How did it take so long to invent zero? Zero took thousands of years to invent because there was no need for it.
There is evidence that people have been counting for 50,000 years. Ancient people counted using tally marks. Cave walls, bones, wood, and stones have been found with tally marks scratched into them. This is a very simple way of counting, and you probably use it yourself when you count. However, it is not that useful if you want to count large numbers. We still count with tally marks, but most cultures break them up into fives. In the west, we write four lines with the fifth line as a diagonal mark through it, like this . In Japan, they use a character that has five lines, like this 正. At a glance, you can quickly multiply the fives and add whatever is left of the last set of tally marks. One thing you will notice with this system is that there is no zero. You are counting things, and you only need to count if you have the thing. There is no need to count if you don’t have the thing. If you were counting the number of sheep each farmer owned and one farmer had none, you simply wouldn’t make any tally marks. There was no need for a symbol to represent nothing.
When civilizations moved from simple tallies to actually writing numbers, there was still no need for a zero in many of them. There are two uses for zero. The first is as a placeholder, and the second is as a value in and of itself. The majority of civilizations had a counting system that didn’t use place values. For example, today, the numbers 23, 203, 2003, and 20003 are obviously different to us. The zeros represent no number in that place. 20003 means 2 ten thousands, zero thousands, hundreds, and tens, and 3 ones. Romans would write these four numbers as XXIII, CCIII, MIII, and III. Any Roman numeral with a line over it multiplied it by one thousand.
The Babylonians were different. It took them 1,500 years, but they gradually started to use a number system where they used two vertical wedges, ∧∧, to show an empty place. The important distinction is that they didn’t think of this as a number, but more as punctuation. It was just a mark to show that there was a space in the number.
Math gradually evolved from simple counting and keeping track of things to being more complex. Although Greek mathematics became extremely sophisticated, Greek mathematicians still didn’t treat zero as a number.
The first time 0 was used as an actual number was in the 7th century AD, roughly 50,300 years after counting was first used. It was used by an Indian mathematician called Brahmagupta. Actually, that is not quite true. Brahmagupta was the first person to write out the rules for how to use zero as a number, but he was not the first person to use it. 0 was being used in Indian mathematics at the time and probably had been for a while. Indian mathematicians made a remarkable leap. If a symbol could represent an empty place in a number, perhaps “nothing” itself could be treated as a number. If you could have 505, then you could have 0 + 5. That led to the next step. If nothing could be used as a number, what were the rules for its use? In 628 AD, Brahmagupta set out those rules. These are his rules:
A number added to zero equals that number. a + 0 = a.
A number minus zero equals that number. a – 0 = a. And, a number minus itself is 0. a – a = 0.
A number multiplied by 0 equals 0. a x 0 = 0.
Division was where he had trouble. When we divide a number, we are really asking what number do I multiply by the divisor to get the dividend. For example, 12 ÷ 3 really means what number multiplied by 3 is 12. The answer is 4, of course. That doesn’t work if you do 12 ÷ 0. What number multiplied by 0 is 12? None of them. Every number multiplied by zero is zero. So, 12 ÷ 0 has no answer. And what about 0 ÷ 0? Every number multiplied by zero is zero, so every number is the answer. Brahmagupta correctly worked out most of the rules for zero, but division by zero remained a mystery. Mathematicians would spend many more centuries trying to understand it. And this is what I learned today.
Sources
https://www.sciencefocus.com/science/resisting-a-new-concept-the-discovery-of-zero
https://en.wikipedia.org/wiki/Counting
https://education.casio.co.uk/blog/a-brief-history-of-numbers
Photo by Magda Ehlers: https://www.pexels.com/photo/0-number-on-red-surface-1339874/