Complex numbers: arithmetic, roots, geometric effects and loci
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The teaching part (1 min) is free to watch. A free account opens the whole lecture: 5 practice questions, 2 real past-paper questions with the answers explained, the mistakes that lose marks, the full-marks tip, a one-card summary and a quick quiz.
What this lecture covers
- A number you cannot see on the line
- Complex arithmetic
- Division and the conjugate
Read the transcript of the teaching part
Last time
Before we start: last time, in Forming and solving differential equations, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.
A number you cannot see on the line
Try to solve x squared equals minus one and the number line says no. So mathematicians invented a new number, i, with i squared equal to minus one. Numbers like 3 plus 4 i live on a plane, not a line, and that plane is where all the geometry of this chapter happens.
Complex arithmetic
A complex number is a plus b i, with a the real part and b the imaginary part. To add or subtract, combine real parts and imaginary parts separately. To multiply, expand the brackets. Replace i squared by minus one. Two plus 3 i, times 1 minus i, gives 5 plus i.
Division and the conjugate
The conjugate of a plus b i is a minus b i. Multiplying a number by its conjugate gives a real number: the modulus squared. To divide, multiply top and bottom by the conjugate of the bottom. For 1 plus i over 1 minus i, multiply by 1 plus i. The top is 2 i and the bottom is 2. The answer is i.
How often it comes up
How often Cambridge has asked each part, counted by ExamYard from every past paper in its question bank: Loci in the Argand diagram.
The whole chapter: Complex Numbers.
Chapter 3.9: Complex numbers
- Complex numbers: arithmetic, roots, geometric effects and loci 6 min
- Modulus and argument, complex quadratics, equations and square roots 6 min
Also for A Level Maths 9709 Paper 3
Which topics come up most in A Level Maths 9709 Paper 3 past papers, counted from every paper since 2004.