If it doesn’t exist, why and how do we use it?

In: Mathematics

It’s not that it doesn’t exist, it’s just not a “real” number, doesn’t mean it’s fake or it’s made up, just it doesn’t belong among regular numbers, they have their own space we call the complex plane

Many things in physics are best described and calculated by using complex numbers, for example electronic circuits with capacitors and inductors are used all the time as oscillators and the equations that calculate that produce square roots of negative numbers, but with complex numbers, we can make sense of it and describe what actually happens

Negative numbers don’t exist either – you can have 3 apples or 5 apples, but you can’t have -3 apples. However, that doesn’t mean they can’t be useful. If I told you that after graduating college that I have a net worth of -£50,000 , you know exactly what that means.

There don’t exist any £-1 coins, and I can’t give you 50,000 of them, but negative numbers are still *useful* in talking about debt because the mathematical rules of debt match those of negative numbers. For example, if I have £5000 of debt and then I manage to earn £6000, I can pay off my debt and have £1000 left over. This matches the fact that -5000 + 6000 = 1000 .

There are many applications of imaginary numbers, depsite the fact that they don’t really “exist”, and they have these applications wherever the mathematical rules of imaginary numbers match those of a real situation. I’m sure there will be other replies to your post giving some example situations like this.

basically you cant say ‘it doesn’t exist’ in maths. It exists because we made it exist. Its like a tool that we figured out how to use and apply in ways that work, and it does work. A stone is a stone but if you hit something with it you can call it a hammer. Same with maths, sqrt(-1) is sqrt(-1) but if i can use it to solve a problem, you see what i mean?

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The beauty of maths is that it is so fundamental and universal. Any civilization that is out there will almost certainly have the same maths as us, because maths is the language of the universe.

Imaginary numbers are used extensively in electrical engineering.

Electrical engineering for AC circuits deals with sinusoidal (cosine and sine) waves. The math for sinusoidal waves can be greatly simplified by using a combination of real and imaginary numbers.

E.g., when an AC voltage is applied to a resistor, the current and voltage have the same phase, that is they are perfectly aligned. If a capacitor is added to the circuit, the current and voltage become out of phase.

The mathematical representation of the phase difference using sine functions is complicated, time consuming and confusing. Simple operations like addition are a nightmare.

Representing the phase difference by using a combination of real and imaginary numbers greatly simplifies the math. After a short while, a person can develop an intuitive understanding of electricity in circuits by using imaginary numbers as a reference.

why shouldn’t they exist?

they exist as much as begative numbers or square roots exist.

Mathmatically they exist as solutions to equations. You could equally say numbers *are* the defining equation, and map all rules of numbers towards thosenof the corresponding equations.

negative numbers are solutions to sime equations like x+5=2.

rational numbers are solutions to equations like 5*x = 3.

irrational numbers are solutions to x^2 + 7 = 10.

complex numbers are also solutions to similar equations like x^2 + 5 = 1.

> how do we use it?

Complex numbers (numbers of the form a+bi for real numbers a and b, where i^(2) = -1) are very useful because they can encode rotation.

When we multiply by a positive real number, we “scale” by that factor. Multiplying by 2 scales up by 2; multiplying by 1/2 scales down by 1/2. Multiplying by a negative real number “scales” but also “reflects” the number from positive to negative. If you think about a number line, multiplying by a negative will “reflect” your number about 0.

Complex numbers extend this geometric intuition. Instead of representing complex numbers on a line, we represent them on a plane, where the x-axis is the real number line. Now, multiplying by a complex number both scales your original number and *rotates* it by a certain amount about the origin.

Multiplying by i, for instance, rotates a number 90 degrees counterclockwise. Multiplying by i^(2) rotates by 90 degrees, twice—so it rotates by 180 degrees. If you imagine starting with a positive real number and multiplying by i twice, then you end up with a negative real number. And this is exactly what you expect, since i^(2) = -1, so multiplying by i twice is the same as multiplying by -1, and we already know -1 flips positive reals to be negative reals.

The idea though is that complex numbers let us rotate by any amount. This makes them very useful for compactly and concisely representing lots of more complex phenomena, since instead of having to write multiple equations to capture some behavior, we can just use complex numbers to capture all sorts of scalings and rotations in single equations.

Another reason i is so useful is because of the formula e^(ix) = cos(x) + i*sin(x). This formula lets us encode all sorts of waves concisely in terms of complex numbers.

Sqrt(-1) does not exist *in real numbers*, that is, there is no real number which produces -1 when squared. This makes the operation of square root feel weirdly incomplete in that you can’t apply it to any number you want, unlike other operations. So mathematicians started thinking: if the square root of a negative number like -1 cannot be found in any real numbers, what if we said that it’s an imaginary (that is, not real) number, namely i? Let’s make this assumption and see where it leads us. They then end up with complex numbers, which are things like A + B*i, that is, they consist of a real part A and an imaginary part B. And it turns out that these complex numbers are very neat, at least in that every complex number has a square root, but not only that. Complex numbers turned out to be so useful that electromagnetism theory, quantum theory and a load of other very important physics stuff simply does not work without them.