3.9 Complex numbers

Learning objectives

3.9.1Complex numbers• understand the idea of a complex number, recall the meaning of the terms real part, imaginary part, modulus, argument, conjugate, and use the fact that two complex numbers are equal if and only if both real and imaginary parts are equal Notations Re z, Im z, |z|, arg z, z* should be known. The argument of a complex number will usually refer to an angle i such that 1 G-rr i, but in some cases the interval 02 1G ri may be more convenient. Answers may use either interval unless the question specifies otherwise.3.9.2Complex numbers• carry out operations of addition, subtraction, multiplication and division of two complex numbers expressed in Cartesian form x + iy For calculations involving multiplication or division, full details of the working should be shown.3.9.3The result that• use the result that, for a polynomial equation with real coefficients, any non-real roots occur in conjugate pairs e.g. in solving a cubic or quartic equation where one complex root is given.3.9.4Complex numbers• represent complex numbers geometrically by means of an Argand diagram 3.9.5Complex numbers• carry out operations of multiplication and division of two complex numbers expressed in polar form coss inrr i ei/ii+ i^h Including the results |z1z2| = |z1||z2| and arga rg argzz zz12 12= +__ _ii i, and corresponding results for division.3.9.6Complex numbers• find the two square roots of a complex number e.g. the square roots of 5 + 12i in exact Cartesian form. Full details of the working should be shown.3.9.7Complex numbers• understand in simple terms the geometrical effects of conjugating a complex number and of adding, subtracting, multiplying and dividing two complex numbers 3.9.8Complex numbers• illustrate simple equations and inequalities involving complex numbers by means of loci in an Argand diagram. e.g. |z - a| < k, |z - a| = |z - b|, arg(z - a) = α.

For $z=x+iy$ with real $x,y$ :

Notation	Meaning
$\operatorname{Re}z=x$	real part
$\operatorname{Im}z=y$	imaginary part (not $iy$ )
$\|z\|=\sqrt{x^2+y^2}$	modulus
$\arg z=\theta$	directed angle from positive real axis
$z^*=x-iy$	conjugate

A non-zero complex number has arguments differing by $2\pi$ . Unless specified, use a consistent principal interval such as $-\pi<\theta\le\pi$ ; $0\le\theta<2\pi$ may also be convenient. $\arg0$ is undefined.

$$x+iy=u+iv$ if and only if $x=u$ and $y=v$. Also$ zz^*=x^2+y^2=|z|^2.

For $z=-1+i\sqrt3$ , $|z|=2$ and a principal argument is $2\pi/3$ ; its conjugate is $-1-i\sqrt3$ .

Equality requires both components, not merely equal moduli. Quadrant determines argument; $\tan^{-1}(y/x)$ alone is insufficient.

Add complex numbers componentwise, multiply by expanding and replace i² with −1, and divide by multiplying numerator and denominator by the denominator’s conjugate.

After simplifying, write the answer in a+bi form. Equality of complex numbers requires equality of both components.

(1+2i)/(3−i) =[(1+2i)(3+i)]/10=(1+7i)/10.

A complex fraction is not simplified by dividing real parts and imaginary parts separately.

If a polynomial has real coefficients and $a+ib$ ( $b\ne0$ ) is a root, then $a-ib$ is also a root. The result is conditional on real coefficients.

$The pair gives the real quadratic factor$ [x-(a+ib)][x-(a-ib)]=(x-a)^2+b^2.

Write the conjugate root immediately, multiply the pair to form a real factor, divide the cubic/quartic polynomial by it, then solve the remaining lower-degree factor and verify the total degree.

$If a real cubic has roots $1+i$ and $2$, it also has $1-i$, so a monic polynomial is$ (x-2)[(x-1)^2+1].

Conjugate pairing is not guaranteed for non-real coefficients. The broad fundamental theorem/root count is not the learning target here.

An Argand diagram has horizontal real axis and vertical imaginary axis. The complex number $z=x+iy$ is represented by point/vector $(x,y)$ from the origin.

Algebra	Argand meaning
$\operatorname{Re}z=x$	horizontal coordinate
$\operatorname{Im}z=y$	vertical coordinate
$\|z\|$	distance from origin
$\arg z$	directed angle from positive real axis

$-2+3i$ is plotted at $(-2,3)$ in quadrant II; its modulus is $\sqrt{13}$ and its argument must lie in the chosen quadrant-II branch.

Label axes, scale and reference points. Read a plotted point back as real coordinate plus $i$ times imaginary coordinate.

This is a coordinate representation, not a graph $y=f(x)$ . Polar multiplication belongs to the next objective.

$Write$ z=r(\cos\theta+i\sin\theta)=re^{i\theta},\qquad r>0.

Operation	Modulus	Argument
$z_1z_2$	$r_1r_2$	$\theta_1+\theta_2$
$z_1/z_2$	$r_1/r_2$	$\theta_1-\theta_2$

[2e^{i\pi/3}][3e^{-i\pi/6}]=6e^{i\pi/6},\qquad \frac{2e^{i\pi/3}}{3e^{-i\pi/6}}=\frac23e^{i\pi/2}.

Operate moduli and arguments separately, reduce the final argument to the requested interval, and convert to Cartesian form only if asked.

Do not add moduli during multiplication. De Moivre powers and general $n$ th roots are not objectives in this syllabus section and must not be imported.

$To solve $z^2=p+iq$, set $z=a+ib$ with real $a,b$:$ (a+ib)^2=(a^2-b^2)+2abi. $Hence $a^2-b^2=p$ and $2ab=q$.$

$Also $a^2+b^2=|p+iq|=\sqrt{p^2+q^2}$. Add/subtract this with $a^2-b^2=p$ to find $a^2,b^2$, then use $2ab=q$ for signs.$

$For $z^2=5+12i$:$ a^2-b^2=5,\quad2ab=12,\quad a^2+b^2=13, $so $(a,b)=(3,2)$ or $(-3,-2)$ and$ z=\pm(3+2i).

Square both answers and show full Cartesian working. The two roots are always opposites for a non-zero complex number.

Do not take square roots of real and imaginary parts separately, and do not report only a principal root.

Operation	Geometrical effect
$z\mapsto z^*$	reflect in real axis
$z\mapsto z+w$	translate by vector $w$
$z_1-z_2$	displacement from point $z_2$ to $z_1$
multiply by $re^{i\theta}$	scale distances from origin by $r$ , rotate by $\theta$
divide by $re^{i\theta}$	scale by $1/r$ , rotate by $-\theta$

Multiplication by $i=e^{i\pi/2}$ rotates every point $90^\circ$ anticlockwise about the origin without changing modulus. Multiplication by $-2$ scales by $2$ and rotates by $\pi$ .

Addition/subtraction use parallelogram/displacement geometry; multiplication/division use origin-centred scale and rotation. Conjugates keep modulus and negate the argument within branch conventions.

Predict the geometric result, then confirm with Cartesian or polar arithmetic.

A transformation effect is not a locus condition. Powers/general roots are not needed for this objective.

Condition	Locus
$\|z-a\|=r$	circle centre $a$ , radius $r$
$\|z-a\|<r$ / $\le r$	interior, boundary excluded/included
$\|z-a\|>\|z-b\|$	points closer to $b$ than $a$ : one side of perpendicular bisector
$\|z-a\|=\|z-b\|$	perpendicular bisector of segment $ab$
$\arg(z-a)=\alpha$	ray from $a$ at angle $\alpha$ , endpoint $a$ excluded

Plot reference points first, draw each boundary, decide included/excluded style from equality, shade the correct side/interior/exterior, then intersect all regions.

$|z-(2+i)|<3$ is the open disc centred at $(2,1)$ with radius $3$ . Adding $\arg(z-(2+i))=\pi/4$ restricts to points on the corresponding ray that also lie inside the disc.

For equal distances, do not draw a circle: the locus is a straight perpendicular bisector. Test one simple point to choose a side for an inequality.

$\arg(z-a)$ is undefined at $z=a$ , and is a direction condition rather than distance. De Moivre powers/roots are unrelated and not required.

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—Cambridge International AS & A Level Mathematics (9709)Read

3.9 Complex numbers

Learning objectives

Name every part of a complex number and fix an argument branch

Complex arithmetic is checked by matching real and imaginary parts

Complete real polynomial roots with the conjugate pair

Place $x+iy$ at $(x,y)$ on an Argand diagram

Multiply and divide complex numbers with a polar ledger

Find both exact square roots in Cartesian form

Read complex operations as transformations in the Argand plane

Translate complex conditions into Argand loci and boundary rules

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