Imaginary and Complex Numbers

Imaginary numbers. It is not something that could be easily visualized – even its very name suggests so – but its presence in mathematics has been undeniable. Today, I will give an introduction to imaginary numbers, and some of its applications.

It is known for a fact that all real numbers, when squared up, yield positive numbers. That is why to have a result, a quadratic equation of the form:

$ax^{2} + bx + c = 0$

must satisfy, given that a is non-zero, the following criteria (non-negative discriminant):

$b^{2} - 4ac \geq 0$

Many equations don’t satisfy this condition, among those the seemingly very simple:

$x^{2} + 1 = 0$

In an effort to work around this drawback, a new concept must be introduced to find the roots of these equations – something that yields negative when squared up. Thus, the imaginary unit is born:

$i^{2} = -1$

So why “imaginary”? It’s simply because it’s not real. For every real number, you can always find a way to “visualize” it. A whole number is easy – for example an apple, two apples,… A fraction, or rational number, is harder, but still doable: for a fraction $m/n$ , just split each apple into $n$ equal parts and take $m$ parts. Real number is hardest to imagine, but recalling a basic theorem gives us the way:

For every real number $L$ , there exists a sequence of rational numbers that takes $L$ as its limit.

Thus, we can imagine the apples being split indefinitely, and the result will approach the real number as it goes on. But you cannot imagine $i$ apples – not even close. That’s because imaginary numbers are so different from the real ones that they, at least in a geometrical perspective, belongs to an entirely different dimension:

The complex number plane. The real number axis is horizontal, negative to the left, positive to the right. The imaginary number axis is vertical, negative downwards, positive upwards.

This concept allows mathematicians to discover a new type of number – complex numbers. They’re “complex” because they have two parts – the real part $a$ and the imaginary part $bi$ in the general representation:

$a+bi$

Geometrically, they have two be denoted in a plane (not an axis as for real numbers only), with two components giving it a two-dimensional nature. The most common one is to use the Descartes coordinates above, then a number of the form $a+bi$ would be at the point $(a, b)$ .

Interestingly enough, the need for complex numbers did not arise due to quadratic equations. That’s because if one is only interested in real roots, then equations will real coefficients either have two roots or none. This is easily understandable: if one root of a quadratic equation is complex, the other must also be a complex root, since the sum of two roots is real according to the Vieta formula. This is aided by the fact that the general formula for the roots does not require complex numbers to be feasible. The case is different with cubic functions. Take the monic trinomial below that can be easily transformed to from the general cubic equation $ax^{3} + bx^{2} + cx + d = 0$ via the substitution $x = t - \frac{b}{3a}$ :

$t^3 + pt + q = 0$

Now, the Cardano’s formula for general roots looks like this:

$t_{k} = \alpha_{k} \sqrt[3]{-\frac{q}{2}+\sqrt{\frac{q^{2}}{4}+\frac{p^{3}}{27}}}+\alpha_{k}^{2} \sqrt[3]{-\frac{q}{2}-\sqrt{\frac{q^{2}}{4}+\frac{p^{3}}{27}}}$

with $\alpha_{k}, k \in \{1, 2, 3\}$ being the three cube roots of $1$ , two of which is complex (the other is the trivial $1$ ). So even if the equation has three real roots (for example, $x^3-4x = 0$ with roots $0, 2, -2$ ), you still have to use complex numbers in order to find them using the generalized form. And that’s the true motive behind the first proper use of imaginary and complex numbers.