Math Core

On a cold morning the thermometer reads 4 degrees below zero. A diver swims 30 feet below the surface. Whole numbers can't describe these amounts, because they stop at zero. To go below zero, you need negative numbers.

Numbers on both sides of zero

Picture a number line. The whole numbers $0, 1, 2, 3, \dots$ march off to the right. Now keep the line going to the left of zero, using the same spacing. The numbers you reach there are the negative numbers: $-1, -2, -3, \dots$

Negative numbers are to the left of 0. Positive numbers are to the right.

A negative number is written with a minus sign in front, like $-6$ , read "negative six." A positive number can be written with a plus sign, like $+6$ , but usually the sign is left off: $6$ means positive six.

Zero is neither positive nor negative. It is the dividing point between the two sides.

Definition

Integers

The integers are the whole numbers together with their negatives:

$\dots, -3, -2, -1, 0, 1, 2, 3, \dots$

The integers keep going forever in both directions.

Numbers like $\dfrac{1}{2}$ , $-3.5$ and $2\tfrac{1}{4}$ are not integers, because they fall between the tick marks. An integer has no fraction or decimal part.

Positive and negative in real life

Positive and negative numbers describe amounts that go in opposite directions from a starting point of zero. Once you decide which direction is positive, the other direction is negative.

situation	zero means	positive	negative
temperature	$0°$	above zero	below zero
elevation	sea level	above sea level	below sea level
bank account	no money	a deposit or money you have	a withdrawal or money you owe
football	starting spot	yards gained	yards lost

Reading a situation

Find the zero point first. Then ask: is this amount above/more/gained (positive) or below/less/lost (negative)?

Worked example: Writing integers for situations

Write an integer for each situation.

A town is 85 feet below sea level.
Maria deposits $40 into her account.
A football team loses 7 yards.
The temperature is 12 degrees above zero.

Solutions.

Below sea level is negative: $-85$ .
A deposit adds money, so it is positive: $40$ .
A loss is negative: $-7$ .
Above zero is positive: $12$ .

Locating integers on a number line

To place an integer, start at $0$ . Count right for a positive number and left for a negative number.

Worked example: Plotting integers

Plot $-7$ , $-2$ and $5$ on a number line.

For $-7$ , start at $0$ and count 7 units to the left.
For $-2$ , count 2 units to the left.
For $5$ , count 5 units to the right.

Common mistake

Watch the direction when you count on the left side. $-7$ is farther from zero than $-2$ , so it sits farther to the left. Students sometimes put $-7$ near $-2$ because "7 comes after 2." On the left side, the numbers grow in size as you move away from zero.

Worked example: Reading a number line

A number line has a dot 3 tick marks to the left of $0$ and another dot 8 tick marks to the right of $0$ . Each tick mark is 1 unit. What integers do the dots show?

The dot 3 units left of zero is $-3$ . The dot 8 units right of zero is $8$ .

Tip

A vertical number line works the same way, like a thermometer. Positive numbers go up from zero, and negative numbers go down.

Practice

Practice 1

A submarine is 15 meters below the surface of the ocean. Write an integer for its position, using the surface as zero.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 2

Jalen earns $25 mowing a lawn. Write an integer for the change in his money.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

Which of these numbers is an integer?

Practice 4

Which statement about zero is true?

Practice 5

What integer is 6 units to the left of $0$ on a number line?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

A number line shows a dot at $1$ . Move the dot 5 units to the left. What integer does it land on?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

In a game, winning points is positive. Which situation is described by $-20$ ?

Practice 8

How many integers are between $-4$ and $4$ ? (Don't count $-4$ or $4$ themselves.)

Enter a number. Fractions like 3/4 and sqrt(2) are OK.