed and been used around us for a long time, even though individuals who employed inventing it were not aware that the systems they were creating to solve mundane problems would in the future be considered the foundation of mathematical concepts such as algebra, geometry, etc. These concepts came from efforts to make day-to-day life easier. For example; the modern day concept of combinatorics evolved from an effort to solve problems of enumeration in medicine and perfumery.

The man who invented the concept of permutations and combinatorics; Bhaskra , a Hindu mathematician, did not at that time know that he was creating what we now call combinatorics. This theme is similar for many inventors of mathematical concepts in history. The math we use today has evolved over time, not only in logic and understanding, but also in methods of interpretation and notation.

Today, we usually think of the Pythagoras’s theorem as a statement about numbers; a2 + b2 = c2, and understand that that if a right-angled triangle has short sides of lengths a and b, and long side (hypotenuse) c, then a2 + b2 = c2 . However, Pythagoras , the Greek philosopher who proved the theorem wrote about it in terms of squares, which was 200 years later written down by Euclid, another Greek mathematician, in his book using actual squares: if you draw the squares on the three sides, then you can cut up the squares on the two short sides and piece them together to make up the square on the hypotenuse. Historians have also found that this theorem was inspired by the works and proofs similar to the theorem that were found in the book Vijaganita, by Bhaskara, and traces of knowledge of the theorem have also been found in Babylonian tablets from circa 1900–1600 BCE. The Pythagoras theorem is something that has changed the dimensions completely. Although, the Greeks claim to be the inventory of this principle , the evidence shows that Mesopotamian and Babylonian mathematicians also contributed in Pythagoras theorem far earlier than the Greek mathematicians. It is fair to say that inspiration came to different mathematicians of different cultures via the flow and exchange of information that mathematicians had been working on in the past.

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Another example of this the evolution of the notation that we use today, and how it shaped different mathematical concepts that we use today. The earliest notation known is the Babylonian sexagesimal system. It was positional, similar to modern notation, the place of the numeral in the sequence defines the value it represents, and they even had some symbol to denote a placeholder , the way we now use 0. The Egyptians had pictograms and the Greek, used their alphabet to denote numbers. Later the Greek had some acrophonic system , the first letter of the word for the numeral represents the numeral, and combined them in a way that is somewhat related to the Roman numerals we know. The Chinese then used a similar system with dots and bars, where they oriented their numerals up or down so that placeholders could be omitted to some extent. However, our familiar numerals, often called Arabic, originate from the Hindu and were indeed passed on to us by the Arabs ,mainly via al-Khwārizmī's al-jabr (825 AD) which also coined the word algebra. Fibonacci introduced these numerals in Europe, which helped spreading the news.

The evolution of this notation came as an effort to work on the disadvantages of the one being previously used. For example, the creation and addition of 0 to the system came as an effort to revolutionize money lending and borrowing issues. The evolution also gave birth to systems such as the decimal system, which is crucial to modern day math. The Babylonian sexagesimal system, gave birth to this as it appears to have been superimposed on a decimal system. In the tablets in which these numbers are written the numbers 1 through 9 are represented by a corresponding number of wedge-shaped vertical strokes, and 10 is represented by a new symbol, a hook-shaped mark that resembles a boomerang. However, the next grouping is not ten groups of 10, but rather six groups of 10. The symbol for the next higher group is again a vertical stroke. Logically, this system is equivalent to a base-60 place-value system with a floating sexagesimal point, which can now be interpreted as the “decimal”. This base-60 system is the reason we conventionally believe that the circle can be divided into 360 degrees. Babylonian mathematicians divided all circles into 360 or 720 equal parts and divided the radius into 60 equal parts. In that way, a unit of length along the radius was approximately equal to a unit of length on the circle. Another thing to be noted is the reason behind why the Babylonians set out to find the degrees in a circle. The measurement of angles, arcs of circles, is essential to observation of the sun, moon, stars, and planets, since to the human eye they all appear to be attached to a large sphere rotating overhead. This astronomical endeavor was crucial to understand in order to predict the weather, season changes, and in turn harvest expectations and preparations.

Through this example we see how from the creation of the sexagesimal notation we gained the idea of a decimal system, and how the use of this notation in understanding astronomy has led to our understanding of geometry, arcs and circles. The idea here is that in order to solve daily issues, and make predictions of the future easier in order to make strategic decisions in the present, mathematicians of the past created ground breaking mathematical concepts and systems that we now use with different notations.

These concepts that were applied to day-to-day tasks, however inch at a bigger and more philosophical understanding of the history of math. Everything we know today came about as a question, and then a discovery. A student can learn a great deal simply by considering the unusual nature of the document and asking some questions, and mathematicians were just that. They were students of the past, looking at different concepts with the lens of how to apply it to the present and predict the future. This solidifies the idea that math is a past, present and future concept, in fact an inquisitive question around the systems we use in the world today could lead to the advancement of theorems that could possibly be proved, or made more concrete in the future. We see this a lot in algebra word problems, often encountered in high school math. Even though they are useless, they hold the same idea which is that complicated mathematical reasoning was not invented in order to find solve questions like when two trains will meet if they set out from different stations at different times, or how many candies Sally has if Bilal takes a few from her, but more to flesh out the subject and paint it in brighter and more realistic colors. This substantiates why we need to keep asking ourselves broad philosophical questions while we are studying the past.