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Number systems and complex numbers: Class 11 Mathematics
Natural, integer, rational and irrational numbers, and complex numbers: the imaginary unit, operations, conjugate, modulus, and the properties of complex numbers.
Key points
The number systems are built step by step. The natural numbers N are 1, 2, 3 and so on. The whole numbers add 0, and the integers Z add the negatives. Rational numbers Q are numbers that can be written as p/q with integers p and q and q not zero. Irrational numbers such as √2 and π cannot be written in this way, and their decimals neither end nor repeat. Rational and irrational numbers together make the real numbers R.
A terminating decimal such as 0.25, or a recurring decimal such as 0.333…, is rational. A recurring decimal can be written as a fraction. Let x = 0.333…. Then 10x = 3.333…, so 9x = 3 and x = 1/3.
The equation x² + 1 = 0 has no real solution. We define the imaginary unit i by i² = −1, so i = √−1. The powers of i repeat in a cycle of four: i¹ = i, i² = −1, i³ = −i, i⁴ = 1. To find iⁿ, divide n by 4 and use the remainder.
A complex number is a number of the form z = a + bi, where a and b are real. a is the real part, Re(z), and b is the imaginary part, Im(z). If b = 0 the number is real, and if a = 0 it is purely imaginary. Two complex numbers are equal when their real parts are equal and their imaginary parts are equal.
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.
- Division: multiply the top and the bottom by the conjugate of the denominator, so that the denominator becomes a real number.
The conjugate of z = a + bi is z̄ = a − bi. The product z z̄ = a² + b² is always real and not negative. The modulus of z is |z| = √(a² + b²), the distance from the origin on the Argand plane. The Argand plane draws a complex number a + bi as the point (a, b).
Properties of the conjugate and modulus:
- z + z̄ = 2a is real, and z − z̄ = 2bi is purely imaginary.
- The conjugate of z₁ + z₂ is z̄₁ + z̄₂, and the conjugate of z₁z₂ is z̄₁ z̄₂.
- |z₁z₂| = |z₁||z₂| and |z₁/z₂| = |z₁|/|z₂|.
- z is real if and only if z = z̄.
If a quadratic equation with real coefficients has a negative discriminant, its roots are a pair of complex conjugates. For x² − 2x + 5 = 0 the discriminant is 4 − 20 = −16, and x = (2 ± 4i)/2 = 1 ± 2i.
Study tip: After every complex number calculation, collect the real and imaginary parts separately and write the final answer in the form a + bi.
Definitions and flashcards
Say the meaning first, then open the card to check.
Rational number
A number that can be written as p/q, with p and q integers and q not zero.
Irrational number
A real number that cannot be written as a fraction of integers.
Imaginary unit
The number i, defined by i² = −1.
Complex number
A number a + bi with a and b real.
Conjugate
The complex number a − bi for the number a + bi.
Modulus
The value √(a² + b²) of a + bi.
Argand plane
A plane on which the complex number a + bi is shown as the point (a, b).
Formula sheet
| Name | Formula | What it means |
|---|---|---|
| Product of complex numbers | (a + bi)(c + di) = (ac − bd) + (ad + bc)i | Use i² = −1. |
| Conjugate product | (a + bi)(a − bi) = a² + b² | A real, non-negative number. |
| Modulus | |a + bi| = √(a² + b²) | The distance from the origin. |
| Division | (a + bi)/(c + di) = (a + bi)(c − di)/(c² + d²) | Multiply top and bottom by the conjugate. |
| Powers of i | i⁴ = 1 | Divide the power by 4 and use the remainder. |
Short questions with model answers
These are the kind of questions asked for 2, 3 and 5 marks. Try each one before opening the answer.
Define a rational number and give an example of an irrational number. (2 marks)
A rational number is one that can be written as p/q with integers p and q and q ≠ 0, for example 3/4. An irrational number cannot be written like this, for example √2 or π.
Find the value of i¹⁹. (2 marks)
19 ÷ 4 leaves a remainder of 3, so i¹⁹ = i³ = −i.
What is the conjugate of a complex number and why is it useful in division? (3 marks)
The conjugate of a + bi is a − bi. The product of a complex number and its conjugate is a² + b², which is real, so multiplying the denominator by its conjugate makes the denominator real.
Show that the sum of a complex number and its conjugate is real. (2 marks)
If z = a + bi then z̄ = a − bi, so z + z̄ = 2a, which is a real number.
When are two complex numbers equal? (2 marks)
a + bi = c + di when a = c and b = d. Both the real parts and the imaginary parts must be equal.
Numericals with worked steps
Simplify (3 + 2i)(1 − 4i).
- 3(1) + 3(−4i) + 2i(1) + 2i(−4i).
- = 3 − 12i + 2i − 8i².
- With i² = −1, −8i² = +8.
- = 3 + 8 − 10i = 11 − 10i.
Answer: 11 − 10i
Express (2 + 3i)/(1 − i) in the form a + bi.
- Multiply top and bottom by 1 + i.
- Numerator: (2 + 3i)(1 + i) = 2 + 2i + 3i + 3i² = 2 + 5i − 3 = −1 + 5i.
- Denominator: (1 − i)(1 + i) = 1 + 1 = 2.
- Result = −1/2 + (5/2)i.
Answer: −1/2 + (5/2)i
Find the modulus of 5 − 12i.
- |z| = √(5² + (−12)²).
- = √(25 + 144) = √169 = 13.
Answer: 13
Solve x² + 4x + 13 = 0.
- Discriminant = 16 − 52 = −36.
- x = (−4 ± √−36)/2 = (−4 ± 6i)/2.
- x = −2 ± 3i.
Answer: x = −2 + 3i or x = −2 − 3i
Write 0.272727… as a fraction.
- Let x = 0.2727…
- 100x = 27.2727…
- Subtract: 99x = 27, so x = 27/99 = 3/11.
Answer: 3/11
Common mistakes
- Writing √−4 × √−9 = √36 = 6. The correct working is 2i × 3i = 6i² = −6.
- Forgetting to change the sign of i² in a product.
- Leaving i in the denominator instead of multiplying by the conjugate.
- Taking the modulus of a + bi as a + b or including i inside the square root.
- Using 0 in place of the remainder when a power of i is a multiple of 4. In that case iⁿ = 1.