Integers
When amounts run both ways from a starting point, how do numbers below zero behave?
Where you are 6 inside
What this area is about
The integers are the whole numbers together with their negatives: \ldots, -2, -1, 0, 1, 2, \ldots They describe amounts measured from a starting point with room on both sides, such as temperatures below freezing or heights below sea level. Each integer has a side of zero and a size, and the four operations have to be made to work with both.
How these fit together
Definitions. Negative numbers name the amounts on the far side of zero. Opposites and zero pairs pair each number with the one the same distance away on the other side, so that a + (-a) = 0. Absolute value keeps only the size: the distance from 0.
A skill. Placing and comparing integers puts them on a line that runs both ways and reads which of two is greater.
Operations. Adding and subtracting integers combines signed numbers, with subtracting as adding the opposite. Multiplying and dividing integers explains why the sign of a product of nonzero integers depends on how many factors are negative.
A path through it
Negative numbers first. Opposites and the number line follow, then absolute value (after the line), then adding and subtracting, and multiplying and dividing last.
Next come rational numbers, where the same rules extend to signed fractions and decimals, and distance on a number line.