The family of numbers
Set | Symbol | Examples |
|---|---|---|
Natural numbers | N | 1, 2, 3, … |
Whole numbers | W | 0, 1, 2, 3, … |
Integers | Z | …, −2, −1, 0, 1, 2, … |
Rational numbers | Q | p/q with p, q integers and q ≠ 0 |
Real numbers | R | all rational and irrational numbers |
Each set contains the one above it.
Irrational numbers
A number that cannot be written as p/q (with p, q integers and q ≠ 0) is irrational. Examples:
√2, √3, √5, π, 0.1010010001…
√n is irrational whenever n is a positive integer that is not a perfect square. So √9 = 3 is rational, but √8 is irrational.
Together, the rational and irrational numbers make up the real numbers. Every real number corresponds to exactly one point on the number line, and every point on the line is a real number.
You can locate √2 on the number line geometrically. Draw a right triangle with both legs 1 unit long. By Pythagoras, its hypotenuse is √(1² + 1²) = √2. Using a compass, swing this length from 0 onto the number line.
Decimal expansions
The decimal form of a number tells you which kind it is:
Decimal expansion | Type | Example |
|---|---|---|
Terminating | rational | 7/8 = 0.875 |
Non-terminating, recurring | rational | 1/3 = 0.333… = 0.3̅ |
Non-terminating, non-recurring | irrational | √2 = 1.41421356… |
Remember: π is irrational. 22/7 is only a convenient rational approximation of it.
Converting a recurring decimal into p/q
Example. Write 0.4̅7̅ (= 0.474747…) as a fraction.
Let x = 0.474747… Two digits repeat, so multiply by 100: 100x = 47.474747…
Subtract: 100x − x = 47, so 99x = 47 and x = 47/99.
Example. Write 1.27̅ (= 1.2777…) as a fraction.
Let x = 1.2777… Then 10x = 12.777… and 100x = 127.777…
Subtract: 100x − 10x = 115, so 90x = 115 and x = 115/90 = 23/18.
Operations with irrational numbers
rational + irrational = irrational. For example, 2 + √3 is irrational.
rational (non-zero) × irrational = irrational. For example, 5√2.
irrational + irrational, or irrational × irrational, may be rational or irrational: √2 × √2 = 2 is rational, but √2 × √3 = √6 is irrational.
Identities for square roots
For positive real numbers a and b:
√(ab) = √a × √b
√(a/b) = √a / √b
(√a + √b)(√a − √b) = a − b
(√a + √b)² = a + 2√(ab) + b
Rationalising the denominator
A fraction like 1/√2 is easier to work with if the denominator is rational. Multiply the numerator and denominator by a suitable number.
1/√2 = (1 × √2)/(√2 × √2) = √2/2.
For 1/(a + √b), multiply by (a − √b), its conjugate.
Example. Rationalise 1/(3 + √2).
Multiply top and bottom by (3 − √2): (3 − √2)/(9 − 2) = (3 − √2)/7.
Laws of exponents for real numbers
For a > 0 and rational exponents p, q:
aᵖ × aᵠ = aᵖ⁺ᵠ, aᵖ ÷ aᵠ = aᵖ⁻ᵠ, (aᵖ)ᵠ = aᵖᵠ, aᵖ × bᵖ = (ab)ᵖ
a^(1/n) is the nth root of a, and a^(m/n) = (nth root of a)ᵐ
Example. 8^(2/3) = (cube root of 8)² = 2² = 4.