Operations with Rational Numbers

What counts as a rational number

A rational number is any number writable as a fraction of two integers:

$$\frac{a}{b} \quad \text{where } a \text{ and } b \text{ are integers, } b \neq 0$$

That covers more than it first appears:

Number	As a fraction	Rational?
$7$	$\tfrac{7}{1}$	yes
$-4$	$\tfrac{-4}{1}$	yes
$0.75$	$\tfrac{3}{4}$	yes
$-2.5$	$\tfrac{-5}{2}$	yes
$0.333\ldots$	$\tfrac{1}{3}$	yes
$\sqrt{2}$	—	no

So every integer is rational, and so is every terminating or repeating decimal. The numbers that are not rational — such as $\sqrt{2}$ and $\pi$ — are irrational, and they are not the subject here.

The sign rules carry over unchanged

This is the point of the lesson. Everything learned about integer signs applies to fractions and decimals with no modification at all:

$$\left(-\tfrac{2}{3}\right)\left(-\tfrac{3}{4}\right) = \tfrac{1}{2} \qquad (-0.5)(4) = -2$$ $$\tfrac{1}{2} - \left(-\tfrac{1}{4}\right) = \tfrac{1}{2} + \tfrac{1}{4} = \tfrac{3}{4}$$

Same signs multiply to a positive. Subtracting is adding the opposite. Two like signs in a row make a plus.

Where the minus sign sits on a fraction

These are all the same number:

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

Writing it in front is the usual choice, because it keeps the fraction itself tidy. Moving the sign to the numerator is the useful move when you are about to add:

$$-\frac{3}{4} + \frac{1}{4} = \frac{-3}{4} + \frac{1}{4} = \frac{-3 + 1}{4} = \frac{-2}{4} = -\frac{1}{2}$$

With a common denominator, the numerators are integers being added, and every integer rule applies to them directly.

Adding and subtracting

Get a common denominator, then treat the numerators as integers.

$$-\frac{1}{3} + \frac{1}{2}$$

Rewrite over $6$:

$$-\frac{2}{6} + \frac{3}{6} = \frac{-2 + 3}{6} = \frac{1}{6}$$

Different signs, so the distances subtract, and the answer takes the sign of whichever was further from zero.

For decimals, line up the point and use the same sign reasoning:

$$-3.4 + 1.2 = -2.2$$

Multiplying and dividing

Multiply the digits, then apply the sign:

$$\left(-\frac{2}{3}\right)\left(\frac{6}{7}\right) = -\frac{12}{21} = -\frac{4}{7}$$

Dividing means multiplying by the reciprocal, and the sign rule is unchanged:

$$-\frac{3}{4} \div \frac{1}{2} = -\frac{3}{4} \times \frac{2}{1} = -\frac{6}{4} = -\frac{3}{2}$$ $$-\frac{1}{2} \div \left(-\frac{1}{4}\right) = -\frac{1}{2} \times \left(-\frac{4}{1}\right) = 2$$

Two negatives give a positive, exactly as with integers.

Why one set of rules covers everything

It would be possible to learn separate procedures for negative integers, negative fractions and negative decimals. Nobody does, and the reason is worth stating.

The sign rules were never about integers specifically. They follow from the distributive property and from what “opposite” means, and both of those hold for every rational number. So the rules could not have come out differently for fractions even in principle.

That is why $\left(-\tfrac{2}{3}\right)\left(-\tfrac{3}{4}\right)$ needs no new technique. Decide the sign — same signs, so positive — then multiply $\tfrac{2}{3}$ by $\tfrac{3}{4}$ as usual. Splitting the problem in two is the method, and it works no matter what the numbers look like.

It also means the arithmetic and the sign can be checked separately, which makes a wrong answer easier to diagnose. A result with the right digits and the wrong sign points at the rules; the reverse points at the fractions.

Multi-step problems

The order of operations does not change: brackets, exponents, multiplication and division left to right, then addition and subtraction left to right.

$$-2 + 3 \times (-4)$$

Multiplication first:

$$-2 + (-12) = -14$$

Doing the addition first would give $1 \times (-4) = -4$, which answers a different question.

Worked examples

Common mistakes

Practice problems