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  • A-Level Further Maths

Complex Numbers, Argand Diagrams & Loci

We know from single maths that if the discriminant of a quadratic equation is positive, it has two solutions, if the discriminant = 0, there is one repeated root, and if it is negative there are no roots.


If the discriminant is negative, the quadratic has two imaginary roots, given in terms of i, the square root of -1:

i = √-1

A complex number is a sum of a real and an imaginary number, given in the form a + b i, where a is the real number and b the coefficient of i. Often, complex numbers are represented by the letter z.

Adding and Subtracting Complex Numbers

To add or subtract complex numbers, treat the real and imaginary terms separately:

(a+b i) + (c+d i) = (a+c) + (b+d)i

A few examples:

  • 3 + (2+4i) = (3+2) + (4i) = 5+4i

  • (1+2i) - 4i = (1) + (2-4)i = 1-2i

  • (2-2i) + (6+i) = (2+6) + (-2+1)i = 8-i

Multiplying Complex Numbers

To multiply an imaginary number by a real constant, just expand the brackets:

k(a+b i) = ka+kb i

For example:

  • 3(2+4i) = 3(2) + 3(4i) = 6+12i

To multiply two imaginary numbers, expand like any quadratic, using the fact that:

i² = -1

For example:

  • (2+3i)(3+4i) = 6 + 8i +9i +12i² = 6 + 17i - 12 = -6+17i

Complex Conjugation

Just like conjugate pairs when rationalising a denominator, the conjugate of a complex number has the opposite sign:

For any complex number z = a +bi, the conjugate is given as z* = a -bi

The complex conjugate is given an asterisk.

For any complex number z, the product of z and z* is a real number