NDA · Chapter

Complex Numbers

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fails when both are negative

Start

Powers of iThe period-4 cycle; reduce the exponent mod 4Locked

Standard form & equality

z = x + iy

, Re and Im; two complex numbers are equal iff both parts are equalLocked

ArithmeticAdd, subtract and multiply in

x + iy

formLocked

Conjugate

\bar z

;

z + \bar z = 2\operatorname{Re} z

; conjugate of a sum or productLocked

DivisionMultiply top and bottom by the conjugate of the bottom; the bottom becomes real, then splitLocked

Purely real / purely imaginarySet the other part to zero;

z = \bar z

vs

z = -\bar z

Locked

Solving for zPut

z = x + iy

and equate parts; equations in

z

and

\bar z

Locked

Square root of a complex numberSolve

(x + iy)^2 = a + ib

Locked

Argand plane $z$ as a point and as a vectorLocked
Modulus as distance $|z|$ , and $|z_1 - z_2|$ is the distance between the pointsLocked
Modulus rules $|z_1z_2| = |z_1||z_2|$ ,
Triangle inequality & boundsMax and min of $|z_1 \pm z_2|$ , and when a bound is reachedLocked
ArgumentPrincipal argument by quadrant, not by $\tan^{-1}(b/a)$ aloneLocked
Polar & Euler form $r(\cos\theta + i\sin\theta) = re^{i\theta}$ Locked
Multiply/divide in polarModuli multiply, arguments add; argument rulesLocked
De Moivre's theorem $(\cos\theta + i\sin\theta)^n$ ; reduce the angle; rewrite $\sin\theta + i\cos\theta$
RotationMultiplying by $e^{i\alpha}$ rotates; rotating about a vertex; triangle problemsLocked

Quadratics with complex rootsConjugate-pair roots (real coefficients only); sum and productLocked
Cube roots of unity $\omega^3 = 1$ , $1 + \omega + \omega^2 = 0$ , $\omega^2 = \bar\omega$ Locked
Using ωReducing powers mod 3, factorising expressions, related rootsLocked
nth roots of unitySum = 0, product, regular-polygon geometryLocked
Polynomial at a complex xUse the minimal quadratic instead of substituting directly; powers of the rootsLocked

Circle locus $|z - a| = r$ : centre $a$ , radius $r$ Locked
Perpendicular bisector $|z - a| = |z - b|$ is a lineLocked
Loci from algebraic conditionsRe or Im of a quotient $= 0$ ; $|z - a| = k|z - b|$ (a circle for
Argument loci $\arg(z - a) = \theta$ is a ray; a fixed argument of a quotient is an arcLocked
Conic loci $|z - a| \pm |z - b| = k$ as an ellipse or hyperbolaLocked
Greatest/least modulusOn a disc: $|a| \pm r$ ; check the extreme point lies in the regionLocked
Regions & areasInequalities as shaded regions; area; lattice pointsLocked

|z_1/z_2|

,

z\bar z = |z|^2

,

|\bar z| = |z|

Locked

first

Locked

k \ne 1

)

Locked