Geometrical meaning of z1+z2?

The fourth vertex of the parallelogram formed with O, z1 and z2.

The distance between the points representing z1 and z2.

Modulus-argument form of z?

z=r(\cos\theta+i\sin\theta) with r=|z|, \theta=\arg z.

Second quadrant, acute angle \frac{\pi}{4}, so \frac{3\pi}{4}.

Third quadrant, so -\frac{3\pi}{4}.

State \left|\frac{z1}{z2}\right| and \arg\frac{z1}{z2}.

\frac{|z1|}{|z2|} and \arg z1-\arg z2.

Which formulae prove the multiplication rule?

The compound angle formulae for \cos(A+B) and \sin(A+B).

What does multiplying by i do?

Rotates by \frac{\pi}{2} anticlockwise, with no change of length.

What do you do if an argument comes out as \frac{4\pi}{3}?

Subtract 2\pi to get the principal argument -\frac{2\pi}{3}.

Argand diagrams and modulus-argument formEdexcel A-Level Further Maths: Flashcards

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How is $z=x+iy$ shown on an Argand diagram?

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How is $z=x+iy$ shown on an Argand diagram?: As the point $(x,y)$ , or the vector from the origin to it.
Geometrical meaning of $z_1+z_2$ ?: The fourth vertex of the parallelogram formed with $O$ , $z_1$ and $z_2$ .
Meaning of $|z_1-z_2|$ ?: The distance between the points representing $z_1$ and $z_2$ .
Modulus-argument form of $z$ ?: $z=r(\cos\theta+i\sin\theta)$ with $r=|z|$ , $\theta=\arg z$ .
Range of the principal argument?: $-\pi<\theta\le\pi$ .
Find $\arg(-1+i)$ .: Second quadrant, acute angle $\frac{\pi}{4}$ , so $\frac{3\pi}{4}$ .
Find $\arg(-1-i)$ .: Third quadrant, so $-\frac{3\pi}{4}$ .
State $|z_1z_2|$ and $\arg(z_1z_2)$ .: $|z_1||z_2|$ and $\arg z_1+\arg z_2$ .
State $\left|\frac{z_1}{z_2}\right|$ and $\arg\frac{z_1}{z_2}$ .: $\frac{|z_1|}{|z_2|}$ and $\arg z_1-\arg z_2$ .
Which formulae prove the multiplication rule?: The compound angle formulae for $\cos(A+B)$ and $\sin(A+B)$ .
What does multiplying by $i$ do?: Rotates by $\frac{\pi}{2}$ anticlockwise, with no change of length.
Convert $2\left(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}\right)$ to $x+iy$ .: $\sqrt3+i$
What do you do if an argument comes out as $\frac{4\pi}{3}$ ?: Subtract $2\pi$ to get the principal argument $-\frac{2\pi}{3}$ .

Exam questions on Argand diagrams and modulus-argument form

The complex number $z=-1+i\sqrt3$ is given.
Find the modulus of $z^2$ and the principal argument of $z^2$ .2 marks
$z_1=4\left(\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}\right)$ and $z_2=2\left(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}\right)$ .
Find $\frac{z_1}{z_2}$ in the form $x+iy$ , giving exact values.2 marks
The complex numbers $z=1+i\sqrt3$ and $w=-\sqrt3+i$ are represented by the points $Z$ and $W$ on an Argand diagram with origin $O$ .
Express $z$ and $w$ in modulus-argument form, with each argument in the range $-\pi<\theta\le\pi$ .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).