Complex number, an introduction

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I don’t remember where I first saw this way of introducing complex numbers, but I have used it ever since.

To start with, if you don’t know anything about complex numbers, then that is fine, and if you do, pretend you don’t.

Let us start with a number line from zero and up. So, we only have 0 and the positive real numbers. Let us indicate the number one by an arrow from the origin to the point at one.

Next, we will define what it means by multiplying by minus one. We define

·(–1) ≡ rotate by 180° in the counterclockwise direction.

So, to accommodate the result of 1·(–1) will need to extend the number line to the left.

We have that 1 times anything is the anything, so we say that 1·(–1)= –1.

But what is so special with 180º? Let us define a new operation, times question mark.

·? ≡ rotate by 90° in the counterclockwise direction.

So, what would 1·? mean? Try to visualize it yourself before continuing.

To be able to visualise this we need to get outside the number line. If we rotate the arrow corresponding to the number one by 90º counterclockwise.

We end up above the real number line. Let us call the new point ? because 1·?=?Now, how about multiplying by ? again? Where would we be now?

I hope you got –1. If we now multiply this by “?” again we get to –1·? = –?. Then once more to get (–?)·? = 1.

So, given this information, what would the square root of –1 be?

Hopefully you have found the answer “?”.

The imaginary unit

So, this new number “?” is the famous number i, and we have that i²= –1. I would argue that the name “the imaginary unit” is a bit of a bad choice. It is not more imaginary than –3, square root of 2 or 3/5.

Complex numbers

Say we multiply by 3 instead of 1. We will get a result three times further away from the origin. It’s reasonable to assume the same would happen if we multiply by 3i, except for that we get a 90-degree rotation first. Ok, so let us define a complex number as a combination of a scale factor, like 3, and an angle, like 90. We may then define a complex number as a pair of numbers as

r∠θ

Here r is the distance to the origin, the magnitude of the number, often denoted the modulus, and θ is the angle, often denoted the argument (why so many silly names when it comes to complex numbers?).

We have, for example,

1=1∠0º,
i=1∠90º,
–1=1∠180º,
3=3∠0º,
3i=3∠90º

These numbers can both be seen as places in a coordinate system, where the modulus is the distance from the origin and the argument is the angle from the positive real (x) axis, but they can also be seen as a stretching and rotation under multiplication.

Now can now define multiplaction of two compex numbers as

r∠α·s∠β = rs∠(α+β)

I.e. a multiplication of their modulus and addition of their arguments. We call this the polar form.

We may also split our complex number into a real part and an imaginary part by simple trig. We get the real part a=r cos θ, and an imaginary part b=r sin θ.

From this we get the classical definition of a complex number in the form a+bi, where a and b are real numbers. We often denote complex numbers with the letters z or w, so, for example, z=a+bi. The number is now written in rectangular form.

So, we have:

Im(z) = b: The number b is called the imaginary part of z.
Re(z) = a: The real part of z.
Arg(z): The angle from the positive real axis to the line from the origin to the number is called the argument (a really strange name). The angle is positive in the counterclockwise direction, and negative in the clockwise direction.
|z|: The distance from the origin to the number is called the modulus (another strange name). This is denoted with the absolute value symbol, |z|.
We can see the number a+bi as two lined perpendicular to each other. This means that we can use the Pythagorean theorem to find the modulus.

$\displaystyle \left| z \right|=\sqrt{{{{a}^{2}}+{{b}^{2}}}}$

A diagram, like the above, with a real axis and an imaginary axis is called an argand diagram.

…and other angles

So, what’s so special with 90°? We can easily multiply our complex numbers to get other angles. Say we would like to rotate by 45°. We may then multiply by the complex number

$\displaystyle \sqrt{2}\angle 45{}^\circ =1+i$

The angle θ=45°. Now what happens if we multiply the number by itself? I.e., what is (1+i)(1+i)? Now, let us just proceed as if i is just any ordinary variable, with the exception that i²= –1. We get 1·1+1·i+i·1+i·i=1+2i-1=2i. So, we have indeed rotated by 45°+45°=90°.

This is the same result as using the first rule for multiplication of complex numbers

$\displaystyle \sqrt{2}\angle 45{}^\circ \cdot \sqrt{2}\angle 45{}^\circ =\sqrt{2}\sqrt{2}\angle \left( {45{}^\circ +45{}^\circ } \right)=2\angle 90{}^\circ =2i$

Compared to 1+i, the length, the modulus, has changed. We could fix that by starting with a number with the modulus 1.

Since

$\displaystyle \left| {1+i} \right|=\sqrt{{{{1}^{2}}+{{1}^{2}}}}=\sqrt{2}$

we have that the number

$\displaystyle \frac{1}{{\sqrt{2}}}(1+i)=\frac{1}{{\sqrt{2}}}+i\frac{1}{{\sqrt{2}}}$

will have the modulus 1. And, indeed if you square that number you will get

$\displaystyle {{\left( {\frac{1}{{\sqrt{2}}}(1+i)} \right)}^{2}}=\frac{1}{2}2i=i$

A number with the modulus 1.

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Last modified: Apr 25, 2026 @ 19:30