Numbers grew as people needed them: counting numbers first, then zero, then negatives, then fractions. Each new kind fixed something the old ones could not do.
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Each family contains the one before it. Knowing which family a number belongs to answers most questions here.
👨👩👧 From counting to rationals
| Family | What it holds | Example |
|---|---|---|
| Natural (N) | 1, 2, 3, … | counting things |
| Whole (W) | naturals + 0 | 0, 1, 2, 3, … |
| Integers (Z) | wholes + negatives | … −2, −1, 0, 1, 2 … |
| Rational (Q) | anything writable as p/q, q ≠ 0 | 3/4, −5, 0.25, 2⅓ |
Every integer is rational. 5 can be written 5/1, so it qualifies. The reverse is not true — 3/4 is rational but not an integer.
These four property names appear directly in exam questions.
📜 Know the names
| Property | Means | Example |
|---|---|---|
| Closure | the answer stays in the family | 2/3 + 1/3 = 1, still rational |
| Commutative | order does not matter | a + b = b + a |
| Associative | grouping does not matter | (a+b)+c = a+(b+c) |
| Distributive | multiplication spreads over addition | a(b+c) = ab + ac |
Two special numbers: 0 is the additive identity (a + 0 = a) and 1 is the multiplicative identity (a × 1 = a). The additive inverse of a is −a; the multiplicative inverse (reciprocal) of p/q is q/p.
Subtraction and division are NOT commutative. 5 − 3 ≠ 3 − 5, and 6 ÷ 2 ≠ 2 ÷ 6. Only addition and multiplication are.
Adding needs a common denominator; multiplying does not.
🧮 The two operations
Add or subtract: make the denominators the same, then add the numerators. 2/3 + 1/6 = 4/6 + 1/6 = 5/6.
Multiply: numerator × numerator, denominator × denominator, then reduce. 3/4 × 2/9 = 6/36 = 1/6.
Divide: flip the second fraction and multiply. 2/3 ÷ 4/9 = 2/3 × 9/4 = 3/2.
Between any two rationals there are infinitely many more. Between 1/2 and 3/4 you can always take the average — (1/2 + 3/4) ÷ 2 = 5/8 — and repeat for ever.
12 questions on rational numbers and their properties. 🍀
