Math Core

Real measurements are rarely whole numbers. A stock drops $2.35, a hiker descends $1\tfrac{3}{4}$ miles, a lake level falls $0.8$ inches. The good news: the rules you learned for integers work exactly the same way for negative fractions and decimals.

What is a rational number?

Definition

Rational number

A rational number is any number that can be written as a fraction $\dfrac{a}{b}$ , where $a$ and $b$ are integers and $b \ne 0$ .

Integers like $-7$ (which is $\dfrac{-7}{1}$ ), fractions like $-\dfrac{2}{3}$ , decimals like $-4.15$ and mixed numbers like $-2\tfrac{1}{2}$ are all rational numbers.

A negative fraction can be written three ways, and they all mean the same number:

-\frac{3}{4} = \frac{-3}{4} = \frac{3}{-4}

The form $\dfrac{-3}{4}$ is usually the easiest to add with, because the sign rides along with the numerator.

The same rules as integers

Adding and subtracting rational numbers

Change every subtraction to adding the opposite: $a - b = a + (-b)$ .
Same signs: add the absolute values and keep the sign.
Different signs: subtract the absolute values and keep the sign of the number with the larger absolute value.

For fractions, get a common denominator first, just as you always have.

Here is $\dfrac{3}{4} + \left(-\dfrac{5}{4}\right)$ on a number line. Start at $\dfrac{3}{4}$ and move $\dfrac{5}{4}$ to the left.

3/4 + (−5/4) = −2/4 = −1/2

In numbers: $\dfrac{3}{4} + \dfrac{-5}{4} = \dfrac{3 + (-5)}{4} = \dfrac{-2}{4} = -\dfrac{1}{2}$ .

Negative mixed numbers

A negative mixed number is negative in both parts:

-2\tfrac{1}{3} = -\left(2 + \tfrac{1}{3}\right) = -2 - \tfrac{1}{3}

It is not $-2 + \dfrac{1}{3}$ . The safest way to work with negative mixed numbers is to change them to fractions first: $-2\tfrac{1}{3} = -\dfrac{7}{3}$ .

Worked example: Fractions with unlike denominators

Find $-\dfrac{2}{3} + \dfrac{1}{4}$ .

Use the common denominator $12$ :

-\frac{2}{3} + \frac{1}{4} = \frac{-8}{12} + \frac{3}{12} = \frac{-8 + 3}{12} = \frac{-5}{12} = -\frac{5}{12}

Does the sign make sense? $\dfrac{2}{3}$ is bigger than $\dfrac{1}{4}$ , and the bigger one is negative, so the answer is negative.

Worked example: Subtracting a negative fraction

Find $\dfrac{1}{6} - \left(-\dfrac{5}{8}\right)$ .

Add the opposite, then use the common denominator $24$ :

\frac{1}{6} + \frac{5}{8} = \frac{4}{24} + \frac{15}{24} = \frac{19}{24}

Worked example: Decimals

$-3.8 + 1.25$
$-0.6 - 2.45$

Solutions.

Different signs: $3.80 - 1.25 = 2.55$ . The number with the larger absolute value is $-3.8$ , so the answer is $-2.55$ .
Add the opposite: $-0.6 + (-2.45)$ . Same signs: $0.60 + 2.45 = 3.05$ , so the answer is $-3.05$ .

Worked example: Mixed numbers in context

At the start of the week a pond's water level was $1\tfrac{1}{2}$ inches above normal. By Friday it had dropped $3\tfrac{3}{4}$ inches. Where is the level now, compared with normal?

Change to fractions with denominator $4$ : $1\tfrac{1}{2} = \dfrac{6}{4}$ and $3\tfrac{3}{4} = \dfrac{15}{4}$ .

\frac{6}{4} - \frac{15}{4} = \frac{6 + (-15)}{4} = \frac{-9}{4} = -2\tfrac{1}{4}

The level is $2\tfrac{1}{4}$ inches below normal.

Using properties to make it easier

Addition is commutative (you can switch the order) and associative (you can regroup). Use that to pair up numbers that are easy to combine, especially opposites.

4.7 + (-9.2) + (-4.7) = \big(4.7 + (-4.7)\big) + (-9.2) = 0 + (-9.2) = -9.2

Common mistake

When you subtract, only the number after the minus sign changes to its opposite. In $-\dfrac{1}{2} - \dfrac{1}{3}$ , the $-\dfrac{1}{2}$ stays negative, and you get $-\dfrac{3}{6} + \left(-\dfrac{2}{6}\right) = -\dfrac{5}{6}$ . A common error is to change both signs and get $\dfrac{5}{6}$ or $\dfrac{1}{6}$ .

Tip

Estimate first. $-3\tfrac{7}{8} + 1\tfrac{1}{10}$ is about $-4 + 1 = -3$ . If your exact answer is not close to $-3$ , look for a mistake.