Imaginary Numbers Are Real [Part 9: Closure]

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

[MindYourDecisions]

This series is great! I really liked the jigsaw puzzle analogy. Is that puzzle board available anywhere?

[MidnightSt]

Amazing series.
In 30 or so minutes you made me understand a topic I've dreaded and avoided for years.

[RapidDominance]

This series was good.
Please tell me something awesome is next

[logansmith]

We need people like this to be the educators in our society. There are topics like this that I struggle with everyday in college that are just brushed over. I'm not being taught to learn, but rather how to do. Thank you for the very informative video. It made a connection in my brain that I have been searching for for a long time!

[NickCybert]

Wow this was a really cool series. I wish imaginary numbers had been introduced to me in the way you just describe, as the natural outcome of trying to complete algebra, and as rotations on the complex plain. Can you believe I was taught what a unit circle was, and absolutely nothing about how it relates to what you went through in this series? Just rote memorization of the coordinates of different positions on the circle.

[MrCritical]

This is something every school needs to show their students learning complex numbers. As an engineering student, I always felt that what is taught is insufficient. It's videos like these that really teach and reinforce my concepts. Kudos to you, for making a video that is still changing lives 4 years later.

[gonzalotapia]

The sentence "Imaginary numbers are the exact missing piece that makes algebra complete" blew my mind.

[akyu]

Outstanding videos.
As an adult trying to repair the fractured mathematical education I received as a child, videos like this are amazing.

[Daruqe]

"...with the notable exception of dividing by zero..."

So...the rationals AREN'T closed under division?

[justinli]

The quality of the video is beyond anything I have seen. It is just amazing how a complex concept can be explained so clearly in just 5 minutes

[SlipperyTeeth]

When you finish this series, how about a quick thing on quaternions?

[legolad]

Do a vid of dividing rational numbers by zero. Open the can!

[asimdeyaf]

Seems more intuitive to represent all numbers with angles. For instance...
Positive real numbers: 1∠0° or 1∠360°
Negative real numbers: 1∠180°
Positive imaginary numbers: 1∠90°
Negative imaginary numbers: 1∠270°
Complex numbers: 1∠45° or 1∠100° or 1∠346.87° or etc...
Does anyone do this, or are there problems that require separating the real component from the imaginary?

[Erik-ywkj]

I just found your channel, watched all your videos, and I can't wait to see more! =)

[NoBlitz]

This entire series really brought me back to my college days. Thanks! Now invent a new number beyond "imaginary"!

[giffyguy]

This series makes it seem very clear, that the set of complex numbers is undeniably NOT closed under division or exponentiation. This is because we do not yet have a way to define 1/0 or 0^0, similar to how they didn't have a way to define the square-root of -1 before complex numbers came around.

[feynstein]

We got all of this from basically addition. Subtraction is the opposite of addition, multiplication is repeated addition and division is repeated subtraction. Perhaps if we were to find a completely new operation, we'd find more types of numbers.

[WithASideOfFries]

Really clever way to end the series! Great job! This deserves so much more views.

[GlorifiedTruth]

I love how they bring CLOSURE to algebra and this video brings CLOSURE to the series. First rate.

[enargins]

Really enjoyed this series. Watched the whole thing straight through (couldn't stop watching :-) ). Thanks for sharing! Subscribed.