The Alphabet of Math: The counting numbers

Just like any language, math has an alphabet. We will build the alphabet in steps. The basic alphabet is the natural or counting numbers. The natural numbers occur in nature. They are also called the counting numbers because they are numbers used to count things. Just like the English alphabet, there is an order. In order, the natural numbers are 1,2,3,4,5,6,7,8,9.

According to The History of Numbers, when math started out, there were only three numbers; one, two and many. There are so called "primitive" cultures alive in the world today where those are still the only numbers used. One, two and many isn't very precise. So many has to be quantified.

According to The History of Numbers, people naturally began counting on fingers. That was good for five or ten things, but what if you had more things to count? Then you could count two more things by counting arms, two more by counting legs, two more by counting eyes or ears. In some primitive cultures, it is possible to count a large number of items by ticking off the fingers and various other body parts. To know if you have lost any sheep during the day, just remember what part of the body you counted up to in the morning and make sure you get that far at the end of the day.

The most commonly developed scheme for keeping track of numbers is using marks on a piece of wood; one mark for each item. Grouping marks, say into fives or tens, makes it easy to see at a glance how many items you have. With practice, you learn to recognize multiples of five or ten. The Sumerians were the first to use symbols for numbers, but they were really only sophisticated groups of tick marks. Counting on body parts is good for a relatively small number of items. The Sumerians were able to represent truly large numbers in the millions to keep track of grain harvests and distribution of resources and even do some astronomy which calls for REALLY big numbers.

The numbers that we use today come from India through Arabia. With the addition of the concept of zero, which we also get from the Indians, we can represent any number. We start counting at 0, denoting an absence of items and count from 1 to 9. To represent larger numbers, we borrow a concept from the Sumerians called positional notation. As we go above 9, we add numbers to the left. Each position to the left represents a multiple of 10. Each position to the left gets a new name. From right to left they go ones, tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions and so on. So 25 represents 2 tens and 5. The number 347 is 3 hundreds, 4 tens and 7.

With the alphabet from 1 to 9 and positional notation, it is possible to represent any number we can count, no matter how large. With this alphabet, we are ready to learn how to combine numbers together to form larger numbers. We'll discuss that in my next blog on basic arithmetic.