Imagine how hard doing math in the 16th century must have been... I'll never cease to be amazed by what ancient mathematicians were able to do without already knowing what to do the way I know it because I've been taught.

While I'm watching these videos I think "Oh, of course, that makes so much sense!" And then immediately after the video is over I think "I have absolutely no idea what I just saw."

Remember when you were 5 and you thought 100 was the most fascinating, complicated number ever?

This seems like the kind of guy who can recall enough maths and physics to send a rocket to the moon but can't remember his own address

This video is an excellent reason why so many people hate mathematics. So many bad teachers just teach mathematics as some rules or patterns. Great teachers like this man tell you why we need imaginary numbers. My highschool teacher just said "imagine you can find the square root of an imaginary number". Throughout my limited mathematics career I was saved by wonderful teachers like this.

So what happens if you're perverse

I love how he says "Unless you're particularly perverse in your choice of cubic polynomial it's going to sort of like that" and the one that he's picked doesn't look like that :')

I basically understood Zero of what was said in the video. I just watch to see how happy these people are to talk about their passion.

Whens the official brown paper going on sale?

3:13-3:17 Definitively the best way to describe complex math problems :)

"oh gosh, this is a valley"

6:44 A sniper is aiming at your head!

The concept of 'i' (or 'j' in my case) became much more friendly when I was told that it's just a vector at 90 degrees... in other words, it's a second dimension of the number line. For those who have taken trig and are wondering what use it is, remember that circle that you probably had tattooed on your arm; if the X-axis is the real number line and the Y-axis imaginary, think of the progression of the unit circle and the sine and cosine values you were made to memorize; cosine is the 'real' component, as it is the X value in the coordinates of the unit circle, while sine is the complex component, as it is the Y value. What is the difference between sine and cosine values? 90 degrees or pi/2 radians. To explain how 'i' can have magnitude, go again to the unit circle; at sin(pi/2), the Y value is 1, even though cos(pi/2) is 0; thus, 'i' is the unit of the complex domain, which is at a phase shift of 90 degrees from the real domain.

Why would anyone think that finding three solutions is easier than one?

3:39 if you didn't show paper change written on a sheet with romantic background music then I would never understand you changed papers. THANKS !!!!

I wish my math teachers throughout school would have added in the history behind the equations we were doing. It would have made me more interested in the subject, but like so many other people I just kind of gave up on math courses in high school. I eventually went back to college and passed all the math requirements, but it was a much longer process that involved taking remedial classes.

I really appreciate these historical videos. Something about the history of math to me is so interesting, perhaps it's the realization that these people were on to something so huge, so enormous, but yet could not harness it's power like we can today.

as i am 3.87298335 (√15) i am extremely offended by this

I'm pretty sure he meant ether x^3 -5x +7 or x^3 +5x^2 + 7 in the beginning. If you have only positive elements and no term in x^2 you don't obtain the shapes he draws during that part and the curve is always incrasing, assuring you have only 1 zero.

When I was taking 400 level math courses in university, this is what my professor said on the first day. "Hi, my name is blah blah blah, and I'll be very honest with you all. Only about 5% of you will understand what I'm talking about right away, about 15% of you will understand it if you try. about 30% of you will understand it, but not soon enough to pass the course this year, and the rest of you will most likely never understand this math" 25% of the class passed the course.

I always think "put the cap back on the marker!" when I watch these videos.

X^3 GON GIVE IT TO YA

The ad was a math problem, what's going on?

I'm surprised no top comments mentioned complex conjugates or the rational roots test.

He must be one of the greatest Christopher Walken impressionists ever if he gave himself a chance.

So the square root of negative fifteen isn't a useless number, it's just the example a guy used.... :|

Of all the video's I have seen on KZfaq that promised to blow my mind, this one really does, so intriguing yet so incomprehensible

Weird. Didn't see this in my subscription box. Only saw the extra footage one.

at about 2:20 he said that you can have either 1 or 3 solutions. Couldn't you possibly have 2 solutions if the x axis goes through the top of the crest or the bottom of the valley, or am I completely wrong?

Me as a number

What a "real" treat it is to hear Professor Mazur provide such a discourse for the masses.

It's a shame that the adjective "imaginary" was chosen for square root of -1. It is quite a misnomer. Mathematicians love to do things first forward, and then backward. Perfectly reasonable. In the beginning we had addition. Then it was desired to un-add, that is to add backwards; subtraction. But we only had the natural numbers to work with. "5 - 3", well that is "2", of course, but what is "3 - 5"? Adding two natural numbers always yielded another natural number, but un-adding, i.e. subtracting, would sometimes yield new, strange things, which we all know today as negative numbers (as well as zero). But, these negative numbers were quite counter intuitive - how could you have less than nothing? [ Parenthetical: I do very much like the idea that |a - b| = |b - a|, analogous to |a + b| = |b + a|. ] The creation of the negative numbers really only required the creation of one new number which completely embodied all the mental constructs necessary to understand and use the negative numbers. That new number was simply -1. Every negative number is just the absolute value or magnitude of said number, i.e. the natural number, times -1. Negative one was the new number needed. Negative one takes our number line in a whole new direction with marvelous and very practical, real world results. But I think perhaps initially those negative numbers seemed rather imaginary to many folk. We have very nearly the same scenario with √(-1). We invented exponents. We liked them, found them useful. Then we tried to do things backward, to "un-exponent" things, i.e. taking roots, and we ran into some confusion. Actually more than one confusion. First of all, the roots of many numbers turned out to be non-expressible as ratios, so the idea of irrational numbers was developed. But then there was this whole class of numbers, quite literally half of our number line, for which un-exponenting (coined term) them made little or no apparent sense, that is the negative numbers. √(-1) embodies all the mental constructs necessary to understand and use the roots negative numbers. Square root of negative one takes our number line in a whole new direction with marvelous and very practical, real world results. √(-1) is a very "real" number, very useful for solving real life problems, as Cardano discovered. √(-1) is no more "imaginary" than any other number. Of course, we know that all numbers and mathematical objects are constructs of our minds, which is to say, "imaginary" (LOL). Cheers to all - hope I wasn't boring.

He should have commented on the fact that imaginary numbers are by far the most useful thing in mathematics today.

This happened to me once in a math test. The bonus problem required me to work backwards from some given roots, and I had to foil complex numbers together (which I'd never done before). It about scared me to death since I wasn't sure if the i's would cancel out XD Happy ending though: they always do.

These videos have that chilling feeling that you gonna see something you already know but when you see it you go like wait a minute I never seen that before.

properly enjoying some of the videos on this channel is the best use of my math class i've found so far

Why hasn't anyone say that there can be 2 solutions not just 1 or 3. If the peak or crest "touches" the x axis, it will have only 2 solutions, not 3. Although one can argue that the "touching" is actually two solutions touching the same spot but can they be really counted as two solutions if they are the same?

as 40 i am offended by this video

A pleasure to see Barry Mazur speaking again. One of my most respected Mathematicians nowadays.

"You'd think this guy has three solutions, so it must be easier to find them" No.. that's not what I would think at all

I'm afraid this video was like watching cricket for me. By that I mean that at no time did I even vaguely understand what was going on.

Dat blue reflection spot on his forehead...

I would just like to add that it is possible to have 2 zeros in a cubic function. The local maximum or the local minimum can be immediately on the x-axis without crossing it, making it have 2.

I like the palpable pause during the paper change. A very nice touch.

remember when you were young? you shone like the sun...

At 3:46 he's written the word EQUATiON" with all CAPS except for the "I". Is this some kind of mathematical Freudian slip or do I win a prize? :).

The Cardano vs Tartaglia dispute over the cubic solution is one of the great stories of early mathematics.

I don't miss solving these in my calculus classes

You can have 2 solutions pretty easily if one of them is a vertex.

I don't really understand how the first part about cubic polynom functions and the second part about sqrt(-15) are connected in any way. Can someone explain please?

The movement of his hand and his gob, are very relaxing, almost therapeutic

I'm not sure whether or not you've covered it but I would love to see a video on the distribution of prime numbers. Also the primorial function, harmonic numbers, and the number-of-divisors function would be amazing.

You can also have 2 solutions: if a top a dip just touch the x axis.

Dear Numberphile, could you please do a video explaining why there cannot exist any general solution for any polynomial of degree 5 or above in the same way that degree 2, 3 and 4 have the quadratic, cubic and quartic equations? Thank you.

I forgot how to do all this stuff after highschool. Thanks for making me spend all my lunch periods studying something I don't need or remember as an adult.

it's amazing watching this video, knowing very well that I am currently doing all of this maths in Maths B and C right now in Grade 11 Maths (Australia). to think that these were difficult to great mathamatitions many years ago, for it now to be taught standard in School classrooms

Weird to imagine a time in which basic Algebra was the forefront of mathematical knowledge.

Fun video. I was listening rather than watching and thought it was Larry David doing math... Good work as usual numberphile!

Ahhhh excellent. This has a variety of engineering applications and Numberphile channel applications. Now all you need is the engineering and the Numberphile channel.

His answer in the trailout is great: "Do we complain bitterly about things that we're on the verge of understanding but not quite?" ... YES

Just amazing! I wish math was always like this!

TFW a video you find in your sub box helps you understand a math lesson you had today. Thanks numberphile! Couldn't you have two zeros by having one of the bumps overlap exactly on the x axis though? (I think he mentioned that this was possible briefly, I could probably come up with an example if I had enough time.)

awesome video! I never whatched one presented by this gentleman before. Good job! I would like to know how to solve those equations though :/

Wonderful explanation.

Imaginary numbers are far easier to accept than say...1 + 2 + 3 + ... = -1/12 :)

0:40 like the higgs field and the false vacuum?

Most of the stuff you guys talk about I don't understand at all but I still enjoy watching because I love math but never learned algebra or calculus. :-)

I love these historical-mathematical lessons.

This guy reminds me of a more nuanced Bernie Sanders.

I feel like I'm in middle school yay

I've felt this way before but can't say how for idea security reasons. Let's just say the Fibonacci sequence was my number 15. I was barking up the wrong tree for way too long (like 5 years too long): finally decided to just re-invent things, looking at the requirements of the answer first and then working my way back from there. I think any Numberphile, or mathematician (take your pick) goes through this realization at some point.

This is teaching me my next unit in algebra 1 probably, my class literally just started common factors and distribution of polynomials.

Isn't is possible to have a cubic function with 2 zero's? I remember having to do these a lot in old precalc classes, also these things popped up in calculus when I had to find maximums/minimums of functions.

If Cardano was still alive, he would have realised that most numbers of that form would be 'useless' to him. Let's say we had 23i: (5 + 23i)(5 - 23i) = 25 + 115i - 115i + 529, and they would have canceled anyway. So a general rule for this would be: (a + bi)(a - bi) = a^2 + b^2 Which apparently is an actual formula that I didn't know about. But then, it was the 16th century, so it's understandable.

Just for the record, Cardan was not the person who discover how to find the solution of a cubic equation. it was Tartaglia.

As someone that came from Computerphile's videos and isn't the best with math or numbers in general, these videos astounds me. I understand most of what's here, but it amazes me how people find this kinda stuff out in the first place.

even for someone who loves numberphile and other math videos, i found this exceedingly bland.

I like this man. He seems like an interesting person to have coffee with. I'd like to see more from him.

he explained something incredibly easy in an incredibly complicated way or as I call it "trivia bombardment" which is the instance when one gives so much trivia about a fact, remembering it is harder than processing the latter

I think it's kind of funny he said it's so subtle as to be useless immediately after using it, thus proving that it is not, in fact, useless

in Highschool the SQRT of NEG ONE was a novelty ; and it was called "i" ; Then I did Elec-Eng and it was called 'J' and it was essential and beautiful.

Kind of makes me wonder what a perverse polynomial does under these circumstances. Is it as bad as it sounds?

This should be part of every introduction to complex numbers..It makes their necessity obvious rather than "just" introducting i = sqr(-1)

I'm in high school math and I really like this stuff, it's super interesting

I don't get why he acts as though you can only get 1 or 3 solutions to a cubic function. You can get anywhere from 1 to 3 solutions, just like you can get up to n solutions to a polynomial of n degree.

Am I very perverse for thinking that it would have two x-intercepts if the x-axis ran tangential to one of the curves?

This video helped me today in a test . THANK U GUYS VERY MUCH

Lol, "paper change" interlude along with some happy music

Dr. Strangelove?

In EE we use complex numbers to represent reactive power, etc. which made me notice that complex numbers feel like they have their own mathematical systems that allow you to do manipulations that would be difficult with real numbers fairly trivially. This is because of the 1/i = -i and i^(n) = i^(n+4) relationships. My question is if anybody knows if this is a unique property of complex numbers or if we've defined other similar number systems? Furthermore, do we use complex numbers because they're useful or and mathematically consistent (Apparent power = |real power + reactive power|), meaning we'd have to do more complex maths if it didn't work out nicely, or because it's a derivation of a true, physical quantity?

Pi: I’m irrational. I: I’m imaginary. Root of -15: amateurs. Pi: what was that punk? Root of -15: amateurs!

And that's why we need mathematicians. To find useless numbers.. and put them to use, even if they hate these "creatures"

"You have to pass through a region out of the familiar mental theater. Let us say imaginary numbers." Is he trying to be confusing with all this? All you need to say is "we need to use imaginary numbers to get the answer".

KZfaqr livedandletdie commented on the Numberphile video "Problems with Zero" "Real numbers are as real as imaginary numbers." I'd only add, "Vice, versa"..

7:17 do we conplain bitterly about things that we're on the verge of understanding but not quite? As an engineering student: YES.

Imagine if Cardano could see our smartphones and computer graphics now, that use 4D numbers to rotate and orientate objects in 3D space. He'd be blown away by that. (also, William Hamilton would also be impressed with the uses we have imagined for his formula)

My complex mathematics teacher taught me you should be wary of writing complex number as the square root of a negative number, as this might fool you. For example you might come to think: sqrt(-1)*sqrt(-1)=sqrt((-1)*(-1)) = sqrt(1)=1 This is of course wrong because {sqrt(-1)}^2 should by definition be -1. What this tells us is that you can't apply all the rules for square roots of positive numbers to square roots of negative numbers . If you use "i" instead of sqrt(-1) you will not accidentally end up fooling yourself.

why does numberphile always use brown paper and a sharpie? Why not copy paper and a pen?

Many surmise that the book Hamlet reads during the play is "Cardano's Comfort."

Cubics may also have two x-intercepts other than one or three as described in the video, it's when the local minimum or local maximum touches the x-axis instead of cutting through. In equation form, it means when you factorise the polynomial and arrive at three factors, two of the factors will be identical, giving the same x-intercept, while the other factor will give you another x-intercept.