1.3.2 The Integers
Integers are the numbers ... -3, -2, -1, 0, 1, 2, 3 ...
Integers expand the counting numbers to include numbers that result from subtracting two equal counting numbers or by subtracting a larger counting number from a smaller counting number.
If we subtract a bigger counting number from a smaller counting number, what does that mean in the physical world? In other words, what would it mean if we have 3 dollars, and subtract 5 of them? This is where the power of abstraction comes in; even if we can't "physically" do this, we can use our imagination to think about it and invent a mathematical creature to describe it. As you know already, 3 - 5 = -2; one could think of the "-" as "not" having 2 of the things.
Again, it's important to realize that negative numbers came out of this subtraction problem. It's a huge conceptual leap forward since you're describing "not" something. Together, the counting numbers, along with 0 and the negative counting numbers, make up the integers. We can notate the integers by writing ... , -3, -2, -1, 0, 1, 2, 3, ... The 3 dots on either end tell us that the pattern continues on forever.
The integers are the numbers ... , -3, -2, -1, 0, 1, 2, 3, ...
What's the first integer that's more negative than -101?
Can you think of a number that's not an integer?
When you add or subtract two integers together, do you get another integer?
When you multiply two integers do you always get an integer?
Do you always get an integer when you divide two integers?
Since we can combine numbers in a way that give us non-integer numbers, it looks like we'll have to invent more types of numbers. In the next section, we'll take a look at rational numbers.