Imaginary Numbers Are Real [Part 7: Complex Multiplication]

Imaginary numbers are not some wild invention, they are the deep and natural result of extending our number system. Imaginary numbers are all about the discovery of numbers existing not in one dimension along the number line, but in full two dimensional space. Accepting this not only gives us more rich and complete mathematics, but also unlocks a ridiculous amount of very real, very tangible problems in science and engineering.

Part 1: Introduction
Part 2: A Little History
Part 3: Cardan's Problem
Part 4: Bombelli's Solution
Part 5: Numbers are Two Dimensional
Part 6: The Complex Plane
Part 7: Complex Multiplication
Part 8: Math Wizardry
Part 9: Closure
Part 10: Complex Functions
Part 11: Wandering in Four Dimensions
Part 12: Riemann's Solution
Part 13: Riemann Surfaces

[hunarahmad]

There should be an Oscar prize for educational videos and this series should win one hands down.

[Sceleri]

Ahhh the maths feel so good when they makes senses

[shortcutDJ]

at 6 i thought the series was done, but no it's still going strong.

[jaeger]

Wow, I just wanted to learn about what complex numbers were, found your first video, got hooked and I have to say this is really really amazing, haven't had such mathemagical moments in a long while :D

[pcmori]

Electric power systems engineers use the angle notation of imaginary numbers as "phasors" to greatly simplify calculations when dealing with reactance, voltage, current and power. Imaginary numbers super practical for supplying electricity to the grid!

[thriftyplus]

This series has been excellent at simplifying imaginary... uh, I mean LATERAL numbers. I've incorporated these videos into my lesson plans.

Another concept that students have difficulty understanding is the irrational number 'e'. Perhaps you can create a set of videos exploring the history and meaning of this number.

[kinshukdua]

One of the best series on Imaginary number on youtube!
Great Work!

[AbhinavRawal]

Wow! The background music, the narration, the writeup, so cool bub. Hats off for making this amazing series.

[kingshuk]

Tears falls from my eyes... after watching this Picture ( time = 2m 12 s). when angles are added and distance are multiplied

[michaelwinter]

Wait ...that's what polar coordinates are?! No wonder they have never made sense. They're in the Shadow Realm of numbers!!!

[fonseps]

Since we can think of complex numbers as vectors, why is their multiplication so different from vector product and scalar product?

[dazenbland]

I just want to say, I've always just accepted certain things in math like this to get through school. Now that I'm out of school I've started asking why and wanting to know all of the proofs behind these math topics. These videos have been incredibly helpful in explaining answers to my questions in simple and easy to understand ways. Thank you

[IngrainedReason]

When I was watching the earlier videos I was trying to think what the square root of i was and I couldn't work it out. When you point out how the angles and distance add up and multiply respectively it gave me a eureka moment as I realised it was simple to work backwards from the angle and distance of i to get the square root. I've been very much enjoying these videos, great work!

[EducatedTiger]

I've been struggling with the phasor domain (engineering term for polar form) for over a year now, and you just explained it beautifully in a way that makes sense. Thank you so much!

[clericbob]

This video (and the last 2 episodes of this series) finally made me understand why quaternians are used for tracking rotation in 3 dimensions - it's just an extension of the way that complex math is useful for rotation in 2 dimensions, raised to an additional degree. I've been working with 3D models (as a hobbyist and professionally) for a few years at this point, and I had never really understood where quaternians came into play before now.

[nirajabcd]

Damn.. How I wish I was taught like this back then, I would have taken math as a full-time career.

[spamdump]

Handy tip: check your scientific calculator for the R-->P and P-->R functions. They can be used to quickly convert between rectangular and polar coordinates. It does the Pythagorean and ArcTan in one step. The only tricky bit to remember is that you need to use the register exchange function "x<-->y" to get both parts of the answer.
Enter as follows:
To convert 3+4i
enter 3 R-->P 4 =
should read 5
Use the register exchange function "x <-->y" to get the angle 53.1
Polar to Rect is same using the P-->R, enter mag then angle.

[anrdaemon]

All good, but you forgot to add Part6 to the playlist.

[mathematicsfanatic]

Indian students perparing for JEE knows all the rules involved in "lateral" numbers multiplication and all BUT never thought how there can be a separate dimensional plane for complex number. Great work 🤓🤓

[NitinChauhan-vhyk]

I just stumbled across your video and decided to watch it out of nostalgia. Thanks for reminding me how magical mathematics is. Amazing job!