Why do complex number seem to be so helpful in real-world problems?

Question

Complex numbers are often used in Physics especially in Electrical Circuits to analyze them as they are easy to move around like phasors. They make the processes easy but it seems kind of amusing to use something which has no other real world analogous meaning to my knowledge being used to solve the most practical real world problems.

What other method were used prior to having developed complex numbers and why were they replaced? For example, can every problem where we use complex numbers also be done using other techniques such as matrices, how did the insight come to use such an obscure entity, or did doing the operations just seem easy with it?

Answer 1

There is a natural sequence in mathematical reasoning, as follows:

1. start with positive integers
2. invent the notion of addition
3. invent subtraction as the reverse of addition
4. realise about here, or perhaps earlier, that zero is also a useful number
5. after playing with subtraction for a while, invent negative integers
6. invent multiplication as repeated addition, or as a good way to find the number of items in a simple rectangular array of items or in a collection of bags
7. invent division as the reverse of multiplication
8. invent rational numbers because they come up in division operations
9. start to explore equations such as $x \times x = 2$ and realise that one also has numbers "in between" the rational numbers which can't be expressed as the ratio of finite integers, so they are called irrational
10. also explore equations such as $x \times x = -2$ and realise that this can be written $x \times x = 2 \times (-1)$ so $x = \sqrt{2} \times \sqrt{-1}$. After playing with this for a while, invent the numbers we now call imaginary numbers, and by adding them to real numbers invent complex numbers.

So far the sequence of ideas is very natural. We just considered the four basic arithmetic operations (add, subtract, multiply, divide) and we were led very naturally to the complex numbers. What happens next is rather interesting. You can now write any equation you like, using just those four operations, and to solve the equation you don't need any new types of numbers. Complex numbers are sufficient. Thus there is a certain sense of completeness once one arrives at complex numbers. Also, as long as the natural world is described by mathematics then one can expect that these numbers are going to crop up all over the place in the analysis of situations in the natural world. As indeed they do---they are highly important and much used in almost all areas of science, not just in circuit theory. For example, anything involving a wave form can be conveniently expressed using $\exp(i(k x - \omega t))$ where $i = \sqrt{-1}$. It doesn't matter what type of wave it is. It could be water wave or a sound wave or a brain wave or a quantum wave or whatever.

Finally, a comment on the way mathematics goes next, after complex numbers. The next step is to introduce the idea of a new type of operation which is like multiplication but is not commutative. This comes up in matrix analysis and also in sequences of actions where the net result depends on the order in which the actions occur. An example is rotations about different axes in three dimensions. One can then invent further things such as vectors, and others which have such names as quaternions, spinors, fields, etc. These can be thought of either as collections of numbers, or as new types of numbers.

Answer 2

"...something which has no other real world analogous meaning ... such an obscure entity?"

Your assumption is mistaken, there is nothing especially different or obscure about the square roots of negative numbers or mixing them with real numbers to create complex ones. "Real" and "imaginary" are just historical epithets, arbitrary jargon, in just the same way that "rational" and "irrational" are.

Consider a complex number in polar form $(A, \theta)$ where $A$ is its amplitude or radius and $\theta$ its angle from the real axis. It is much easier to multiply complex numbers in this form, while in electrical engineering $\theta$ is referred to as the phase angle, so this form is often used.

Euler discovered that you can express this algebraically as an exponential: $Ae^{i\theta}$. This led to what has been declared the most beautiful equation of all time, $e^{i\pi} + 1 = 0$. Beauty and obscurity lie only in the eye of the beholder.

In geometry, complex numbers are treated as an alternative metric or ruler, which you can lay down on some topological manifold to create a familiar surface or space. Surfaces with a complex metric are known as Riemann surfaces. The significant point here is that the surface can exist as a physical entity, such as a sphere, and the metric is just an afterthought that helps you navigate it. You can alternatively use a real (x, y) grid or, as we do on Earth, real polar coordinates. The choice depends on the kind of maths you want to do, such as what rule you have for "multiplying" one coordinate pair with another. Neither real nor complex arithmetic is any more "obscure" than the other.

Complex numbers are used in electrical and electronic engineering, and many other areas of physics, because the arithmetic accurately predicts circuit and other physical properties. The numbers were discovered first, so there was no "prior technique". The older Newtonian mechanics does not generally involve complex numbers. The parts of it which do, such as wave mechanics, were mysteries until complex numbers came along.

In other disciplines, such as taking square roots of negative numbers, there was previously assumed to be no solution. Imaginary and complex numbers not only provided solutions but closed the hierarchy of natural < integer < rational < real < complex such that the number system was now complete, with no unsolvable loose ends. Another answer now expands on this.

Perhaps others know of an area which once found solutions a different way?

Answer 3

How do you describe a rotation? One approach is to do

$$x' = x\cos\theta - y\sin\theta$$ $$y' = x\sin\theta + y\cos\theta$$

That's unwieldy. Another approach is to declare: I use a complex coordinate $r$ where $x$ equals the real part, and $y$ equals the imaginary part. Now we can write the rotation much more simply:

$$r' = re^{i\theta}$$

Physicists hate unwieldy math expressions, so they usually opt to describe anything that has to do with rotations, like waves and vibrations, with complex numbers to keep their formulas simple. It makes reasoning about the physical contents of the equations so much easier.

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TL;DR:
Complex numbers are so helpful because they concisely describe rotations, and rotations crop up just about everywhere in physics.

Answer 4

Physics is replete with second-order differential equations, resulting e.g. from an action of the form $\int L(t,\,q,\,\dot{q})dt$ being stationary. If they are linear, or we approximate them as such close to an equilibrium, we get something like $\ddot{q}=Aq$ for some constant $A$, which is typically nonzero. A stable equilibrium requires $A<0$, in which case the functions $\exp\pm i\sqrt{|A|}t$ span the solution set. So the real question is why equilibria are often stable. Presumably, it's because they minimize energy rather than maximizing it. For example, a pendulum has a low-energy stable equilibrium & a high-energy unstable equilibrium, the latter achieved by inverting it.