- 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.

Wrong.

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+bi) + (c+di) = (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+bi) = ka+kbi

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 numberz=a+bi, the conjugate is given asz* =a-bi

The complex conjugate is given an asterisk.

For any complex numberz, the product ofzandz* is a real number