One or Many? A History of Numbers – Part I

Sheamol Obeda

From school, we’ve always been taught that the square roots of a negative numbers do not exist, but then the concept of imaginary numbers blows our minds! How can we possibly have a square root of -9?! Does this number really exist? Where is it?! How can we see it?! To understand imaginary numbers, we must take a look at the history of numbers themselves and ask questions about life before modern numbers. These articles will cover the origins of numbers such as 0, or the square root of 2, as well as explore the discovery of the well-known numbers used in everyday mathematics, from e and π to the lesser used φ.

In the beginning

Before the human race had all of the modern numbers that we use today, we had two “numbers”: one and “many.” We soon adopted a new number, the number 2, but there were still no numbers greater than 3. As time passed, new numbers and number systems were introduced and used. The ancient Sumerian and Babylonian civilizations started using systems for measurement so that landowners could be fairly taxed, but it wasn’t until the Egyptians’ creation of pyramids that the world saw the real use of mathematics and geometry. With their remarkable measurement system of using a man’s elbow to the fingertip plus a width of their palm, or one cubit, they were able to construct pyramids with astonishing accuracy!

The Egyptians and Babylonians were also the first cases in history to use the constant now known as π (even though the term π did not exist at that point in time). Both the Babylonians and Egyptians had rough numerical approximations to the value of pi and, later, mathematicians in ancient Greece (particularly Archimedes) improved on those approximations. By the start of the 20th century, about 500 digits of pi were known. With computation advances thanks to computers we now know more than the first six billion digits of pi.

The Sumerians are also believed to have been the first people to create calculation methods. They would use sticks or bone with notches in for basic calculations, but it wasn’t until the Chinese that we saw the first abacus. Since then, we’ve seen many calculation methods, from base 12 addition on your fingers used by some Indian cultures (which is still taught today!), to slide rules, mechanical calculators, and, finally, a computer! But more about this later…

There is geometry in the humming of the strings, there is music in the spacing of the spheres

In around 520 BCE, Pythagoras founded his school of mathematics, observing the harmonies of whole numbers and what he could do with them. He taught his students that all numbers are either whole or can be written as a division of whole numbers (i.e. all numbers are rational). He was fascinated with the triangle– specifically finding triangles that had integer values for sides. The Pythagorean theorem that we all learned at school is credited to Pythagoras. He was the one to use it and teach it to the wider community, although he was not the one who discovered it. The Indian texts found in Sulva Sutras and the Shatapatha Brahmana, almost 2000 before the birth of Pythagoras, clearly showed the theorem that we all associate with him.

One of Pythagoras’s followers found a problem with his view of rational numbers. There is a story that Hippasus found a number that was irrational. Hippasus was an excellent mathematician, and he noticed something about the pentagram. When a pentagram is divided up, there is a ratio between the lengths of the pieces. Hippasus took the measure of the length of the red side divided by the green side. It’s equal to the length of the green side divided by the blue side, which is also equal to the length of the blue side divided by the purple side. However, none of these things are expressible as the ratio of two whole numbers. They form the golden ratio, which, in decimals, is approximately 1.61803. Hippasus was clever, but not that clever. He announced that he had found a way to demolish Pythagoras’s ideology. However, this was done on a boat populated only by himself, other Pythagoreans, and Pythagoras. The story goes that Pythagoras pushed him overboard, drowning him, and forced his other followers to swear to secrecy. It’s doubtful this actually happened, especially since there’s another rumor that Hippasus was killed not for discovering the golden ratio, but with the square root of two, another irrational number (he realized there was no rational way to express the diagonal of a square with sides one unit long).

This was the first case of people not understanding a new type of number. There are plenty more examples of this in history, which include the use of the number 0, negative numbers, and eventually modern mathematical concepts such as complex numbers.

I, for One, like Roman Numerals

Later on, during the Greek period, Archimedes, a renowned scientist that loved to play games with numbers, entered the realm of the unimaginable. He posed questions such as: “how many grains of sand could fit into the universe?” and “how would we turn the surface of a sphere into a cylinder?” His methods are still used today. For example, his methods are used to plot the globe on a map. However, the Romans ended the time of Archimedes.

Romans invading Greece were interested in power, not abstract mathematics. They killed Archimedes in 212 BCE and impeded the development of mathematics. They brought new systems of numbers, the Roman Numerals, but they were too complicated for calculations, so actual counting had to be done on a counting board, an early form of the abacus. Although the use of the Roman numeral system spread all over Europe and remained the dominant numeral system for more than five hundred years, not a single Roman mathematician is celebrated today. The Romans were more interested in using numbers to record their conquests and count dead bodies, not for the curiosity of learning about the universe.

Edited by: Kaylynn Crawford and Karen Yung