Math & Money

A brief history of numerical systems

One, two, three, four, five, six, seven, eight, nine, and zero. With just these ten symbols, we can write any rational number imaginable. But why these particular symbols? Why ten of them? And why do we arrange them the way we do? Numbers have been a fact of life throughout recorded history. Early humans likely counted animals in a flock or members in a tribe using body parts or tally marks.

But as the complexity of life increased, along with the number of things to count, these methods were no longer sufficient. So as they developed, different civilizations came up with ways of recording higher numbers. Many of these systems, like Greek, Hebrew, and Egyptian numerals, were just extensions of tally marks with new symbols added to represent larger magnitudes of value. Each symbol was repeated as many times as necessary and all were added together. Roman numerals added another twist.

If a numeral appeared before one with a higher value, it would be subtracted rather than added. But even with this innovation, it was still a cumbersome method for writing large numbers. The way to a more useful and elegant system lay in something called positional notation. Previous number systems needed to draw many symbols repeatedly and invent a new symbol for each larger magnitude. But a positional system could reuse the same symbols, assigning them different values based on their position in the sequence.

Several civilizations developed positional notation independently, including the Babylonians, the Ancient Chinese, and the Aztecs. By the 8th century, Indian mathematicians had perfected such a system and over the next several centuries, Arab merchants, scholars, and conquerors began to spread it into Europe. This was a decimal, or base ten, system, which could represent any number using only ten unique glyphs.

The positions of these symbols indicate different powers of ten, starting on the right and increasing as we move left. For example, the number 316 reads as 6×10^0 plus 1×10^1 plus 3×10^2. A key breakthrough of this system, which was also independently developed by the Mayans, was the number zero. Older positional notation systems that lacked this symbol would leave a blank in its place, making it hard to distinguish between 63 and 603, or 12 and 120.

The understanding of zero as both a value and a placeholder made for reliable and consistent notation. Of course, it’s possible to use any ten symbols to represent the numerals zero through nine. For a long time, the glyphs varied regionally. Most scholars agree that our current digits evolved from those used in the North African Maghreb region of the Arab Empire. And by the 15th century, what we now know as the Hindu-Arabic numeral system had replaced Roman numerals in everyday life to become the most commonly used number system in the world. So why did the Hindu-Arabic system, along with so many others, use base ten?

The most likely answer is the simplest. That also explains why the Aztecs used a base 20, or vigesimal system. But other bases are possible, too. Babylonian numerals were sexigesimal, or base 60. Any many people think that a base 12, or duodecimal system, would be a good idea. Like 60, 12 is a highly composite number that can be divided by two, three, four, and six, making it much better for representing common fractions.

In fact, both systems appear in our everyday lives, from how we measure degrees and time, to common measurements, like a dozen or a gross. And, of course, the base two, or binary system, is used in all of our digital devices, though programmers also use base eight and base 16 for more compact notation. So the next time you use a large number, think of the massive quantity captured in just these few symbols, and see if you can come up with a different way to represent it.

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