complex numbers

imaginary number

a number that can't exist, usually in the form ki where k is a real number

complex number

a number with real and imaginary parts, usually in the form z = a + bi where a is real

i

√-1

surds in terms of i

√-x = i√x

complex conjugate

when a complex number has the sign changed of the imaginary part ie

where z = a + bi, z* = a - bi and vice versa

realising complex denominators

a complex denominator is realised much like surds using its complex conjugate, and the result is often simlified to x/c ± (y/c)i

polynomials with complex roots

if polynomials have complex roots, they always occur as complex conjugate pairs

complex quadratic equations

for quadratic equations with no real roots, roots occur as complex conjugate pairs

complex cubic equations

cubic equations can have one real root and a complex conjugate pair

complex quartic equations

quartic equations can have a pair of real roots and a complex conjugate pair, or two complex conjugate pairs

finding complex roots of cubic equations

-given a real root c, find (z ± c) that equals 0

-divide the cubic equation by it to produce a quadratic equation

-solve the quadratic equation

finding complex roots of quartic equations

-given a complex root (a ± bi), the other root is equal to its complex conjugate

-perform (z - (a + bi))(z - (a - bi)) to find a quadratic factor of the quartic

-find the second quadratic factor either by long division or by observation of the coefficients

-find the roots of the second quadratic factor