When are two complex numbers equal?

When their real parts are equal and their imaginary parts are equal.

How do you divide \frac{z}{w} in the form a+bi?

Multiply numerator and denominator by the conjugate of w.

What does a negative discriminant tell you?

The quadratic has no real roots, but two complex roots.

Complex number arithmeticAQA A-Level Further Maths: Flashcards

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What is $i^2$?

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What is $i^2$ ?: $-1$
What are $\mathrm{Re}(3-2i)$ and $\mathrm{Im}(3-2i)$ ?: $\mathrm{Re}=3$ and $\mathrm{Im}=-2$ .
When are two complex numbers equal?: When their real parts are equal and their imaginary parts are equal.
$(a+bi)+(c+di)=$ ?: $(a+c)+(b+d)i$
$(a+bi)(c+di)=$ ?: $(ac-bd)+(ad+bc)i$
What is the complex conjugate of $x+iy$ ?: $x-iy$
What is $(x+iy)(x-iy)$ ?: $x^2+y^2$ , which is real.
How do you divide $\frac{z}{w}$ in the form $a+bi$ ?: Multiply numerator and denominator by the conjugate of $w$ .
What does a negative discriminant tell you?: The quadratic has no real roots, but two complex roots.
Roots of $z^2-6z+13=0$ ?: $3\pm2i$
Find $(1+i)^2$ .: $2i$
Find $\frac{1+i}{1-i}$ .: $i$

Exam questions on Complex number arithmetic

The complex numbers $z$ and $w$ are given by $z=3-2i$ and $w=1+4i$ .
Find $\frac{z}{w}$ in the form $a+bi$ , where $a$ and $b$ are real.2 marks
The quadratic equation $z^2-6z+13=0$ has roots $\alpha$ and $\beta$ , where $\alpha$ has positive imaginary part.
Find the value of $\alpha^2+\beta^2$ .2 marks
The complex number $z=x+iy$ , where $x$ and $y$ are real, satisfies $(2+i)z=7+i$ .
Find $z$ in the form $x+iy$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).