Real and complex numbers: Class 9 Mathematics

Key points

Numbers grow in steps. The natural numbers N are 1, 2, 3 and so on. Adding 0 gives the whole numbers W. Adding the negatives gives the integers Z. A rational number can be written as p/q, where p and q are integers and q is not zero, so Q contains all integers and all fractions. Numbers that cannot be written like this, such as √2, √3 and π, are irrational. Rational and irrational numbers together form the real numbers R.

Every rational number has a decimal that either terminates (0.75) or recurs (0.1666…). Every irrational number has a decimal that neither terminates nor recurs, for example √2 = 1.41421356… A recurring decimal can be changed into a fraction. For x = 0.7777…, multiply by 10 to get 10x = 7.7777…, subtract to get 9x = 7, so x = 7/9. If two digits repeat, multiply by 100.

Real numbers are placed on the number line, with one point for each real number. Between any two real numbers there are infinitely many rational and irrational numbers. Positive numbers lie to the right of zero and negative numbers to the left. The sign < means less than, and > means greater than.

Properties of real numbers under addition and multiplication:

• Closure: a + b and ab are real numbers.
• Commutative: a + b = b + a and ab = ba.
• Associative: (a + b) + c = a + (b + c) and (ab)c = a(bc).
• Identity: a + 0 = a, so 0 is the additive identity. a × 1 = a, so 1 is the multiplicative identity.
• Inverse: a + (−a) = 0. For a not zero, a × (1/a) = 1.
• Distributive: a(b + c) = ab + ac.

Properties of equality and inequality:

• Reflexive a = a, symmetric (if a = b then b = a), transitive (if a = b and b = c then a = c).
• Additive property: if a = b then a + c = b + c. Multiplicative property: if a = b then ac = bc.
• Trichotomy: for any a and b exactly one of a < b, a = b, a > b is true.
• If a < b, then a + c < b + c. Multiplying both sides by a negative number reverses the inequality.

A radical is an expression of the form ⁿ√a, where n is the index, a is the radicand and √ is the radical sign. The nth root of a is the number whose nth power is a. Radicals and exponents are linked by ⁿ√a = a^(1/n). For example ³√8 = 8^(1/3) = 2. When the index is 2 it is not written.

Laws of exponents and radicals:

• aᵐ × aⁿ = aᵐ⁺ⁿ and aᵐ ÷ aⁿ = aᵐ⁻ⁿ.
• (aᵐ)ⁿ = aᵐⁿ and (ab)ⁿ = aⁿbⁿ.
• a⁰ = 1 for a not zero, and a⁻ⁿ = 1/aⁿ.
• √(ab) = √a × √b, and √(a/b) = √a / √b for positive a and b.

Rationalisation removes a radical from the denominator. For 1/√a, multiply top and bottom by √a. For a denominator like √a + √b, multiply by its conjugate √a − √b, because (√a + √b)(√a − √b) = a − b, which has no radical. The conjugate of a binomial surd changes only the sign between the terms.

The equation x² + 1 = 0 has no real solution, because the square of a real number is never negative. We therefore define the imaginary unit i with i² = −1, so i = √−1. The powers of i repeat every four steps: i = i, i² = −1, i³ = −i, i⁴ = 1. To find iⁿ, divide n by 4 and use the remainder.

A complex number has the form z = a + bi, where a and b are real. a is the real part and b is the imaginary part. If b = 0 the number is real, and if a = 0 it is purely imaginary. Two complex numbers are equal only when both real parts and both imaginary parts match. Real numbers are a special case of complex numbers.

Operations on complex numbers:

• Addition and subtraction: (a + bi) ± (c + di) = (a ± c) + (b ± d)i.
• Multiplication: (a + bi)(c + di) = (ac − bd) + (ad + bc)i, using i² = −1.
• Conjugate of a + bi is a − bi. The product (a + bi)(a − bi) = a² + b² is real.
• Division: multiply the top and bottom by the conjugate of the denominator.

Study tip: Practise converting recurring decimals into fractions and rationalising surds, because these appear in almost every paper. In complex number questions always replace i² by −1 at once, then collect the real and imaginary parts separately. Check each answer by substituting a simple value or by reversing the step.

Definitions and flashcards

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Formula sheet

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Short questions with model answers

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Numericals with worked steps

Common mistakes

• Saying that a non-terminating decimal is always irrational. A recurring decimal such as 0.333… is rational.
• Writing √(a + b) = √a + √b. This is false: √(9 + 16) = 5, but √9 + √16 = 7.
• Forgetting to flip the inequality sign when multiplying or dividing by a negative number.
• Writing i² = 1 instead of i² = −1, which spoils every multiplication of complex numbers.
• Multiplying only the numerator by the conjugate when rationalising. The same factor must multiply the denominator too.

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