What Is It About Numbers, Anyway?

What is a number? If someone came up to you and claimed to have a completely new kind of numbers, how would you know if they were making sense? It's all in the operations.

Numbers are useful from the point of view of what you do with them, and what you do is often to operate. Some important operations are:

• addition (+)
• multiplication (X)
• subtraction (-)
• division (/)
• counting

When we describe operations, we can discuss their properties.

• If you can switch the order of the numbers you operate on, and it never affects the result, then the operation is called commutative.
• If you can change the order of operating on numbers, and it never affects the result, then the operation is called associative.
• If there is a number which, when combined using the operation, leaves the other number unchanged, that number is called the identity.
• If every number has an opposite, the operation has the inverse property. This means that you can operate with a number, then operate with its opposite, and get back to the number you started with. For example,

      (2 + 5) + (-5) = 2
  

  or

      (2 * 3/5) * (5/3) = 2
  

  Thus, negatives give you the inverses for addition. And reciprocals give you the inverses for multiplication.