Rational Numbers & Their Operations

A rational number is any number you can write as a fraction — an integer over an integer, as long as the bottom isn't zero. That makes the family bigger than it first looks: every whole number is rational, because 7 is 7/1. Every integer is rational. −1.5 is rational — it's −3/2. Even recurring decimals are rational, as you proved last lesson. The only numbers left outside are ones like π and √2, whose decimals never end and never repeat.

Rationals fill in the gaps between the integers. To place one, convert it into a decimal or a mixed number and find its integer neighbours. −3/2 is −1.5, so it sits halfway between −2 and −1 — note it's to the LEFT of −1, because more negative means smaller. 3/4 sits three-quarters of the way from 0 to 1. When you're comparing negatives, the line is the honest referee.

Adding and subtracting fractions needs matching pieces — a common denominator. −2/3 + 1/2: the denominators 3 and 2 both fit into 6, so rewrite as −4/6 + 3/6. Now only the numerators matter, and they follow the integer rules from Lesson 1: −4 + 3 = −1. Answer: −1/6. Nothing new is happening — it's integer addition wearing a fraction costume.

Multiplying fractions is the friendly one — no common denominator needed. Handle the sign first, using Lesson 1's rules: negative times positive is negative. Then multiply straight across, tops together and bottoms together: 6/36, which simplifies to 1/6. So −3/4 × 2/9 = −1/6. The professional move is to cancel BEFORE multiplying — the 3 into the 9, the 2 into the 4 — so you multiply tiny numbers and never face an ugly simplification at the end.

To divide by a fraction: keep, change, flip. Keep the first fraction, change the division to multiplication, flip the second fraction to its reciprocal. −5/6 ÷ (−2/3) becomes −5/6 × (−3/2). Signs first: negative times negative is positive. Cancel the 3 into the 6, multiply across: 5/4, or 1¼. And notice the sense of it — asking how many two-thirds fit into something is the same as multiplying by 3/2.

Now everything joins up: 1/2 − 3/4 × (−2/3). BODMAS says multiplication first: 3/4 × (−2/3) — sign is negative, cancel and multiply to get −1/2. Substitute back: 1/2 − (−1/2). Tidy the signs: minus a negative is plus. 1/2 + 1/2 = 1, exactly. One line of working that uses every idea in this chapter.

Let's zoom out. This chapter looked like four topics, but it's one system. The rational numbers are all the fractions, and that includes the integers and every terminating or recurring decimal. Across all of them, the same machinery applies: BODMAS decides the order, the sign rules handle the negatives, place value runs the decimal arithmetic, and everything lives together on one number line. If you can run that machinery reliably — one operation per line, signs tidied first — you're ready for the algebra ahead, where letters take over from numbers but the rules stay identical.