How do you multiply two complex numbers?

Expand the brackets and replace i^2 with -1.

How do you divide by a complex number?

Multiply numerator and denominator by the conjugate of the denominator.

Geometrical meaning of w1+w2?

The fourth vertex of the parallelogram OACB with A=w1, B=w2.

Which vector is w2-w1?

The vector from the point w1 to the point w2.

What does multiplying by i do?

Rotates the point 90^\circ anticlockwise about O.

What does multiplying by a positive real k do?

Enlarges from O by scale factor k.

-i, so dividing by i rotates 90^\circ clockwise.

Arithmetic of complex numbersEdexcel International A Level Further Maths: Flashcards

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How do you add two complex numbers?

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How do you add two complex numbers?: Add the real parts and add the imaginary parts separately.
How do you multiply two complex numbers?: Expand the brackets and replace $\mathrm{i}^2$ with $-1$ .
$(3+2\mathrm{i})(1-4\mathrm{i})$ ?: $11-10\mathrm{i}$ .
How do you divide by a complex number?: Multiply numerator and denominator by the conjugate of the denominator.
$(c+d\mathrm{i})(c-d\mathrm{i})$ ?: $c^2+d^2$ , a real number.
$\dfrac{3+2\mathrm{i}}{1-4\mathrm{i}}$ ?: $\dfrac{-5+14\mathrm{i}}{17}$ .
What does purely imaginary mean?: The real part is zero.
Geometrical meaning of $w_1+w_2$ ?: The fourth vertex of the parallelogram $OACB$ with $A=w_1$ , $B=w_2$ .
Which vector is $w_2-w_1$ ?: The vector from the point $w_1$ to the point $w_2$ .
Distance between points $w_1$ and $w_2$ ?: $|w_2-w_1|$ .
What does multiplying by $\mathrm{i}$ do?: Rotates the point $90^\circ$ anticlockwise about $O$ .
What does multiplying by a positive real $k$ do?: Enlarges from $O$ by scale factor $k$ .
$\dfrac{1}{\mathrm{i}}$ ?: $-\mathrm{i}$ , so dividing by $\mathrm{i}$ rotates $90^\circ$ clockwise.

Exam questions on Arithmetic of complex numbers

The complex numbers $z_1=3+2\mathrm{i}$ and $z_2=1-4\mathrm{i}$ .
Given that $z_1+\lambda z_2$ is purely imaginary, where $\lambda$ is a real constant, find $\lambda$ .2 marks
The complex numbers $w_1=4+\mathrm{i}$ and $w_2=1+3\mathrm{i}$ are represented by the points $A$ and $B$ on an Argand diagram. The origin is $O$ and $OACB$ is a parallelogram.
Find the exact length of the diagonal $AB$ .2 marks
The complex number $z=1+2\mathrm{i}$ .
Find $z^2$ and $z^3$ , each in the form $a+b\mathrm{i}$ .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).