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Complex numbers

الأسبوع 1 · الأعداد المركبة: الوحدة التخيلية ت وقوى ت والصورة أ + ب ت وتساوي عددين مركبين والعمليات والعددان المترافقان والقسمة · 3 دروس
3

Adding, subtracting and multiplying complex numbers, conjugates, and dividing by a complex number

Mathematics Sec 1 Term 1, Unit 1, Lesson 1
GoalAdd, subtract, multiply and divide complex numbers, and use the conjugate.
New words
conjugate · a − bi is the conjugate of a + bi (only the sign of the imaginary part changes)rationalising the denominator · multiplying top and bottom by the conjugate so the denominator becomes real
Explanation

Complex numbers follow the usual rules of algebra, with one extra fact: i² = −1. To add or subtract, combine the real parts and the imaginary parts separately. To multiply, expand the brackets and then replace i² by −1. The conjugate of a + bi is a − bi. A number and its conjugate have a real sum 2a and a real product (a + bi)(a − bi) = a² − b²i² = a² + b². This is the key to division: to divide by c + di, multiply the top and the bottom by the conjugate c − di. The bottom becomes the real number c² + d², and then you write the answer as a + bi. Be careful with (3 + i)²: it is 9 + 6i + i² = 8 + 6i, not 9 + 1.

OperationRuleExample
add(a + bi) + (c + di) = (a + c) + (b + d)i(6 − 2i) + (1 + 5i) = 7 + 3i
multiplyi² = −1(3 + 2i)(4 − i) = 14 + 5i
conjugate pair(a + bi)(a − bi) = a² + b²(5 − 2i)(5 + 2i) = 29
Worked examples
Add, subtract and multiply. (6 − 2i) + (1 + 5i) = 7 + 3i. (4 + 3i) − (1 − 2i) = 3 + 5i. (3 + 2i)(4 − i) = 12 − 3i + 8i − 2i² = 12 + 5i + 2 = 14 + 5i.
Divide (5 + 5i) by (2 − i). The conjugate of 2 − i is 2 + i. Top: (5 + 5i)(2 + i) = 10 + 5i + 10i + 5i² = 5 + 15i. Bottom: (2 − i)(2 + i) = 4 + 1 = 5. So the answer is (5 + 15i)/5 = 1 + 3i. Check: (1 + 3i)(2 − i) = 2 − i + 6i − 3i² = 5 + 5i ✓.
Find (1 + i)⁸. First (1 + i)² = 1 + 2i + i² = 2i. Then (1 + i)⁸ = (2i)⁴ = 16 × i⁴ = 16.
Class activity

Find the mistake. Hana writes (3 + i)² = 9 + 1 = 10. Expand it correctly with three terms and say which fact she forgot.

Practice
1
Calculate (2 − 3i) + (5 + 4i).
2
Calculate (6 + i)(2 − 3i).
3
Write the conjugate of 4 + 7i and the product of the two numbers.
4
Calculate (10 − 5i)/(1 − 2i).
5
Expand (3 + i)².