Imaginary Numbers Are Real [Part 3: Cardan's Problem]

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

[magnuswright]

This is what I love about math. If you have a problem that you know is solvable but you can't solve it, you can say "let there be another type of number."

[mohamad.abdelrida]

This really shows how amazing math mathematicians really are. These guys took 12 or so episodes just to prove one point. I love you.

[sajjadulhaq]

Words are just not enough to explain how awesome these series are. Only true mathematician can understand that how difficult it is to prepare such lectures.
I have been searching for such kind of study from many years.
I am truly great-full for this series.

[ExperimentarEnCasa]

beautiuflly explained videos... true masterpieces !!

[frzferdinand]

Videos are long enough to get me hooked, short enough to feel not long enough. Very, very well done.

[rossboyd]

This is a really well produced series.

[antonmoric]

Really great series - the best that I have ever seen on lateral/negative numbers!

[ShekaranJagadeesan]

This is what should have been taught in "Introduction to Complex Numbers"
HISTORY of either Science and Maths plays a very crucial role in learning. What our textbooks have become is what i now reffer to as "L.A.M.E- Lost And Mug-up Era"

[StrzalaOstryPazur]

25 years ago I've lerned this in high school .... now I'm starting to understand. Tkanki you

[srivatsan]

you should do a couple of videos on calculus(limits and stuff)the Italians had their share of fighting over math so did Newton and Leibnitz!(kinda) plus your narration is awesome.

[kingshuk]

Love you
I am just a Simple Sound Engineer and a Musician....
You teach here so Well and So Simply .. that i understand Everything....
And i feel like i am also a Mathematician
Thank you very Much.. I wish if you ware my math teacher in my High
thank you... love you brother.. ♥

[fullmetalflix]

great video....
circuits brought me hear although i have completed differential equations and never have i heard such a brilliant explanation

[lodgechant]

Even though I don't yet fully understand the algebra you're working with in these videos, you present the story in such a thrilling way that I CAN'T WAIT to see the next video!

[eeltauy]

Thank you so much for this series. It's really great. specially with the accompanying pdf. just one small problem:
the picture used for Rafael Bombeli is actually François Viète. It appears in both the video and the pdf, so I though I would mention it.
Thanks again.

[GGs-cu]

I'm not even a math major or anything but you make learning so fun I just wanna watch it

[jadenfox]

4:05 those numberphile videos in the recommended though...

[sanelprtenjaca]

1:25 That's image of F. Vieté, not Bombelli.
Yes, negative and complex numbers perfectly makes sense in geometrical (positional) interpretation. Great videos!

[Saltofreak]

awesome video man, i'm glad i found this channel.
one minor thing, i noticed that the picture used for Rafael Bombelli is actually Francois Viete. i guess someone who uploaded it mixed the names up.

[hatoriyoshiyuki]

Yours videos are the best about imaginary numbers I've ever seen!! 🤩
Thank you very much to make this topic so interesting to listen and learn. 😄

[stephenzhao]

0:25 Cardan's student, Rafael Bombelli made some incredible insights about what's really going on here. Let's remember why Cardan was stuck. The square roots of negative numbers ask us to find a number, that when multiplied by itself will yied a negative. Neither positive nor negative numbers will work. Bombelli's first big insight was simply to accept that if positive numbers won't work and negative numbers won't work, then maybe there's some other kind of number out there that will. Now if there is some other kind of number out there, a good follow-up question is, "What are we going to call it?" After all, we need to use it in our equations, Bombelli's approach was a very practical one. Rather than dream up a new name and symbol, Bombelli simply let the square roots of negatives be their own thing. In the past, mathematicians would have thrown in the towel here and declared the problem impossible, but Bombelli was able to press on simply by allowing the square roots of negatives to exist. 1:17 1:55 Is '√-1' a "real" thing? However, before we dismiss the square root of minus 1 as some abstraction invented to torture students, let's review what we've learned so far. ... 2:24 Let's make sure we're clear about what it means for the square root of negative 1 to be its own number. If our new number is truly a discovery and not an invention, it should behave like the other numbers we already know about. It should follow the established rules of algebra and arithmetic, and it turns out the square root of minus one does, for the most part. Just as we can split apart the root of the product of two positive numbers, we can also split apart the square roots of negatives. The square root of minus 25 splits into the square root of 25 times the square root negative 1. This process is important because it allows us to express the root of any negative using the square root of minus 1. The square root of minus 25 becomes 5√(-1). We can use this process to expand the root of any negative number, writing it as some number we already know about, thimes the square root of minus one. Let's quickly make sure that our new numbers follow the same algebra rules as our old numbers. In algebra problems with x, only like terms can be added and subtracted: 2x+3x = 5x, but 2+3x = 2+3x. Likewise, 2√(-1) + 3√(-1) is equal to 5√(-1), but 2+3√(-1) is just 2+3√(-1). Finally, unlike terms can be multiplied just as in algebra with x. 5 times x is just 5x, and 5 times √(-1) is just 5√(-1). Now, there are some cases where our new numbers behaves a little strangely, but these can often be avoided by first separating out the square root of minus one. √(-5) x √(-2) = √5 √2 (√1)^2 = -√10.