Counting Before Numbers Existed

  • by Brett Berry - Wed, 06/01/2016 - 06:26

What is the first lesson in math you remember?

You probably don’t remember distinctly but we can guess you were first tasked with learning to count. You were most likely very young, which is a shame because this first lesson is more critical than you may realize.

Everything you do in mathematics will be based upon an understanding of the system you’re working with. So let’s begin at the beginning with the basis of the system: counting.

Counting before numbers existed. Counting before counting was a thing.

Counting Sheep

Here’s a thought problem:

Pretend you are a sheep herder charged with keeping track of a herd of sheep. Trouble is, you have never been taught numbers or any sort of symbolic representation akin to numbers. Every morning and evening you need to ensure that none of your sheep have wandered away. How do you do this?

Did you give it a try? What did you come up with?

Here are a couple possibilities:

  1. Possible solution one: Obtain a large supply of stones. Group all of your sheep together. Move each sheep by you one-by-one into an enclosed area. Each time a sheep passes, place a stone in a pile. Once all sheep have passed, discard of excess stones. Now each morning and evening thereafter, pass the sheep by you removing the stones from your pile. If you are short stones or are left with excess, then you know that you have gained or lost sheep.
  2. Possible solution two: Repeat the above process, but this time instead of using stones get a large rock and make a mark on it for each sheep as it passes by. Each morning and evening, retrace the marks. If the marks match up, all your sheep are accounted for.

 

What do these solutions have in common?

In each case we’re making a correspondence between one sheep and one of something else (a stone, a tally). In mathematical terms, we have created a one-to-one correspondence. This is the most basic form of counting.

For thousands of years, humans used various forms of tallies and stones to count things. But maintaining large piles of stones and long lists of tallies became cumbersome. So grew the need to group numbers together.

One ancient civilization located in modern-day Iran developed a system of clay tokens based upon finger counting. They had a symbol representing one unit that they would bake into a clay token, another symbol for ten units, and another for 10 ten-units (a hundred), and so forth. They would string these clay tokens together to create large numbers.

This clay token system was great for counting, but it had its drawbacks. The main drawback was that it was not a positional system, meaning that so long as the tokens were strung together it didn’t matter what order they were placed in.
To effectively apply operations to numbers we need a place-value system.

The First Place Value System

The first place-value system was developed by the the Babylonians.

They had two cuneiform symbols used for counting: a vertical line to represent one unit, and a chevron to represent ten units. Different combinations of lines and chevrons were arranged in precise positioning to create unique number representations up to the value of sixty.

This is an example of a sexagesimal or base 60 system since they had 60 unique glyphs.

So if you have ever wondered why minutes and seconds are measured in units of 60, or why shapes like circles, rectangles and triangles are multiples of 60 degrees (at least in Euclidean geometry), you can thank the Babylonians for their influence.

Of course there were many number systems between then and now, and I won’t bore you with all of them. Most of them struggled to develop into an arithmetic system because of one main reason: they lacked a symbol for the concept of zero.

The Discovery of Zero

Finally in 500 A.D., a number system cropped up that included a symbol for zero.

The Hindu-Arabic numerals are the basis of our modern number system and were truly monumental. Up to this point, humans hadn’t a reason to count nothing. But now with the inclusion of a place-holder zero along with distinct symbols for the numerals 1–9, we had a decimal number system that could be adapted into modern mathematics.

For the next thousand years the Hindu-Arabic numerals spread in popularity. Eventually they adopted latin script and morphed into the digits we know commonly today as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

The addition of zero to the natural numbers was such a big deal, we have a separate term for the counting numbers including zero, the whole numbers.

So there it is.

Ten digits. The alphabet of mathematics. The first thing you learn in math. And it took humans a really, really, very long time to create.

Sometimes we take for granted the foundations of our knowledge. We dismiss it as easy, simple, even obvious, but truth is we should be in awe of the fact that out of complete nothingness humanity created a system that would stand for centuries as the foundation of modern mathematics, engineering, physics, computer science and technology.

How amazing is that!

Brett Berry is a math evangelist who writes a math blog for Medium.com called Math Memoirs.