Complex numbers explained: from i to exponential form

Complex numbers are one of those topics that frighten you by their name before they do by their substance. “Complex” sounds difficult, and then there is that $i$ which “is the square root of minus one”, a sentence many students are told once and never have explained. The result: most learn to do the sums mechanically, but if you ask them what a complex number represents they freeze.

That is a pity, because complex numbers are actually one of the most elegant topics you meet between the final year of high school and your first university exams, and once you have the right idea they become almost fun. They turn up everywhere: in the study of differential equations, in alternating-current circuits, in signal analysis, in quantum physics. In this article I start from where $i$ really comes from, go through the three forms in which a complex number can be written, arrive at Euler’s formula (the one “most beautiful in mathematics”) and finish with the mistakes I see most often when I prepare students for the admission test or the exam.

Where i really comes from

The story usually told, ”$i$ is the square root of $-1$”, is true but misleading, because it makes complex numbers look like a trick invented to make the sums work. The reality is more interesting.

Complex numbers were not born to solve $x^2=-1$ (who cares about an equation that “has no solutions”?). They were born in the sixteenth century, in Italy, while mathematicians such as Cardano and Bombelli were trying to solve cubic equations. The surprising fact is this: to find real solutions of certain cubics, the solving formula forced you to pass through square roots of negative numbers that then, as if by magic, cancelled out and left a correct real result. They were “imaginary” numbers that served as an intermediate step to reach a concrete answer.

So the working definition is simply:

From here everything is built. A complex number is something of the form:

where $a$ and $b$ are real numbers. $a$ is called the real part ($\text{Re}(z)$), $b$ is the imaginary part ($\text{Im}(z)$; careful: it is $b$, not $bi$). If $b=0$ you have an ordinary real number: complex numbers do not replace the reals, they contain them. If $a=0$ you have a purely imaginary number.

The idea that changes everything: the complex plane

The leap in quality comes when you stop thinking of $z=a+bi$ as a formula and start thinking of it as a point (or a vector) on a plane.

Imagine a Cartesian plane where the horizontal axis is the real part and the vertical axis is the imaginary part. The number $z=3+2i$ is simply the point with coordinates $(3,2)$. This is the Argand-Gauss plane (the complex plane), and it completely changes the way you see things: complex numbers are not abstract entities, they are points of the plane, and all the operations have a geometric meaning.

• The sum of two complex numbers is the sum of the vectors: head to tail, exactly as in physics.

• The modulus:

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

  is the distance of the point from the origin (it is Pythagoras’ theorem, nothing more).

• The conjugate $\bar{z}=a-bi$ is the point reflected in the real axis.

Keep this picture in mind, because it is the key to understanding the trigonometric form and why multiplication does what it does.

Algebraic form: everyday calculations

The form $z=a+bi$ is called algebraic and it is the one you use to do sums, differences and products “by hand”. The rules are those of polynomials, with the one addition that every time you meet $i^2$ you replace it with $-1$.

Sum: you add real parts to real parts, imaginary parts to imaginary parts.

Product: you multiply term by term like two binomials.

Note the step: $-10i^2$ becomes $+10$ because $i^2=-1$. This is where people who go on autopilot fall.

Division is the only point that deserves a trick. To divide

you multiply top and bottom by the conjugate of the denominator: this way the denominator becomes real (because $w\cdot \bar{w}=|w{|}^2$) and you can separate real and imaginary parts again.

This “rationalising with the conjugate” is exactly the same mechanism you use to rationalise

: same idea, different context.

Trigonometric form: when geometry comes into play

If a complex number is a point of the plane, I can pin it down in two ways: with the Cartesian coordinates $(a,b)$, or by saying how far it is from the origin and which direction it points in. The second way gives the trigonometric form:

where

is the modulus (the distance from the origin) and $\theta$ is the argument (the angle the vector makes with the positive real axis). The link with the algebraic form is direct: $a=\rho \cos \theta$ and $b=\rho \sin \theta$.

Why complicate your life with sines and cosines? Because the trigonometric form makes multiplication trivial and geometrically transparent. When you multiply two complex numbers:

In words: the moduli multiply, the arguments add. Multiplying by a complex number therefore means scaling and rotating. Multiplying by $i$ (which has modulus 1 and argument 90°) is exactly a rotation by 90° anticlockwise. That is why applying $i$ twice ($i^2=-1$) rotates you by 180° and you end up on the negative side of the real axis. The “mysterious” definition $i^2=-1$ becomes something you see.

From here comes De Moivre’s formula for powers:

which makes computing powers immediate and, a crucial fact for tests and exams, lets you find all the nth roots of a non-zero complex number. A non-zero complex number has exactly $n$ nth roots, placed at the vertices of a regular polygon on the plane: this is one of those things that sounds abstract in words and becomes obvious with a drawing.

Exponential form and Euler’s formula

Here comes the most beautiful part. Euler’s formula says:

At first sight it looks absurd (what is an exponential doing with an imaginary exponent?), but it makes perfect sense and is proved by comparing the series expansions of $e^x$, $\cos x$ and $\sin x$ (which you will do in Calculus 1). The practical consequence is that the trigonometric form can be written very compactly:

This is the exponential form, and it is the favourite of engineers and physicists because it makes everything clean. Multiplication becomes the usual rule for exponents:

“The moduli multiply, the arguments add” falls out by itself.

And as a special case, taking $\theta =\pi$:

Euler’s identity, which links in five symbols the most important numbers in mathematics ($e$, $i$, $\pi$, $1$, $0$). It is often called the most beautiful formula in mathematics, and when you understand where it comes from (it is not magic, it is Euler’s formula evaluated at one point) you really appreciate it.

To sum up, the three forms are the same number written in three different languages:

1. Algebraic $a+bi$ → for sums, differences and for “reading off” the real and imaginary parts.
2. Trigonometric $\rho (\cos \theta +i\sin \theta )$ → for understanding the geometry and for roots.
3. Exponential $\rho e^{i\theta }$ → for multiplications, powers and for physics/engineering.

Being able to move quickly from one to another is half the job.

Where they are really needed (and why to study them seriously)

I understand the scepticism: “when will I ever use the square root of minus one?” The answer is: constantly, if you carry on in science.

In physics and electrical engineering alternating-current circuits are analysed with complex numbers (“phasors”): impedances, phase shifts, everything becomes much simpler than with sines and cosines by hand. In signal analysis the Fourier transform, the heart of MP3, JPEG and digital communications, is built on complex exponentials. Linear differential equations, which describe oscillations and waves, are solved by looking for solutions of the form $e^{\lambda t}$ with $\lambda$ complex. And in pure mathematics complex numbers are what makes the fundamental theorem of algebra true: every polynomial of degree $n$ has exactly $n$ roots (counted with multiplicity) if you look for them among the complex numbers.

In other words: complex numbers are not a whim, they are the natural language of everything that oscillates, rotates or propagates.

The typical mistakes I see

1. Confusing the imaginary part with an imaginary number. In $z=3+2i$ the imaginary part is $2$, not $2i$. It is $\text{Im}(z)=b$, a real number. It seems a detail, but getting it wrong ruins the calculations of modulus and argument.

2. Forgetting to substitute $i^2=-1$. In products, every $i^2$ must be turned into $-1$. Leaving it there or treating it like any other letter is the number one mistake in multiplications.

3. Dividing without using the conjugate. Division between complex numbers requires rationalising with the conjugate of the denominator. Anyone who tries to “divide part by part” gets meaningless results.

4. Getting the quadrant of the argument wrong. The angle $\theta$ is not found with $\arctan (b/a)$ alone: the calculator does not know which quadrant you are in. You must always look at the signs of $a$ and $b$ and adjust $\theta$ accordingly, otherwise you end up with the opposite angle. Drawing the point on the plane fixes 90% of these mistakes.

5. Thinking there is only one root. The nth root of a complex number has $n$ values, not one. “The” cube root of 8 in the complex numbers is three numbers, not just the real 2. Forgetting the other two is a classic at tests and exams.

6. Treating $|z|$ as a signed value. The modulus is a distance: it is always $\ge 0$. There is no negative modulus, just as there is no negative distance.

7. Studying the three forms as three separate topics. This is the costliest strategic mistake. Algebraic, trigonometric and exponential are the same object: if you keep them in watertight compartments you double the work and miss the sense of it all. They must be learned as translations of one another.

How we work in lessons

On complex numbers I always start from the drawing, not the formula. The first thing I do is put a point on the Argand-Gauss plane and show that $a+bi$ “is there”: from that moment modulus, argument, conjugate and multiplication stop being rules to memorise and become things you can see. When a student sees that multiplying by $i$ is rotating by 90°, De Moivre’s formula no longer frightens them: they have understood it, not just learned it.

From there we build fluency in moving between the three forms, because that is what the admission test and the exam really ask: they give you a number in one form and you must be able to rewrite it in another, calculate powers and roots, and recognise when the exponential is convenient. I do a lot of exercises out loud together with the student, always asking the why of every step: in the Calculus oral exam what counts is how you get there, not just the result.

We work online with a graphics tablet and a shared screen, writing the steps by hand as in the test and drawing each number on the plane. The 10-hour package comfortably covers a solid block of algebra and analysis (complex numbers, polynomials and the link to differential equations), while for a specific doubt or a revision before a test a single lesson is fine; the price overview is always on the pricing page.

If you want to understand where you stand, I suggest a free 30-minute introductory call: you show me a couple of exercises that do not work for you, we work out together whether the problem is the geometric idea or the calculation, and I tell you honestly how I would set up the work. No hard sell.

For the full picture of how I work on maths there is the maths page; if instead you are preparing an admission test for engineering or a science degree, complex numbers often appear in the algebra section. And if this is the right moment to put your calculus in order, the natural steps next to complex numbers are limits, derivatives and integrals: it is all the same story, told one chapter at a time.

To practise the topic in exam style there is the Calculus 1 practice test; if you are in high school, complex numbers are in the Year 4 test on complex numbers and space. The link with engineering is in the automatic control exam, where poles and zeros are complex numbers. And if you study in English, for example with the IB, the choice of track and the bilingual glossary are in IB Maths AA or AI.