Define the real and imaginary parts of a+bi.

Re(z)=a and Im(z)=b; both are real numbers.

When are two complex numbers equal?

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

a-bi; its point is the reflection in the real axis.

|z|=\sqrt{a^2+b^2}, the distance from the origin.

zz^=a^2+b^2=|z|^2, which is real.

What is the principal argument?

The argument \theta with -\pi<\theta\le\pi.

\arg z for z in the second quadrant?

\pi-\tan^{-1}\left|\frac ba\right|.

\arg z for z in the third quadrant?

-\pi+\tan^{-1}\left|\frac ba\right|, a negative angle.

Modulus-argument form of z?

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

Where is a+bi on the Argand diagram?

At the point (a,b): real axis horizontal, imaginary axis vertical.

Principal argument of -3+4i?

\pi-\tan^{-1}\frac43=2.21 radians.

Complex numbers and the Argand diagramEdexcel International A Level Further Maths: Flashcards

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What is $\mathrm{i}^2$?

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What is $\mathrm{i}^2$ ?: $\mathrm{i}^2=-1$ .
Define the real and imaginary parts of $a+b\mathrm{i}$ .: $\text{Re}(z)=a$ and $\text{Im}(z)=b$ ; both are real numbers.
When are two complex numbers equal?: When their real parts are equal and their imaginary parts are equal.
Conjugate of $a+b\mathrm{i}$ ?: $a-b\mathrm{i}$ ; its point is the reflection in the real axis.
Modulus of $a+b\mathrm{i}$ ?: $|z|=\sqrt{a^2+b^2}$ , the distance from the origin.
Simplify $zz^*$ .: $zz^*=a^2+b^2=|z|^2$ , which is real.
State the product rule for moduli.: $|z_1z_2|=|z_1||z_2|$ .
What is the principal argument?: The argument $\theta$ with $-\pi<\theta\le\pi$ .
$\arg z$ for $z$ in the second quadrant?: $\pi-\tan^{-1}\left|\frac ba\right|$ .
$\arg z$ for $z$ in the third quadrant?: $-\pi+\tan^{-1}\left|\frac ba\right|$ , a negative angle.
Modulus-argument form of $z$ ?: $z=r(\cos\theta+\mathrm{i}\sin\theta)$ with $r=|z|$ , $\theta=\arg z$ .
Where is $a+b\mathrm{i}$ on the Argand diagram?: At the point $(a,b)$ : real axis horizontal, imaginary axis vertical.
Principal argument of $-3+4\mathrm{i}$ ?: $\pi-\tan^{-1}\frac43=2.21$ radians.

Exam questions on Complex numbers and the Argand diagram

The complex number $z=-3+4\mathrm{i}$ .
Find $z^*$ and show that $zz^*=|z|^2$ .2 marks
The complex numbers $z_1=5-12\mathrm{i}$ and $z_2=1+\sqrt3\,\mathrm{i}$ .
Find $\arg z_1$ in radians to 3 significant figures.2 marks
Real numbers $x$ and $y$ satisfy the equation $(2x+y)+(x-3y)\mathrm{i}=7+7\mathrm{i}$ .
Find the values of $x$ and $y$ .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).