The real number system
Every real number is either rational, a fraction of two integers, or irrational, a decimal that never ends and never repeats.
The family of numbers
| Set | Symbol | Examples | Note |
|---|---|---|---|
| Natural numbers | ℕ | 1, 2, 3, 4, … | Counting numbers. Some books include 0; say which you use. |
| Whole numbers | W | 0, 1, 2, 3, … | Natural numbers plus 0. |
| Integers | ℤ | …, −3, −2, −1, 0, 1, 2, 3, … | Whole numbers and their negatives. |
| Rational numbers | ℚ | 3/4, −5, 0.25, 0.333… | Can be written p/q, q ≠ 0. Decimal ends or repeats. |
| Irrational numbers | - | √2, √5, π | Cannot be written p/q. Decimal never ends, never repeats. |
| Real numbers | ℝ | All of the above | Rationals and irrationals together. |
Even, odd and prime
- Even number: can be written 2k, where k is an integer (…, −2, 0, 2, 4, …).
- Odd number: can be written 2k + 1 (…, −1, 1, 3, 5, …).
- Prime number: a natural number with exactly two factors, 1 and itself.
- 2 is the only even prime. 1 is neither prime nor composite.
- Composite number: more than two factors, e.g. 4, 6, 9, 12.
- The 10 primes below 30: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
Rules and formulas
Classify the numbers
Classify each number: −7, 0, 3/4, 0.333…, √16, √5, π, 2.5.
- −7 is a negative integer, so it is not natural or whole. It is also rational, since −7 = −7/1.
- 0 is a whole number and an integer.
- 3/4 is already p/q, so it is rational.
- 0.333… repeats, so it is rational: 0.333… = 1/3.
- √16 = 4, a natural number.
- √5 is not a whole number because 5 is not a perfect square, so it is irrational.
- π = 3.14159… never ends or repeats, so it is irrational.
- 2.5 = 5/2, so it is rational.
Natural: √16. Integers: −7, 0, √16. Rational: −7, 0, 3/4, 0.333…, √16, 2.5. Irrational: √5, π. All eight are real.
Recurring decimal to a fraction
Write (a) 0.454545… and (b) 0.1666… as fractions in lowest terms.
- (a) Let x = 0.454545… Two digits repeat, so multiply by 100.
- 100x = 45.454545…
- Subtract: 100x − x = 45.4545… − 0.4545…, so 99x = 45.
- x = 45/99. Divide top and bottom by 9: x = 5/11.
- (b) Let y = 0.1666… Then 10y = 1.666… and 100y = 16.666…
- Subtract: 100y − 10y = 16.666… − 1.666…, so 90y = 15.
- y = 15/90 = 1/6.
0.454545… = 5/11 and 0.1666… = 1/6.
Prime factors, HCF and LCM
A Cargills outlet has 84 tea packets and 90 biscuit packets. It wants identical gift boxes with nothing left over. (a) What is the largest number of boxes? (b) What is in each box? (c) Find the LCM of 84 and 90.
- Prime factors: 84 = 2 × 2 × 3 × 7 = 2² × 3 × 7.
- 90 = 2 × 3 × 3 × 5 = 2 × 3² × 5.
- HCF = common primes, lowest powers = 2 × 3 = 6.
- Each box: 84 ÷ 6 = 14 tea packets and 90 ÷ 6 = 15 biscuit packets.
- LCM = all primes, highest powers = 2² × 3² × 5 × 7 = 4 × 9 × 35 = 1,260.
- Check: HCF × LCM = 6 × 1,260 = 7,560 and 84 × 90 = 7,560. ✓
(a) 6 boxes. (b) 14 tea and 15 biscuit packets each. (c) LCM = 1,260.
A Cargills shelf counts stock in natural numbers, a month's result can be a negative integer (a loss of Rs. 25,000 is −25,000), and prices such as Rs. 99.50 are rational. Packing and delivery plans use HCF and LCM.
Short answer: “Is √2 rational or irrational? Give a reason.” Irrational. It cannot be written as p/q; its decimal never ends and never repeats.
Short answer: “Express 0.272727… as a fraction in lowest terms.” Let x = 0.2727…; 100x − x = 27, so x = 27/99 = 3/11.
Structured question: “Find the HCF and LCM of 84 and 90 by prime factors.” 84 = 2² × 3 × 7, 90 = 2 × 3² × 5. HCF = 6, LCM = 1,260. Check 6 × 1,260 = 84 × 90.
Natural ⊂ Whole ⊂ Integer ⊂ Rational ⊂ Real. Irrationals sit beside the rationals; together they make the reals. 2 is the only even prime.