Topic 4 of 12

Negative numbers and the number line

Step 8 · Ratios and negative numbers topics

Numbers continue below zero. Learn to place them on a number line, find opposites and absolute values, and compare them.

Numbers below zero

Some quantities go below zero: a temperature 5 degrees below zero, a parking level 2 floors below the ground, owing someone 10 dollars. We write these with a minus sign: $-5$ , $-2$ , $-10$ . Numbers greater than zero are positive. Zero is neither positive nor negative.

The number line

On a number line, zero sits in the middle. Positive numbers go to the right and negative numbers go to the left.

−6

−5

−4

−3

−2

−1

A number line from −6 to 6, with −4 and 3 marked. −4 is four steps left of zero and 3 is three steps right.

Every number is greater than the numbers to its left. So $-2 > -5$ , because $-2$ is to the right of $-5$ . A temperature of $-2$ degrees is warmer than $-5$ degrees.

Opposites

Two numbers that are the same distance from zero on opposite sides are called opposites. The opposite of 3 is $-3$ , and the opposite of $-7$ is 7. Zero is its own opposite.

−6

−5

−4

−3

−2

−1

−3 and 3 are opposites: each is three steps from zero, on opposite sides.

Absolute value

The absolute value of a number is its distance from zero. A distance is never negative. Absolute value is written with two bars:

|{-6}| = 6 \qquad |6| = 6 \qquad |0| = 0

Order and absolute value are different ideas. $-8$ is less than $-3$ , but $|{-8}|$ is greater than $|{-3}|$ . A debt of 8 dollars is a bigger debt than a debt of 3 dollars, even though $-8$ is the smaller number.

Example: Putting numbers in order

Order from least to greatest: 2, $-5$ , 0, $-1.5$ , 4.

Picture each number on the number line.
The negatives are left of zero. $-5$ is furthest left, then $-1.5$ .
Zero comes next, then the positives in their usual order.

Answer $-5$ , $-1.5$ , 0, 2, 4

Example: Below sea level

Sea level is 0. A diver is at $-12$ meters and a fish is at $-7$ meters. Which is deeper, and how far apart are they?

$-12$ is to the left of $-7$ on the number line, so $-12$ is lower. The diver is deeper.
Use distances from zero: the diver is 12 meters down and the fish is 7 meters down.
Subtract the distances: $12 - 7 = 5$ .

AnswerThe diver is deeper, by 5 meters.

Try it yourself

Work each one on paper first, then open the answer to check.

Which is greater, $-6$ or $-9$ ?
Show answer
$-6$ , because it is to the right of $-9$ on the number line.
What is the opposite of $-15$ ?
Show answer
15.
Find $|{-23}|$ .
Show answer
23, because $-23$ is 23 steps from zero.
Order from least to greatest: $-3$ , 5, $-8$ , 0, 1.
Show answer
$-8$ , $-3$ , 0, 1, 5.
The temperature was $-4$ degrees in the morning and 3 degrees at noon. Which was colder, and by how many degrees did it rise?
Show answer
The morning was colder. It rose 7 degrees: 4 degrees up to zero, then 3 more.
True or false: $|{-10}| < |4|$ .
Show answer
False. $|{-10}| = 10$ and $|4| = 4$ , and 10 is greater than 4.