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Where the Babylonian number system differs most from ours is that it had no zero—a true zero would not arise until much later. This meant that Babylonians would often have to work out the size of a number from context. If they saw the cuneiform symbol for 42, for instance, they would have to infer whether that meant 42, or 42×601, 42×602, or 42/60, to name just a few of the options. Though this did sometimes lead to mistakes, it's not as unreasonable as it may first seem. If you heard someone say that a house costs "300" of a particular currency, depending on where in the world you are you could probably work out if it meant 300; 300,000; 3 million; or more.

Base-60 may initially seem complicated compared to base-10, but it gave the Babylonians a mathematical edge. The number 60 is a superior highly composite number, meaning that it has many factors—it can be divided by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60. This makes it easy to work with, particularly when writing fractions.

Recall that just as positions going left from a decimal point represent units, tens, hundreds, and so on, when going right after the decimal point they represent tenths, hundredths, thousandths, and so on. The number 0.347, say, is really shorthand for (Not Shown)

We're so used to writing a third like this in decimal that its recurring nature seems normal, but it is a quirk of our number system. It comes from the fact that 10 cannot be divided by 3. However, 60 can. A third is the same as 20/60, meaning that, in sexagesimal, it could simply be written as .20 or, in other numbers:

As 60 is a superior highly composite number, there are more fractions that can be expressed nicely in base-60 than in base-10.

The ancient Egyptians made similar advances around this time. From around 3000 BCE, the people there had specific symbols to represent different numbers as part of a base-10 system. A single line represented the number 1, two lines the number 2, and so on, up to the number 9. There were then specific hieroglyphs for numbers such as 10, 100, 1000, and so on, as well as symbols for fractions. To write a given number, ancient Egyptians simply listed the correct combination of hieroglyphs.

Much of this is collated in the Rhind papyrus, a manuscript written by a scribe called Ahmes. It is the oldest surviving mathematics textbook we know of and has this extraordinary opening: "Accurate reckoning. The entrance of knowledge of all existing things and all obscure secrets." Ahmes wrote the manuscript in around 1550 BCE and says that he used texts from around 2000 BCE to put it together. That the mathematics it contains could be at least four thousand years old is hard to fully appreciate, especially considering that so much of what it contains resembles mathematics as we know it today.

The textbook contains eighty-four mathematical problems and ways to solve them. Six of the problems are about calculating the slope of a pyramid from its height and width using ideas akin to trigonometry. Mathematics is shaped by the people who develop it, so it is no surprise that Egyptian mathematicians were interested in the mathematics of the pyramids when the pharaohs were so obsessed with building them. But mathematical ideas are also universal. Many other cultures independently uncovered the mathematics of trigonometry, from ancient China to Renaissance Europe, just with different motivations. The papyrus also includes division and multiplication tables, as well as explanations of how to calculate volume and area. Many of our modern-day concepts and ideas around arithmetic, algebra and geometry appear in one form or another.