A rational number is any number that can be written as p/q with q not zero. Every whole number, every fraction and every terminating or repeating decimal is one.
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Each family sits inside the next.
👪 Who is inside whom
| Family | Members | Example |
|---|---|---|
| Natural numbers | 1, 2, 3, … | 7 |
| Whole numbers | naturals together with 0 | 0 |
| Integers | whole numbers and their negatives | −4 |
| Rational numbers | anything of the form p/q, q ≠ 0 | −3/7, 0.25, 5 |
Every integer is rational — 5 is 5/1. But not every rational is an integer. Olympiad papers love the question "which of these is not a rational number?" and the answer is usually something like √2 or a division by zero.
q can never be zero. 3/0 is not a number at all, so it is not rational. 0/3, on the other hand, is simply 0 and is perfectly rational.
Which operations keep you inside the family, and which rules you may use.
⚖️ Property table for rational numbers
| Property | + | − | × | ÷ |
|---|---|---|---|---|
| Closed | ✅ | ✅ | ✅ | ❌ (÷ 0) |
| Commutative | ✅ | ❌ | ✅ | ❌ |
| Associative | ✅ | ❌ | ✅ | ❌ |
⚖️ Identities and inverses
0 is the additive identity: a + 0 = a.
1 is the multiplicative identity: a × 1 = a.
The additive inverse of a/b is −a/b, because they add to 0.
The multiplicative inverse (reciprocal) of a/b is b/a, because they multiply to 1. 0 has no reciprocal.
The distributive law is the one that saves time: a × (b + c) = a×b + a×c. In an Olympiad, 17 × 99 + 17 × 1 = 17 × 100 = 1700 in one step.
Between any two rational numbers there are infinitely many more.
To find a rational number between two others, take their average. Between 1/2 and 1/3: (1/2 + 1/3) ÷ 2 = (5/6) ÷ 2 = 5/12. Repeat and you can find as many as you like — which is why there are infinitely many.
To compare fractions quickly, cross-multiply. For 3/7 and 4/9: 3 × 9 = 27 and 4 × 7 = 28. The bigger product sits above the bigger fraction, so 4/9 is larger.
With negatives the order reverses. −1/2 is smaller than −1/3, even though 1/2 is bigger than 1/3. Draw the line if you are unsure.
12 Olympiad questions on rational numbers and their properties. 🍀
