Math & History: Integers: Intro

I. Intro

Integers are the set of numbers that can be written without a fractional component that consist of positive numbers, zero and negative numbers: {-4, -3, -2, -1, 0, 1, 2, 3, 4}.

Integers can be illustrated using a number line such as this.

Absolute Value
The absolute value of a number is the distance between zero and the number on the number line. As such, the absolute value is always positive.

For instance, the absolute value of 6 is 6 since it is 6 units to the right of 0 on the number line. The absolute value of -6 is 6 since it is 6 units to the left of 0 on the number line.

Negation (Opposites)
The negation of a number a is the number that, when added to a, yields zero. Negation is also known as the additive inverse or opposite of a number.

For example, the negation of 5 is -5 since 5 + (-5) = 0. Of course this also means that the negation of -5 is 5 since -5 + 5 = 0.

Negation and Subtraction
Subtracting a number is the same as adding it's opposite.

For example 5 - 3 = 5 + (-3) and 5 - (-2) = 5 + 2.

Negation and Multiplication
Negation is the same as multiplication by -1.

For example, -3 = (-1) x 3 and 2 = (-1) x (-2)

Opposite of Opposite
The opposite of the opposite of a number is the number itself.

For example, -(-3) = 3

II. Conceptualizing Integers

Real world representations of integers include:

1. Temperature (temperature above 0 degrees is positive, below 0 is negative)