Imaginary numbers

Imaginary numbers are numbers defined in terms of their relation to $\sqrt{-1}$, which is known as $i$. Some equations which seemed impossible before are now completely doable. For example:

$$x^2 = -4$$

To solve this, you just work out the square root of 16 (4) and make it a coefficient of i:

$$x = \sqrt{4}i = 2i$$

Complex Numbers

Complex numbers are numbers with both a real and imaginary part. They can be squared and manipulated just like any other binomial - just keep in mind that while you are expanding these. They are expressed as:

$$x + yi$$

In this equation:

• The real part is the part without an $i$ coefficient: x.
• The imaginary part is the part with an $i$ coefficient: y.

Handy Expressions

Addition

Addition is the same as ever, the real part is the addition of the real parts while the imaginary part is the addition of the imaginary parts.

$$(a + bi) + (c + di) = (a + c) + (b + d)i$$

Subtraction

Subtraction, similarly, has the real part as the subtraction of the real parts and the imaginary part is the subtraction of the imaginary parts.

$$(a + bi) - (c + di) = (a - c) + (b - d)i$$

Multiplication

$$(a + bi)(c + di) = (ac - bd) + (bc + ad)i$$

Division

$$\frac{a + bi}{c + di} = (\frac{ac + bd}{c^2 + d^2}) + (\frac{bc - ad}{c^2 + d^2})i$$

Complex Conjugates

A complex conjugate to a complex number has the same real part but the opposite imaginary part to that number, as in, if:

$$z = x + yi$$

Then its complex conjugate

$$\overline{z} = x - yi$$

The important equations formed from these properties are that:

$z + \overline{z} = 2x$ $z \times \overline{z} = x^2 + y^2$

Equating real and imaginary parts

If…

$$a + bi = c + di$$

.. then due to the nature of complex numbers it is definite that:

$a = c$ $b = d$