Basic Math and Pre-Algebra

PART 1. The World of Numbers

CHAPTER 3. Order of Operations and Integers

The Integers

We call the set of whole numbers and their opposites integers. The set of integers can be written like this: {..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...}. The order in which the numbers are written conveys their value. Any negative number is less than zero. The more you owe, the less you have, so -5 is less than -3. Just as the counting numbers, or positive numbers, go up forever without end, the negative numbers go down forever.

DEFINITION

The integers are the set of numbers that include all the positive whole numbers and their opposites, the negative whole numbers, and zero.

The Number Line

One way to visualize the integers is to place them on a number line. Usually the line is drawn horizontally, but you could make a vertical number line if you chose. Zero is marked at a point on the line, and then the line is broken into sections of equal length, to the left and the right of zero.

The positive numbers are placed on equally spaced marks to the right of 0, getting larger as you move to the right. The negative numbers are placed on equally spaced marks to the left of 0, with -1 at the first mark left of 0, then -2 at the next mark to the left, and then -3 and so on. The arrows on the ends of the line remind you that the numbers keep going. (We’ll fill in the spaces between the integers soon.)

When you compare two numbers, remember that the number to the left is the smaller one. The number 5 is to the left of 9 on the number line, so 5 is less than 9, and you know that -8 is smaller than -1 because -8 is to the left of -1 on the number line. Any negative number is smaller than any positive number, and the negatives are to the left of zero and the positives are to the right.

DEFINITION

The number line is a line divided into segments of equal length, labeled with the integers. Positive numbers increase to the right of zero, and negative numbers go down to the left.

Absolute Value

The integers came into being because people wanted a way to express opposite ideas like owing and having, or winning and losing. The number line lets you do that by assigning directions. Positive numbers go to the right (or up, on a vertical line), and negatives go left (or down).

But sometimes you don’t really care about the direction. You just want to know “how far?” The absolute value of a number is how far from 0 the number is, regardless of direction. The positive number 4 is 4 steps away from 0. The absolute value of 4 is 4. The negative number -4 is also 4 steps away from 0, but in the other direction. Absolute value doesn’t care about direction, so the absolute value of -4 is also 4.

DEFINITION

The absolute value of a number is its distance from zero, without regard to direction. Absolute value cannot be negative.

The symbol for the absolute value of a number is a set of vertical bars that surround the number. The absolute value of -9 is written |-9|, so you can write |-9| = 9. If you write |12| = 12, you’re saying that the absolute value of 12 is 12, or that the number 12 is 12 steps away from 0. The absolute value symbols can act like parentheses, so if you see arithmetic inside them, do the arithmetic first, then find the absolute value of the answer.

Find each absolute value.

11. |-19| =

12. |42| =

13. |0| =

14. |5 - 3| =

15. 7 + |5 - 3| =