Math Core

Which is colder, $-3°$ or $-8°$ ? Which is the lower score in golf, $-2$ or $-1.5$ ? Comparing negative numbers can feel backward at first. The number line makes it simple.

Rational numbers

Fractions and decimals have opposites too, so there are numbers like $-2.5$ and $-\dfrac{1}{4}$ between the integers.

Definition

Rational number

A rational number is any number that can be written as a fraction $\dfrac{a}{b}$ of two integers, with $b \ne 0$ . All integers, positive and negative fractions, and positive and negative decimals like $-0.75$ are rational numbers.

Every rational number has a spot on the number line. For example, $-2.5$ is halfway between $-3$ and $-2$ .

-2.5 is halfway between -3 and -2. 1.5 is halfway between 1 and 2.

Left is less, right is more

On a number line, numbers increase as you move right and decrease as you move left.

Comparing on a number line

Of two numbers, the one farther to the right is greater. The one farther to the left is less.

Every positive number is greater than every negative number.
Zero is greater than every negative number and less than every positive number.

Look at $-8$ and $-3$ :

$-8$ is to the left of $-3$ , so $-8 < -3$ . You can also write $-3 > -8$ . Both statements say the same thing, just read from different ends.

Common mistake

Don't compare negative numbers by their digits alone. $8$ is bigger than $3$ , but $-8$ is less than $-3$ . The negative number that is farther from zero is the smaller one. Think of temperature: $-8°$ is colder than $-3°$ .

Tip

For two negative numbers, the one with the greater absolute value is the lesser number. $|-8| = 8$ is more than $|-3| = 3$ , so $-8 < -3$ .

Comparing fractions and decimals

To compare negative fractions or decimals, first compare their sizes as if they were positive. Then flip the result, because being farther from zero on the negative side means being smaller.

Worked example: Negative decimals

Compare $-1.4$ and $-1.25$ . Use $<$ or $>$ .

Line up the decimal places: $1.40$ and $1.25$ . Since $1.40 > 1.25$ , the number $-1.4$ is farther left of zero.

So $-1.4 < -1.25$ .

Worked example: Negative fractions

Compare $-\dfrac{2}{3}$ and $-\dfrac{3}{4}$ .

Use a common denominator of 12: $\dfrac{2}{3} = \dfrac{8}{12}$ and $\dfrac{3}{4} = \dfrac{9}{12}$ . So $\dfrac{3}{4}$ is the bigger size, which puts $-\dfrac{3}{4}$ farther left.

So $-\dfrac{3}{4} < -\dfrac{2}{3}$ , or $-\dfrac{2}{3} > -\dfrac{3}{4}$ .

Ordering and real-world statements

Worked example: Putting numbers in order

Order from least to greatest: $0.5$ , $-2$ , $-\dfrac{1}{2}$ , $1$ , $-3.5$ .

Start with the negatives. The one farthest from zero is least: $-3.5$ , then $-2$ , then $-\dfrac{1}{2}$ . Then come the positives: $0.5$ , then $1$ .

From least to greatest: $-3.5$ , $-2$ , $-\dfrac{1}{2}$ , $0.5$ , $1$ .

Worked example: Explaining a comparison

At 6 a.m. the temperature in Fargo was $-12°\text{F}$ . In Duluth it was $-7°\text{F}$ . Write an inequality and explain what it means.

$-12 < -7$ , so Fargo's temperature was lower. Fargo was colder than Duluth by 5 degrees.

Practice

Practice 1

Which statement is true?

Practice 2

Which symbol makes this true? $\quad -4.5 \; \square \; -4.2$

Practice 3

Which number is the least? $\quad -9, \quad 2, \quad -1, \quad 0, \quad -5$

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

Practice 4

Which number is greater than $-\dfrac{1}{2}$ ?

Practice 5

Order from least to greatest: $0.5$ , $-1$ , $2$ , $-3$ , $-1.5$ .

Separate answers with commas, e.g. 2, -5

Practice 6

A desert trail starts at elevation $-40$ meters (below sea level). A cave entrance is at $-65$ meters. Which statement is correct?

Practice 7

Which is true?

Practice 8

Nia's account balance is $-15$ dollars and Omar's is $-9$ dollars. Which statement is true?