How Imaginary Numbers Make Real Physics Easier to Understand

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In this video, we'll look at the basics of complex and imaginary numbers, and how they are used in physics!

To begin with, we define the "imaginary number", i, as being the square root of -1. We're often told that negative numbers cannot have a square root, but imaginary numbers are based on the idea that they can. Engineers often use j to represent the imaginary number but we'll stick with i.

An imaginary number can be added to a "real" number (one which does not have a factor of i) in order to create a "complex" number. We look at how two real numbers can be added together, as well as multiplied together.

Imaginary numbers do not fall on the (real) number line, but we instead are found on a perpendicular axis to the number line. That way, we have a real axis and an imaginary axis creating an abstract space. This graph/space is known as an argand diagram, and can be used to represent any complex number. The way to do this is to start at the origin, move as many units in the real direction as the real component, and then as many units in the perpendicular, imaginary direction as the imaginary component. The point we end up at represents our complex number.

The complex number can also be represented with a vector from the origin to the corresponding point on the argand diagram, so its horizontal component is the real part, and its vertical component is the imaginary part. Using this knowledge, as well as basic trigonometry, we can define two new quantities known as the absolute value, or modulus (length) of the vector, and the argument (angle from the real axis). These two pieces of information are equally as good at defining a complex number as knowing its real and imaginary parts.

We can take this information to write a complex number in terms of its absolute value, and the sines and cosines of its argument. However this last part can be converted to a much simpler complex exponential using Euler's identity (https://en.wikipedia.org/wiki/Euler's_identity). We cover the basics of the exponential function as well as how much easier it is to deal with complex exponentials than sines and cosines (as exponentials are easier to multiply).

We then look at two scenarios in physics where we need to represent systems by using sines and cosines. The first is a mechanical harmonic oscillator, such as a mass oscillating on a spring. Instead of dealing with the sine (or cosine) representing the motion of the mass, we can represent it using a complex number evolving over time, do any calculation necessary, and then simply take the real part of the complex number. Taking the real part involves just reading the real part and ignoring the imaginary part. This works because the two components are separate from each other (or perpendicular on the argand diagram). The same logic can be used to represent electric circuits with a sinusoidal input potential difference. This is useful when we have capacitors, inductors, or resistors in our circuit as the voltage and current are not always in phase.

Finally, we look at how quantum wave functions are complex. Although the square (modulus) of a wave function relates to real, measurable probabilities, and the square modulus is not complex, the complex nature of the wave function can be measured in more subtle and indirect ways in effects such as the Aharonov-Bohm effect. Check out the links below for more info, as I've made a full video discussing it.

Videos linked in Cards:
https://youtu.be/Zao9JV1BLg8
https://youtu.be/w9Kyz5y_TPw
https://youtu.be/YMjD8jevTUw

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Timestamps:
0:00 - What are Imaginary, Real, and Complex Numbers, and How Do We Add Them?
3:18 - Representing Complex Numbers on an Argand Diagram
5:08 - The Modulus and Argument of a Complex Number
6:10 - Trigonometric Identities and Exponential Functions
7:59 - Euler's Identity (and Why We Bother With It)
9:28 - Oscillating Mass on a Spring and Complex Numbers
10:23 - Alternating Current Power Sources
12:19 - Quantum Complex-ness
14:31 - Big thanks to Squarespace for Sponsoring!
15:27 - Outro

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Видео How Imaginary Numbers Make Real Physics Easier to Understand канала Parth G