Complex numbers

Complex numbers extend the real numbers. With real numbers we can do most of the usual operations, such as addition, subtraction, multiplication and division (except division by zero). We can also take roots of real numbers, but only of non-negative ones. That leaves a gap, for example when we solve a quadratic equation and the discriminant comes out negative. Complex numbers fill that gap.

What is a complex number

Complex numbers differ from ordinary numbers mainly in having two parts – a real part and an imaginary part. A complex number is an ordered pair of numbers (x, y), where x is the real part and y is the imaginary part. If the real part is zero, we call it a purely imaginary number. The numbers x and y themselves are real.

The set of complex numbers is written with a capital C in blackboard bold: ℂ.

Equality of complex numbers: unlike ordinary numbers, complex numbers have two components. So for two complex numbers to be equal, they must be equal in both components. These complex numbers are therefore all different from one another: (2, 3) ≠ (0, 3) ≠ (2, 0). The complex numbers z₁ = (1, 2) and z₂ = (1, 2) are equal: z₁ = z₂.

Cartesian form and the imaginary unit

Complex numbers are more often written in Cartesian form, which looks like this. The complex number (x, y) has Cartesian form x + yi, where i is called the imaginary unit. For the square of the imaginary unit there is a very important relationship:

$$i^2=-1$$

We will use this equation often from now on, together with other powers of the imaginary unit. Higher powers can be worked out in the usual way. For example, to simplify i³, we can use the laws of indices to split it into i² · i. We already know that i² is equal to minus one. The remaining factor cannot be broken down any further, so we get i³ = −i.

$$i^3=\underset{-1}{\underbrace{i^2}}\cdot \underset{i}{\underbrace{i}}=-i$$

Similarly for i⁴. We can split it into i² · i², where i² = −1, so we get −1 · − 1 = 1.

$$i^4=\underset{-1}{\underbrace{i^2}}\cdot \underset{-1}{\underbrace{i^2}}=-1\cdot -1=1$$

So i⁴ = 1. We can use this fact when working out even higher powers of the imaginary unit. For example, to work out i⁷, we can split it into i⁴ · i³. We know that i⁴ = 1, so we get 1 · i³. And we know that i³ = −i. The result is 1 · (−i) = −i.

Addition and multiplication

We can add and multiply complex numbers. To add two complex numbers z₁ and z₂, we simply add the real parts and the imaginary parts separately:

$$z_1+z_2=(x_1+y_{1}i)+(x_2+y_{2}i)=(x_1+x_2)+(y_1+y_2)i$$

Example: z₁ = 3 + 7i and z₂ = 5 + 8i. The sum looks like this: z₁ + z₂ = (3 + 5) + (7 + 8)i = 8 + 15i.

The product of two complex numbers is a little more complicated, but it can be worked out by expanding brackets in the usual way. The basic formula looks like this:

$$z_1\cdot z_2=(x_1+y_{1}i)\cdot (x_2+y_{2}i)=(x_1{x}_2-y_1{y}_2)+(x_1{y}_2+x_2{y}_1)i$$

How did we get there? Let’s multiply z₁ · z₂ just as we would expand a pair of brackets. We get:

$$z_1\cdot z_2=x_1{x}_2+x_1{y}_{2}i+x_2{y}_{1}i+y_1{y}_2{i}^2$$

Since i² = −1, after simplifying the last term we get:

$$z_1\cdot z_2=x_1{x}_2+x_1{y}_{2}i+x_2{y}_{1}i-y_1{y}_2$$

Now we just collect the terms without the imaginary unit and the terms with it:

$$z_1\cdot z_2=(x_1{x}_2-y_1{y}_2)+(x_1{y}_2+x_2{y}_1)i$$

Example: let’s multiply 5 + 6i and 4 + 7i. We get:

$$(5+6i)\cdot (4+7i)=20+35i+24i+42i^2=20+59i-42=-22+59i$$

Negative, reciprocal and complex conjugate

• The negative of a complex number x + yi is −x − yi. As with real numbers, we get the negative by multiplying the complex number by minus one. Example: the negative of 2 + 7i is −2 − 7i. The negative of −5 + 8i is 5 − 8i, and so on.

• The reciprocal of a complex number x + yi is $\frac{1}{x+yi}$. The reciprocal of 4 − 2i is therefore $\frac{1}{4-2i}$.

• The complex conjugate of x + yi is x − yi. It is usually written with a bar, like this: $\bar{z}$, or sometimes with an asterisk, z*. This is the complex conjugate of the complex number z. The complex conjugate of 2 + 9i is 2 − 9i.

Subtraction and division

Now that we have defined the negative and the reciprocal, we can define subtraction and division too. To subtract two complex numbers, z₁ − z₂, we add the negative of z₂ to z₁. In practice this gives a simple formula:

$$z_1-z_2=(x_1+y_{1}i)-(x_2+y_{2}i)=(x_1-x_2)+(y_1-y_2)i$$

Similarly, we turn division z₁ / z₂ into multiplication by multiplying z₁ · z₂′, where z₂′ is the reciprocal of z₂.

Modulus

We work out the modulus of a complex number z using this formula:

$$|z|=\sqrt{x^2+y^2}=\sqrt{z\cdot \bar{z}}$$

(In the square root, the second z is the complex conjugate of z.) The meaning of this formula is easy to see in the geometric representation of complex numbers.

Every complex number whose modulus is equal to one is called a unit complex number. Examples of unit complex numbers: 1, i, $\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i$.

Properties of the modulus:

• The modulus of a complex number is a real number.
• |z| ≥ 0
• |z| = |−z| = |z*|, where z* is the complex conjugate.