How Imaginary Numbers Make Real Physics Easier to Understand

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#imaginarynumber #complexnumbers #physics

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 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:

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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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Комментарии

As always, let me know what other topics to cover in future videos :)

ParthGChannel

Yes, certainly interested in a lecture about voltage and current, thanks.

MegaSquiff

I read once that Gauss wanted to call them “lateral” numbers which, given the complex plane, makes a lot of sense.

TheWyrdSmythe

Thank you parth I'm really glad you made this video because I was always intrigued on how imaginary numbers would be used I've heard of them in my math classes but never used them to solve a physics problem so it's nice that you gave me your window onto how that would work

dogood

Consensus: Imaginary numbers
Gauss: Lateral numbers
Parth: JEFF!

WilliamDye-willdye

Electrical circuits series all the way! Great video as always

zaynbashtash

I really enjoy your content. I initially saw some of your videos when I was getting into physics a while back. I really couldn’t understand the math but was able to pick up on your enthusiasm for the subject as well as gain insight with a layer of abstraction. I think personalities like yours are crucial to spreading these physical/mathematical ideas effectively. Upon coming across your channel now after having brushed up on some math, I’m left much more fulfilled and informed with the concision of your explanations, as well as your taste in content.

uncle-ffjq

I wish I had math and physics teachers like you during my school days... You make learning science more fun and beautiful! ❤

sandeepkrishnar

“Fundamentals of Electric Circuits” 5E is a good read, rather an easy to follow textbook on things that concern phasors, circuits and complex numbers too. Mathew Sadiku is an excellent writer on the topic of Electromagnetics too. These helped me a lot during my college days..and Parth does well in providing a lot of insight to young students these days..👍🏻👍🏻

dean

Man !!! You just saved my whole Classical Mechanics <3 thanks

malikasim

Awesome video! Would love to see more on this topic. I’m currently in quantum 2 and still don’t fully understand the interpretation of imaginary numbers in a system lol

kadind

Good Morning!
The correct is ì² = -1
And
V-1 = { - i ; i }
The principal Square of -1 in The set complex numbers is i
We write V-1 = i ; so you must to indicate That this is a principal Square of -1 and that exist a second Square that is this - i.
I wish you a Good Day!

OscarMorales-wnql