## Counting Systems

Why do we count to 10? That question might not seem to be worth asking because it is so engrained into our brains. Most people don't stop to ask why, but an obvious answer might be because we have 10 fingers. In the early days people kept track of things on their fingers. This seems natural and kids do it automatically. We have 10 digits to represent 10 different things:

**0, 1, 2, 3, 4, 5, 6, 7, 8, 9**

If we have more than 10 items and we want to represent that number we just combine the 10 digits we already have to make new numbers. So after we have counted to 9 and have exhausted all of our possible digits, we go back to 0 and add a 1 in front of it to represent 10. Then we continue counting as normal until we reach 19 at which point we increment the 1 to a 2 and so on to infinity.

**0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ..., 19, 20, 21, 22, ..., 29, 30, 31, 32, ... ,99, 100, 101, 102, ...**

A more interesting question is what if we evolved in such a way where we only had 8 fingers instead of 10? Most likely our counting system would change as well. We would naturally just count up to 8 things:

**0, 1, 2, 3, 4, 5, 6, 7**

If we want to continue counting then we do just as before and combine the digits we already have. So after we have counted to 7 and have exhausted all possible digits, we continue with 10 and then go on from there.

**0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, ..., 17, 20, 21, 22, ..., 27, 30, 31, 32, ..., 97, 100, 101, 102, ...**

If we only had 4 fingers, then perhaps our counting system would look like this:

**0, 1, 2, 3, 10, 11, 12, 13, 20, 21, 22, 23, 40, ..., 92, 93, 100, 101, ...**

We can go to the extreme and say that we only have 2 fingers. In which case the only digits available are 0 and 1.

**0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, ...**

Notice that the value 10 does not mean the same thing in all counting systems. When we are talking about multiple counting systems and we say a number like 10, we need to specify the counting system to which we are referring. Lets say we physically have 10 coins (10 in the decimal system that is).
Now lets say that we have 4 different people who have lived their whole lives using one of the four counting systems presented. A person like you and me have lived our whole lives using the decimal counting system or base-10. So we look at these coins and say that there are 10 coins. However, a person who has lived his whole life using the octal counting system or base-8 would count the coins and say there are 12 coins. A person who uses a quaternary system or base-4 would say there are 21 coins. Finally, a person who uses the binary or base-2 counting system would say there are 1010 coins. They are all correct and they are all referring to the same number of coins. It can be shown that any number can be converted to other counting systems. |

There's nothing special about our decimal counting system. Other counting systems are just as valid, and we can apply arithmetic operations to the other counting systems just as easily. Here are the addition tables for all four counting systems.

Similar tables can be made for multiplication, but we won't get into those here. If you ever want to experience what kids are going through with these tables, then try to memorize a table that isn't decimal.