Integer

An integer is a positive or negative whole number. -4, 0, and 1,003 are all integers. 499.99 is not an integer. In general integers real numbers; they are rational numbers; and they are not fractions or decimals.

A Mathier Definition

Instead of describing the properties of an integer we'll describe the membership rules for the set $Z$. After we're done, anything that's been allowed into $Z$ counts as an integer.

Start by putting $0$ and $1$ into $Z$. Now, pick an element of $Z$, pick another element of $Z$, and add them together (you can pick the same element twice). Is that number in $Z$ yet? No?! Well let's put it in there fast. We can do the same thing as before except instead of adding, we subtract, and if the difference isn't in $Z$ yet, we put it in there. Anything that could be let into $Z$ with these procedures is an integer.

This is not an efficient algorithm for building out $Z$, but it does show the primary motivation for having integers in the first place. Natural numbers (positive integers) are closed under addition, meaning that if you add any two elements in the set, the sum will be in the set, but natural numbers are not closed under subtraction. Integers are what you get when you expand natural numbers to make a set that is closed under subtraction as well.