Without resorting to QM how do I understand the real world uses of imaginary numbers

Question

I am trying to understand the real world uses of imaginary numbers. I have been told many times that it is not just a convenient tool mathematicians invented but has its places in some fundamental part of physical reality.

But when I research its real life application, without resorting to quantum mechanics, I often just find its usage as something along the lines like, imaginary numbers are the ideal way to analyze the natural swinging motion of objects such as pendulum, or imaginary numbers can be used to describe opposition to changes in current within an electrical circuit.

I can't find the further explanation for those simple words. So image I only have the knowledge of high school of physics, can someone explain the the real world uses of imaginary numbers to me ? Thanks!

Answer 1

We use imaginary numbers for three reasons :

• The mathematics forces us to sometimes.
• They are convenient mathematically sometimes.
• After some practice you forget they're there

What we do not do is use imaginary numbers to mean anything in the real world.

It is important to note that imaginary numbers are a necessary part of the set of numbers in mathematics. If you have not done much mathematics at this level you will not understand that they turn up everywhere because in mathematics everything is connected to everything else. The set of numbers we need is not complete without complex numbers.

So even if you try and avoid imaginary numbers, quite often advanced mathematics will drag you back to them.

In the case of quantum mechanics you can, if you want, construct the entire thing without imaginary numbers. But it feels very odd doing it that way if you are used to advanced mathematics. It is contrived. Using imaginary numbers turns out to be "natural" when you create the formalism of quantum theory.

But when you make a calculation of something that's real (a length, an energy level) because of the way the formalism is defined you will find that the imaginary numbers happily go away and your final result is real.

But the mathematics requires you to do "the stuff in the middle" with imaginary numbers or to use very artificial ways to avoid them. The mathematics "flows naturally" with imaginary numbers in quantum theory. After a while you hardly notice you're using imaginary numbers, just symbols. Imaginary numbers kind of scare beginners, but they're just numbers and blend into the mathematics after you get used to it.

You can think of imaginary numbers as giving you the entire set of all possible numbers. Every mathematical operation has a version than can transform one complex number and convert it to another one. So you cannot get the square root one minus one as a real number - it's an imaginary number. You would be stuck here if you did not use imaginary numbers, but with complex numbers you can keep going and do all sorts of mathematical operations using complex numbers and get a final result.

Answer 2

From the point of view of physics, it is merely a convenient tool mathematicians invented. But that's true of all mathematics: it's extremely useful for modeling physical phenomena, but the phenomena are fundamental. Mathematics is not fundamental.

Electrical engineers are perhaps the most common users of complex numbers to model physical reality. Start with Ohm's Law, $V=IR$. That works for resistors, which dissipate energy as heat. But there are also capacitors, which store energy in electric fields, and inductors (coils), which store energy in magnetic fields. We may generalize resistance $R$ as impedance $Z$. $Z=1/i\omega C$ for a capacitor, and $Z=i\omega L$ for and inductor ($\omega$ is $2\pi f$, where f is the frequency of a sinusoidal signal component). This turns out to be an extremely useful tool for linear circuit analysis.

A physicist will see this as a specialized application of the technique of using Fourier analysis to transform differential equations into algebra, yielding insights into the phenomena, and making solutions to the equations easier to find.

From the Fundamental Theorem of Algebra, you might (correctly) expect that imaginary numbers will often be used when using algebra or math related to it to model physical phenomena.

Answer 3

It is more elegant (i.e. the math ends up being much simpler) to describe sinusoidal oscillations (these may be physical oscillations of a pendulum or mass on a spring, or temporal or spatial oscillations of the direction or amplitude of a vector field such as the electromagnetic field) in terms of complex numbers rather than sines and cosines. This is because in the former you case you need only keep track of an amplitude and phase and addition and multiplication are simple addition and multiplication of complex numbers. If instead you want to use sines and cosines (i.e. avoid complex numbers) you will find yourself having to memorize and apply numerous trig identities and your algebraic expressions will quickly blow up in complexity. It turns out that differential equations are similarly simplified when moving from sinusoidal to complex mathematical approaches.

Can all physics (including QM) be done without complex numbers? yes. Would the mathematics (addition and multiplication of signals and differential equations) get painful fast because trigonometry is ugly? yes.

Answer 4

Imaginary numbers are associated with any transformation that when applied twice, reverses the direction of everything. Or alterrnatively, a transformation that swaps two values and changes the sign of one of them. There are lots of these, many of them so obvious we don't even notice what they're doing, or that there is anything strange about it.

The most familiar is probably rotation. Rotating a plane shape $90^\circ$ seems like a trivial operation. Do it twice, and the resulting $180^\circ$ rotation negates both coordinates. It is the same as multiplying by $-1$.

With oscillations, the process is differentiation. Differentiate the position to get the velocity, differentiate the velocity to get the acceleration, and this is proportional to the position, negated. The force is opposite the displacement, pushing it back towards the centre. $\frac{d}{dt}\frac{d}{dt}x=-\alpha x$. And when moving in a circle, the velocity is at right-angles to the position, and the acceleration is at right-angles to the velocity, pointing back to the centre. So again, we have an operation applied twice giving the same result as multiplying by a negative number.

It's just part of the algebra of 2D space. (Or more generally, of linear algebra, of which multidimensional space/geometry is the most obvious physical example to us.) We can make up any 2D linear transformation from linear combinations of four basic transformation types:

Scaling:$\pmatrix{\alpha & 0 \\ 0 & \alpha}$

Reflecting: $\pmatrix{\beta & 0 \\ 0 & -\beta}$, $\pmatrix{0 & \gamma \\ \gamma & 0}$

Rotating $90^\circ$: $\pmatrix{0 & \delta \\ -\delta & 0}$

The 'numbers' we identified first in geometry were the scaling transformations. We got them from Euclidean geometry, as the 'lengths' of lines, but even given the massive clue this provided, we threw away all the information about a line's direction and concentrated on its magnitude. We only looked at the scalar part, and called these 'the numbers'. But they are really only part of a bigger structure of 'geometrical numbers' (mathematicians call it a Clifford algebra), and the other parts sometimes play a role in physics too.

Particularly interesting are combinations of the first and fourth matrix above.

$\pmatrix{\alpha & \delta \\ -\delta & \alpha}$ which we write as $\alpha+\delta i$.

These transformations scale and rotate through any angle. And any scale-rotation followed by another scale-rotation yields a scale-rotation. They form a closed algebra, another complete number system inside the geometric numbers. We did not, at first, recognise them as such. We only knew about the scalars, and these strange new things didn't quite follow the same rules, so at first we questioned whether they were really 'numbers'. But the maths turned out to be so elegant and so useful that we eventually came to accept them as a sort of convenient fiction. We still didn't believe in their physical reality, though, they were seen as just an elegant calculational tool. It took a very long time to realise they were actually an old friend in disguise - that they are those bits of plane geometry we threw away when we dropped directions to concentrate on just the magnitudes of lengths and lines as our 'numbers'.

(Those other two matrices above that I didn't mention have applications too - that path leads to the world of vectors, which are also 'numbers'. But that's another story...)

There are lots of applications of linear transformations that negate when applied twice. Sometimes, we use complex numbers to represent them, but very often we don't. We treat the magnitudes and directions separately, and combine them with some different machinery (e.g. using coordinates and matrices as I did above). Usually when we do that, we don't even notice them. So we can continue to believe that the world is made up only of 'real' numbers, and the imaginary ones are truly imaginary, or tucked away out of sight in some Platonic ideal realm of mathematics, that we can only perceive indirectly by the shadows they cast.

Complex numbers (and their generalisations to higher dimensions) are involved in anything that uses the geometry of more than one dimension, which is pretty much everything. But because we can always take out the directional bits of geometry and treat them separately, it is completely optional whether to use them and so we usually don't notice they're there.

Answer 5

Arguably, all of physics is convenient mathematical tools. Are negative numbers real? Is zero real? Do numbers with an infinite number of decimals exist since we could never measure all of those? We could do mathematics without ever having to define negative numbers but it turns out it is very convenient. They came from the need to solve equations of the form $x+c=0$ and by formalizing this concept it turns out negative numbers have many uses of their own. Similarly, complex numbers came from the need to solve equations of the form $x^2+c=0$ and they turned out to be useful on their own$^\dagger$. Two important reasons that complex numbers matter are

1. Complex numbers allow for more general solutions to equations. Any polynomial with a term $x^n$ has at least $n$ (complex) roots$^{\dagger\dagger}$. Remember that a quadratic polynomial of the form $ax^2+bx+c=0$ has between 0 and 2 solutions, depending on whether $D=b^2-4ac$ is larger than, equal to or smaller than zero. Using complex numbers there are always 2$^{\dagger\dagger}$. This generalization extends beyond just polynomial equations. Let's say you want a function that solves the following equation: $$\frac{d^2f}{dx^2}=kf(x)$$ It turns out that any function of the form $f(x)=c_1e^{\sqrt{k}\cdot x}+c_2e^{-\sqrt{k}\cdot x}$ solves this. If instead we wanted to find a solution to $$\frac{d^2f}{dx^2}=-kf(x)$$ we would find $f(x)=c_1\sin(\sqrt{k}\cdot x)+c_2\cos(\sqrt{k}\cdot x)$. These solutions look completely different. But, if we allow for complex numbers we can use Euler's formula, $e^{ix}=\cos(x)+i\sin(x)$, and it turns out these completely different solutions are actually the same! Complex numbers simplify math and by doing so simplify physics.

2. Complex numbers are very good at representing 2D rotations. That is why they often pop up in areas that are even vaguely connected to rotations. Waves that propagate can be expressed as sines, but also using rotating complex numbers (see point 1). Because quantum mechanics is all about waves, complex numbers play an important role there. You could write the Schr"odinger equation using only real numbers, but that would only obscure the fact that solutions to the Schr"odinger equation really like to do one thing: rotate.

$\dagger$ Actually complex numbers were discovered when studying the roots of cubic polynomials, see this.

$\dagger\dagger$ You can write a quadratic polynomial in the form $P(x)=(x-r_1)(x-r_2)$ which makes it clear the roots occur at $r_1,r_2$. We say the multiplicity of this polynomial is 2 because there are 2 terms. When for example $r_1=r_2$ the polynomial becomes $P(x)=(x-r_1)^2$ which has only 1 root but we still say the multiplicity is 2. So solutions to a quadratic polynomial can have less than 2 roots but their multiplicity is always 2.