The Useless Number - Numberphile
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- Опубликовано: 24 фев 2015
- One of the first people to stumble over imaginary numbers (namely the square root of -15) thought it was "subtle" and "useless".
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Featuring Barry Mazur.
Extra footage from this interview: • Elegant Confusion and ...
Barry's book on this topic: bit.ly/RootMinusFifteen
Barry on scale and similarity: • A Mathematical Fable -...
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Наука
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.
Exactly what i was thinking man.
+MichaelKingsfordGray certainly a nice thought, but I don't believe in it. Maths literacy generally seems to go down with improvements in IT
+Markus Jaeger not generally true. a lot of things become a lot easier to grasp, because we deal with them daily, even though we don't know the theory behind everything :)
+Matthew Helm Well, the big difference between those people and us learning in schools is that they wanted to discover new things.. we just want to know what to do to get that job that supports our life. Maybe a few ever discover something new, but how to do that is not part of what we learn in school ;)
MichaelKingsfordGray
true.. we're still stuck in that post WW2 'rebuild the world" program.. and they're keeping the wars going, so we can keeep rebuilding, so they can have more robots. People who wake up will be eradicated, or disabled some how.
Remember when you were 5 and you thought 100 was the most fascinating, complicated number ever?
+Elphaba Crux not even kidding, i used to think that was where numbers ended. then one day, i saw street numbers in the 300s and i was like "mum, you've lied to me!"
lol really?
+Oli Patts its not ?
i still dont know man
For me it was 144, because that was the end of the multiplication table at school. 😂
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."
This is exactly how I feel studying math (and that's why learning is necessary).
R0DisG0D
It's when I read the comments below that my head really starts spinning. Those are some intelligent people making these videos, but the people subscribed are downright brilliant, many of them... They sometimes give me better ways to understand what I just saw. Or point out errors they made which lead to some of my confusion. Don't ever stop at just the video, scroll down...
That seems to happen a lot to me, too.
But eventually, after watching whatever it is several times, I finally understand & remember.
That happens when I learn maths sometimes. Hence why I failed A-level maths.
Honestly, Complex numbers are no where near as bad and confusing as mathematics gets. I'm currently in my second year of my Physics degree and, as a requirement of the degree I have to do some maths units. Throughout all of the maths stuff I have done, the complex number stuff is some of the easiest. Yeah, if you think philosophically about the meaning of a negative root, it can be pretty counter-intuitive but if you ignore that and just treat it as another number, they all work out to be pretty simple contrary to their name haha
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
Yeah, he is smart enough to make calculations and build a rocket and fly it to the sun but couldn't cook an egg. I mean he can build you a special rocket that holds the egg, fly it to the sun and cook it, and bring it back to earth, but he can't turn a stove on and throw one in a skillet. He could probably build a solar array out of coffee tins and broken glass, but couldn't figure out how to screw a light bulb in.
I need further examples...
J.D. Salinger I see what you did there 😁
You seem like the kind of guy who lives in a world full of stereotypes to fight with
😂🤣
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.
All of what he said is taught in middle school, or depending on your country, high school.
master3243 Teaching mathematics to me is like rading Shakespeare to a 2 years old. He appreciates the effort, but gets none of it.
It's contagious isn't it?
I often feel like there aren't enough people like this around (me) anymore.
This video's summary: 16th century (Italian) mathematicians knew of the existence of complex numbers as a concept, knew they needed them to solve algebraic equations, but didn't understand them. Kinda like how I am regarding quantum anything :P
As also evidenced in the video about similar blobs (polygons?), this presenter tends to meander when explaining a concept or getting to a point, so I had a hard time seeing where he was going with this one too.
master3243 I learned in school that it is impossible to get the square root of negative numbers. And if I ask calculators, I get either "invalid input", "error" or "not a number" which confirms this knowledge. Yet this video claims that if I multiply those invalid results, I somehow get 15 as if that is the most normal thing in the world...
So what happens if you're perverse
+Taraalcar You teach college Calculus 2.
+Taraalcar then algebra becomes more fun B==3~(Y)
+MultiJebusChrist that makes you a sadist. a mankind hating sadist.
+Taraalcar Instead of 1 or 3 solutions where it crosses the x-axis, there are 2 places where it intersects the x-axis, one of which is crossing it and the other of which the curve is tangential to the x-axis when they intersect but do not cross. This thing of being tangential to the x-axis and intersecting but not crossing is rather perverse to inflict upon 16th century mathematicians, kind of like asking Euclid to deal with non-Euclidean geometry or asking Isaac Newton what he thinks of Einstein's Theory of Relativity.
Yikes.
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.
@@jumblejumbo I'll just watch this video again 6 years later and see what you're talking about.
A square root of an imaginary number doesn't make much sense in the context of this video, as this would be something like ✓(✓(-15)).
@@Zeewman apparently I made a typo 5 years ago and you didn't notice
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 :')
Yes. His perversion is that all the terms in his polynomial (more specifically, all the non-constant terms) had the same sign.
The particular perversion he mentioned was likely a reference to the rare cubic polynomial with _two_ real roots, rather than one with one or three
@@thalianero1071 Not actually that weird, though.
Whens the official brown paper going on sale?
The scribes of the great library of Alexandria would have loved that brown paper. I might just "scroll" up all my maths notes this way ......
I think it is kraft, looks like alios or something
It's cartridge paper. You can buy it at any craft store, or Amazon.
@@tobiasgertz7800 hey cool snake avatar my man
Just make do with cardboard
Why would anyone think that finding three solutions is easier than one?
I think it's because if there's only one real solution then you know for sure that the other two must be complex. where as just looking at the 3 solutions you can see they're all real
Xtremesheep I have no idea what you just said. how could there be any other solutions for x when the graph crosses x coordinate once ?
INameIsGood
Think of a quadratic equation with no real roots i.e. b^2 - 4ac < 0
The reason we can say no real roots exist is that in the quadratic formula:
(- b +/- √(b^2 - 4ac)) / 2a
You must take the square root of a negative number. However, allowing complex numbers gives us two solutions anyway; even though it does not cross the x-axis for real values of x.
As an example take y = x^2 + 1. Roots occur at y = 0:
x^2 + 1 = 0
x^2 = -1
x = i, - i
The same applies to cubics or higher polynomials.
Avis Mercurii If you know one solution (say a) to a cubic, you can divide the cubic by (x - a) and the resulting quadratic equation has the other two solutions.It's like searching for three Easter eggs in a big lawn instead of just one, and finding one tells you where the other two are.I know it doesn't really work that way but some people might thing It does.
***** I know about quadratic equations and about imaginary numbers, but i still don't understand how could there be more answers to x if b^2 - 4ac < 0.
In your example there is only 1 answer, where are those two complex ones come from ?
6:44 A sniper is aiming at your head!
Tyran Tyran Why is there a small blue light on his forehead?!
+RedsBoneStuff
Is his intellect shining at you!
+RedsBoneStuff i thought you were talking about how the wall has separations and lines in the background XD
+I Tolerate Everything Your names is my personality definition.
lulz :)))
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.
+Mike Seo i want that course!
Beats our technical school's four year 50-60% fail rate
So, 5% of people proved his statement to be incorrect? Or at least evidence enough to show his hypothesis was wrong?
Mike Seo So 5% passed the course by chance?
Peter D Morrison probably yeah
3:13-3:17 Definitively the best way to describe complex math problems :)
lol
Gabriel Davis 😓😓🖕🖕
lol
Jake Appleton hahahahahaha
Lol
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.
The concept of "angle" is simply a way to "extend" outside of the curnent dimension one is operating in.
Most of that time, that dimension is the 1st dimension, because in the first dimension we have lines, and when we have lines, we can have "numbers".
The imaginary plane has NO NUMBERS, but rather identities. Identities are not linear, therefore they can't extend in a linear manner.
Therefore, we have to "extend" them, using "angle", because "angle" leads to no linear concept. It is simply all the infinite points of a circle. Points that are identities. We relate these points to linear numbers, so we can derive some sort of information from them.
The complex plane is like creating a "plane of names". We can give each number an infinite variety of "names", which creates a matrix of identities.
You cannot "square root" a line. A line is a concept of 1 dimensional "extension", not 2 dimensional.
A "line" has no length. It only has two points. The line itself simply represents a "connection" between these two points.
The complex plane is simply a "connection" between an identity and a number. That's why you can't find the square root of -1, because squaring 1 or -1 is equally nonsensicle.
Rooting a 2 is equally nonsensicle. Rooting a 3 is equally nonsensicle.
You can only "extend" a line of two into a four. So four is the minimum you can "root".
If you root a 1, 2 or 3, you are literally falling into the 0 dimension of "point" or "identity", and that dimension can only be expressed as "angle" not "line".
And if you "root" any of them, you end up with the same angle. The angle of an equilateral triangle. 60 degrees.
But this can only be rational in the complex plane to 180. And then we double that to 360, to give us extra room to work.
as i am 3.87298335 (√15) i am extremely offended by this
+wasd2333 3.87298335 isnt the square root of 15.
Maltager i know, i go on forever.
This took me a while, to copy myself but the first 500 digits (including decimal point) is 3.872983346207416885179265399782399610832921705291
59082658757376611348309193697903351928737685867351
79163022068609496470131895404391636496156798917461
21203511068754791013493581283919531228889292565846
41702806250919317341265986481845546462855131594026
26176726405086463004505778106319728809397819251883
00355301530010748354438098719014432460775585023104
81171709060628415739758158290101333032744473055806
67026371791243800477125912479424727638166001355365
27229535679457030784654661053577240260167033763749
wasd2333 actually, you cannot go forever, so no, you will never be able to recite the square root of 15 :).
+wasd2333 Yeah, but what's the square root of negative fifteen?
JustSomeGuy i * root(15)
"oh gosh, this is a valley"
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.
+Ch0plol I've heard of college courses like that, but you can say the same about science or most other content-based subjects. There wouldn't be enough school days for that. You could have schools specialize in some subjects, but then it sounds like you're the kind who wouldn't have chosen a math-concentrated school. Plus you've got increasingly heavy curricula like the Common Core that dictate both what to teach almost every day and even how to teach it, and it's impossible to fit anything extra into the class.
As a math teacher, I'd love to spend days telling stories like these, but literally the only thing that matters these days are standardized test scores, and weekly paperwork for the district and administrators that really no one even looks at.
We talked about everything he said in this video as an introduction for the complex numbers chapters in high school, but rather than the teacher just telling us about it like in this video, We did it in a sort of exercise
@smurfyday: But the best way to do well on tests is to understand the material well.
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.
sqrt(-1) or any other complex number is not a number. It's an error.
@@groszak1 Please do explain. Imaginary and complex numbers follow all of the rules, unlike, say, 1/0.
@@snbeast9545 that's because math isn't in an imaginary number system, while 1÷0 is Infinity and is a valid number
@@groszak1 Imaginary is, as op says, a gross misnomer, as it has several applications (like with studys of electricity).
1/0 is not infinity. For the function f(x) = 1/x, if you have x approach 0 from the positives, f(x) tend to infinity, whereas if you have x approach 0 from the negatives, you tend to negative infinity. Since these two results don't match, the limit as x approaches 0 does not exist, so 1/0 cannot be evaluated to infinity.
Algebraically, if we let x = 1/0, then multiply both sides by 0, we get 0x = 0 * 1/0. The zero and 1/0 multiplicatively cancel, so we get 0x = 1. This cannot be evaluated to anything because 0 times anything is 0, so 1/0 cannot be evaluated. (Note: The 0*inf indeterminate form can be evaluated to be 1, but so can 0*-inf, so no dice.)
Lastly, infinity is not a number, so saying 1/0 is infinity is meaningless. Both because infinity is a concept, and because there are several types of infinity.
@@snbeast9545 Infinity and -Infinity are as close as 0 and -0 are. Making one of those pairs of numbers equal also means making the other equal. Floating point makes neither equal, so 1÷0 is Infinity and 1÷-0 is -Infinity. And Infinity is a number, not a "concept".
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.
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.
Neither of those have three roots. x^3 - 5x^2 + 7 does though.
+Jonathan Slavin It only needs 2 extrema, not 3 roots.
Jonathan, read my whole comment and you'll understand what I meant. The equation he wrote doesn't fit with the fonctions he drew.
Étienne Massé He didn't try drawing the graph if the actual equation, it was just an example,, though they did show it later.
yes
Me as a number
lol
And still better than me :(
It doesn't say who is better than you.
Better than you zX Z x zxzx
o Not useless, just misunderstood?
So the square root of negative fifteen isn't a useless number, it's just the example a guy used.... :|
The square root of Negative 15, or of any negative number for that matter, equals the complex number i, which is not useless. Without it, we would not have lights, phones, etc. Something to do with the amount of electrons passing through wires...? Someone correct me on that.
+Ryan Stenger That's not true at all. The square root of -15 is not i; it's a very different number from i, which is by definition the square root of -1. i squared is always -1 and never -15, just like 2 plus 2 is always 4 and never -15.
i is beautiful to both mathematicians and engineers, but it wouldn't render engineering impossible. But it sure does make certain solutions a lot easier to construct!
According to Cardano, square root of -15 was useless.
Without imaginary unit people would still be able to create many things but they couldn't tell why and how. It is just the base of the quantum world.
Kevin VanOrd Thank you for telling me this, I do not know much about this stuff :P
What would √-15 be, then? I assumed since the square root of -1 is i, and it's impossible to get a negative number from something squared (for example, 2^2 is 4, and similarly -2^-2 is also 4. It wouldn't be considered squaring it if you did 2^-2, which is -4), that √-15 is also i.
+Ryan Stenger It's simply √15*i because if you square it, you get 15*i^2 which equals to -15.
Of all the video's I have seen on RUclips that promised to blow my mind, this one really does, so intriguing yet so incomprehensible
I always think "put the cap back on the marker!" when I watch these videos.
Hello Monica😂
He must be one of the greatest Christopher Walken impressionists ever if he gave himself a chance.
Imaginary numbers are far easier to accept than say...1 + 2 + 3 + ... = -1/12 :)
I just designed and ordered a shirt with this on it and I'm very excited to get it :)
that equality is incorrect.
+Alvin the Chipmunk it's right, find numberphile's video on it
Kelvin Porter No, it isn't. A divergent series doesn't converge. This -1/12 is obtained with incorrect assumptions.
There are series that are conditionally convergent.
X^3 GON GIVE IT TO YA
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? :).
You win an eggplant.
Lockirby2 hauterbeysegg detected!
EQUATİON
thank me later.
I'm surprised no top comments mentioned complex conjugates or the rational roots test.
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.
The ad was a math problem, what's going on?
they know how to target their audience?
veggiet2009 They pay Google to do it for them.
as 40 i am offended by this video
as i*sqrt15, I am extremely offended by this video
Sin Midani if your 5 im 5
+arachnid14 When I turned 20 I started to just change the base each year instead of counting my years. I'm now 21 base 17.
I'm not 5, I'm 13
Why would you be?
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?
You could (y)
+astickydog I was literally about to comment the same thing :)
That's right in that case. I'm not sure with my mathematical English, but unless I'm mistaken, it's called the double root (if you factorize the equation, you get this root truly twice). While only a single point, for the sake of root counting, there is added a clause for consistency that directly states that multiple roots count multiple times (in this case, twice).
You could, but I think that would constitute a 'perverse' situation. Either way, it's solved the same way as the 3-intersection one.
As others have said it is a 'repeated root' and the local maximum or minimum touches but doesn't cross the x-axis. And example of this is y=(x-2)(x+2)^2. The repeated root is the factor that is squared, so the graph touches at x=-2 (Yes, the negative of the squared bracket). This touching but not crossing also occurs for other brackets that have an even power. An odd power is an inflection point.
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.
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?
He did say, "unless you're particularly perverse in your choice of cubic polynomials..." :o
In Precalc or perhaps Alg II you learn the terminology to describe that: that's a solution with a multiplicity of two. If you're doing a list of solutions, you would list such a solution as, for example, "x=0 (mult. 2)" if the graph bounces off the x axis at 0.
In higher-degree equations, you'll find solutions with multiplicity of three, four, or higher: if the multiplicity is even, it bounces off the x axis. If it's odd, it passes through it.
EDIT: Solutions with multiplicity of 3 definitely do exist in cubic equations, but not in the examples the video or superjugy were talking about.
All cubic equations always have 3 roots. Some may be the same, some may be complex but always 3.
Of course, I know there are three roots, but Graphically (as he was explaining) there are two points that cut the x axis. I don't know why would explaining that would be difficult for people to understand. That is what I meant when I said that you can argue that that point is two solutions in one point but still is basically just two solutions.
CVGTI
Yeah, but having 3 roots isn't the same as having 3 solutions. If two solutions are the same, they can't both contribute to the number of solutions. In that sense, I could say the equation x - 5 = 0 has 3 solutions, 5, 5, and 5. I think what you mean is that you can only determine an exhaustive list of solutions by setting each of its three roots equal to 0 and solving, and that's not even true. Since two equivalent linear equations necessarily have the same solution, you could also obtain an exhaustive list of solutions by showing that two of the roots are equivalent, then just solving one of them. 3 roots, 2 calculations plus the conclusion of that proof, 2 solutions.
He should have commented on the fact that imaginary numbers are by far the most useful thing in mathematics today.
+Greg “Satan's little helper” Jacques Wow, you realize modern science and life would be impossible without these useless people, right? This RUclips thing wouldn't exist without relativity and all kind of useless math.
Some idiot, who by the name, sounds like a religious nut. Those two things often go together.
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.
Do you not study maths
Samuel Tan No. I kinda missed the boat on maths at school. I do have an interest in the subject though, which is why I enjoy Numberphile and often learn from it. But this particular video went way over my head I'm afraid. :(
grahamlive Pick calculus. It´s fun. (Evil laugh here)
grahamlive You should buy an algebra book and read some of the problems if youre interested
Well you shouldn't be watching this if you didn't do well in high school maths.
properly enjoying some of the videos on this channel is the best use of my math class i've found so far
A pleasure to see Barry Mazur speaking again.
One of my most respected Mathematicians nowadays.
Weird. Didn't see this in my subscription box. Only saw the extra footage one.
The same thing happend to me
RUclips is a lot like Facebook it only delivers certain content. Unless you religiously watch a channel there is a chance you may never see one of their videos again.
Oliver Jackson that's only really true if you use the "what to watch" page (the default page). If you use the "My Subscriptions" page, you'll always see every video from everyone you're subscribed to, even if you haven't watched one of their videos in a year.
The subscription feed doesn't seem to update in real time though, so often new videos won't appear until 5 minutes after they're uploaded/processed. The system does seem to occasionally glitch (or the uploader forgets to make the video public), making it take even longer for the video to appear.
There's also the bug some people have experienced where they'll be automatically unsubscribed from one or more of their subscriptions. I've never had it happen to me, so I couldn't tell you much about it.
Ben Rowe I thought everyone used the my subscription page... I have over a hundred subscriptions so I kind of have to.
cortster12 Some people apparently don't. I sometimes wonder how they manage... then I realise they probably don't live on youtube like me.
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?
In order to solve cubic polynomials you have to use imaginary numbers.
mrwho995
Clearly not for all cubic polynomials. Take x^3 - x^2 - x = 0. The solutions are 0, .5(1 - rt(5)), .5(1 + rt(5)). Which particular kinds of cubics require imaginary numbers?
Taylor Bennett What you solved was a 2nd degree polynomial.. Try x^3+x^2+x+1=0
Taylor Bennett he says it in the video: the ones with 3 "0-points"
***** That was a depressing (though excellent) answer.
"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
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.
remember when you were young? you shone like the sun...
Nah, Except for Dark skins like me.
Is this a joke? It's everywhere and I do not get it
I don't know why people post it, but it's the lyrics to Shine on You Crazy Diamond from the best rock band ever, Pink Floyd. And the letters SYD were for Syd Barrett, one of the members who wandered off.
This song has to be discovered by the most people possible! We have to spread Shine on Your Crazy Diamond beyond our galaxy!
Well radio transmissions are presently carrying the song outwards from Earth and eventually (presumably) beyond the galaxy.
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 like the palpable pause during the paper change. A very nice touch.
What a "real" treat it is to hear Professor Mazur provide such a discourse for the masses.
I see what you did there.
I don't miss solving these in my calculus classes
You can also have 2 solutions: if a top a dip just touch the x axis.
That would still be three, the max or min would be a double root
Collin Luthman what? How?
When the max or min hits the x-axis but it doesn't pass it, it bounces off and resulting is more than one root
exactly, the root would be squared; it would have a multiplicity of 2.
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.
The Cardano vs Tartaglia dispute over the cubic solution is one of the great stories of early mathematics.
Dat blue reflection spot on his forehead...
You can have 2 solutions pretty easily if one of them is a vertex.
But then you'd be being perverse.
Yeah come on weren't you taught manners? Stop being perverse!
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.
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
Fun video. I was listening rather than watching and thought it was Larry David doing math...
Good work as usual numberphile!
Just amazing!
I wish math was always like this!
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
awesome video! I never whatched one presented by this gentleman before. Good job! I would like to know how to solve those equations though :/
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.)
Take any quadratic function where b² - 4ac equals zero (ex. 3x² - 18x + 27) multiply it by any binomial (ex. x + 1). There you have it (3x³ - 15x² + 9x + 27). Verified at WolframAlpha: www.wolframalpha.com/input/?i=graph+3x%C2%B3+-+15x%C2%B2+%2B+9x+%2B+27
Dishmopo
Thanks. Also cool.
jacketsj: the guy with a long name that will wipe you off the earth because I eat children Very intriguing name you got there
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.
+Stefan Alecu You can do a few of those things:
a^2 + 2ab + b^2 = (a+b)^2
a^2 - 2ab + b^2 = (a-b)^2
a^2 - b^2 = (a+b) (a-b)
But when we get to a^2 + b^2, then it's tricky, because the factorization of that is, as you said. (a+bi) (a-bi).
true, but I can understand him, since imaginary numbers weren't discovered then I think, nevermind the formula, so I guess he discovered that the hard way :)
LOOL I was not expecting the useless number will be the imaginary/complex numbers. When today, complex numbers are the reasons why we have cruise control on cars and segways. :D
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.
Weird to imagine a time in which basic Algebra was the forefront of mathematical knowledge.
0:40 like the higgs field and the false vacuum?
Someone's fedora is showing.
Ahhhh excellent. This has a variety of engineering applications and Numberphile channel applications. Now all you need is the engineering and the Numberphile channel.
This should be part of every introduction to complex numbers..It makes their necessity obvious rather than "just" introducting i = sqr(-1)
This guy reminds me of a more nuanced Bernie Sanders.
Hello comrade
even for someone who loves numberphile and other math videos, i found this exceedingly bland.
It should have been called "The Useless Video"
weesh ful man, I have to agree . I found myself losing interest multiple times which is very rare for all of Brady's videos
really? I mean can you imagine what would have happened if the negative roots didn’t cancel? it’s a humbling moment to realize the tools to deal with the problem more generally don’t exist...it’s fascinating really
A lot of people are saying they don't understand the material in the video. I find that odd - this is basically the sort of thing we learned about in standard ninth grade algebra this year.
Just for the record, Cardan was not the person who discover how to find the solution of a cubic equation. it was Tartaglia.
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.
Yes, but I suspect they require the same or similar work to 3 zeros.
Silly phone... And they weren't part of the story.
Logically I would say no. Because if the "valley" hits the x axis exactly this would produce two zeroes
Kallum Noon But imagine if the bottom part of the valley was exactly tangent to the x axis, that would mean there are 2 zeroes. (Actually there are still 3 zeroes but two of them are coincident)
(x-3)^2 * (x-5) is a cubic polynomial where the only two zeros are x=3 and x=5.
The only thing special about this curve is that x=3 is a so-called "double zero". It is represented in the shape of the curve where a critical maximum or minimum lies on the x-axis. For the sake of this video, it is irrelevant.
Kind of makes me wonder what a perverse polynomial does under these circumstances. Is it as bad as it sounds?
That's what I was thinking, what's the deal with perverse cubic polynomials?
Not really that "perverse." His example is one. They don't have a hill and valley.
Think he might be talking about polynomials specifically chosen to have repeated roots.
Giddefication He said: "unless you use very perverse equations, there will be 1 or 3 solutions". So i think he means it is so when there are 2 solutions. So the minimum is on the x axis (broken english but true xD)
JNCressey Or with repeated roots where a local max/min touches the x axis.
This is teaching me my next unit in algebra 1 probably, my class literally just started common factors and distribution of polynomials.
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.
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.
A cubic function has ALWAYS 3 root solutions.
Either 3 real solution, or 2 real and one imaginary or one real and 2 imaginary
Texan PlayeR
I think you have only two possible scenarios: three real solutions or one real solution, one imaginary solution and its conjugate. Please correct me if i'm wrong.
You can have two real solutions if either the local minimum or maximum intersects the x-axis as well. The speaker was well aware of this, which is why he kept referring to "perverse" functions--perverse because someone would need to take the time to build such a function with two real solutions.
One real and two imaginary is not possible. Imaginary comes always in pairs. Thus it is always 1 real and two imaginary or three real and in this case also double solutions are possible.
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.
+Muzz TheGreat then I read your comment and replied "k"
+Matt Derry then I was compelled to make a quick joke and so I said L out loud
+Khorps Then I broke this chain. :trollface:
Tommy Thach then i got pissed off
Well I M shocked by this sudden change of events N'd will instantly make up for the 2 lost chain parts. Are you O kay with that?
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
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.
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.
Piggybread
You may very well actually be aware of this, but:
What you said there is not quite right. If we put imaginary numbers aside, the square root of a real number is by definition a positive number. This is practical because for most intents and purposes you don't want mathematical functions that have more than 1 possible answer.
So sqrt(1) equals 1 exclusively by definition.
What this means in practice is just that the solutions to
x^2 = a are x = sqrt(a) or x = -sqrt(a)
A root is just an exponent, and the laws of exponents are pretty clear about negative bases and fractional exponents. As a consequence, the square root function is undefined for negative numbers, so claiming that sqrt(-1) = -1 is wrong, per definition.Piggybread is right if he applies your logic, yet you tell him he's wrong. I'm wondering what it is exactly what you're trying to achieve here.
flopski
In my post on march 11th I corrected piggybread's incorrect statement that sqrt(1) = plus minus 1. I interpreted his post as a general statement, not a comment using my logic.
In my original post I claim that anyone who is new to complex numbers could benefit from using "i" consequently instead of sqrt(-1).
I never claimed sqrt(-1) =-1. Where did you take that from?
Actually the reason why the conclusion in the original post is wrong, is because of this rule:
sqrt(a) *sqrt(b)=sqrt(a*b) HOWEVER only if a>=0 and b>=0
Many people which did not applied this rule say the definition of i is i^2=-1 and not i=sqrt(-1) because of the conclusion you mentioned in your first post. !Because of this rule the definition of i=sqrt(-1)!
ImGonnaShout2000 that is incorrect. You can't just define the square root to be positive "by definition". If you could, then you could "prove" that +1=-1
Am I very perverse for thinking that it would have two x-intercepts if the x-axis ran tangential to one of the curves?
Yep, I was coming down here to say this :P
Josephiroth
I was hoping that would be mentioned, but the point of the video seems to be that bizarre concepts like "imaginary" numbers are at times necessary to solve very real problems.
Videos can only be so long, after all, and there are probably entire books dedicated to complex numbers.
Josephiroth the point which the x-axis is tangent to has multiplicity and counts as 2 points
William Pfeiffer Okay... that does sound familiar, but I haven't exactly retained all of my math knowledge.
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. :-)
Pi: I’m irrational.
I: I’m imaginary.
Root of -15: amateurs.
Pi: what was that punk?
Root of -15: amateurs!
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?
Ovenman940 Complex numbers are imaginary, as the name suggests. They aren't real physical things that exist, and we use them only to make certain calculations more convenient.
Lets say you have a 2-dimensional vector. Normally you'd need to have 2 components (x and y) to represent it, which turns a single vector into two scalar quantities. This can make some math more complicated or less complicated depending on what kind of math you'd like to do with it.
Using complex numbers, you can turn a 2-dimensional vector into a scalar by setting the x-components as real and the y-components as imaginary, which means instead of working with 2 real scalar equations, you're working with 1 complex scalar equation.
Exact examples of when this is useful are hard to come up with, but it is generally easier to deal with less equations, especially considering how easy complex numbers are to work with in general.
***** not actually true, complex numbers are used in Quantumn Mechanics
Much simpler to work in complex exponentials than with sin waves.
***** I guess complex numbers are about as real as negative numbers.
"You are right that the real numbers do not possess the necessary structure to model electrical circuits in a simple way."
You only use complex numbers in electrical engineering in order to get around solving differential equations ((the potential over a coil/inductor equals L(constant) times dI/dt, current through a capacitor equals C(constant) times dU/dt)). You could do it all with classical numbers, but it's far far more convenient with complex numbers.
all of the properties you said are because i is kind of like 1 1 has all the properties you said also
Dr. Strangelove?
If I am not mistaken, the solution to cubic equations was found by Tartaglia, not Cardano. Cardano just tricked Tartaglia to tell it to him and Cardano published it in his Ars Magna. Piece of trivia: during his time Cardano was more famous for being a ... doctor than a mathematician.
Wonderful explanation.
I feel like I'm in middle school yay
I'm likin' your enthusiasm!
:D
+Autodidactus Communitati I'm sorry sir. Do you have an authorization pass to time travel?
"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".
The point was that imaginary numbers were not an existent concept at the time
Leo Anbu Well obviously but he doesn't have to make it so obtuse.
gckbowers411
Maybe that's just his style of speech.
Would you criticize his style of speech?
Would you criticize a man's style to his face?
Leo Anbu No, I would introduce him to Google Translate.
gckbowers411 /watch?v=nDwQJ64UpFQ
As far as I know, there are not 3 but 5 cases: in 2 cases there are 2 solutions instead of 1 or 3. Example:
x³-11x²+39x-45 = (x-3)(x-3)(x-5)
You always have three solutions to cubic equations. What you meant are two *unique* solutions.
What I don't understand is that he appeared to say the cubic equations with three real solutions are the most difficult, but then his example of a tough equation involves a cubic with only one real solution and two complex solutions...
I like this man. He seems like an interesting person to have coffee with. I'd like to see more from him.
RUclipsr livedandletdie commented on the Numberphile video "Problems with Zero"
"Real numbers are as real as imaginary numbers."
I'd only add, "Vice, versa"..
This made me very curious! If x is cubed, does that still make that equation (xcubed+5x-8) a quadratic equation? I know you need your x to be squared for it to be considered a quadratic, but is there an exception?
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
why does numberphile always use brown paper and a sharpie? Why not copy paper and a pen?
They use them for style points.
cred
For artistic differentiation purposes.
Is recicled paper:
·
Tree hungers maybe? XD
Walter Daniels Vi Hart use notebook paper and Sharpies, but uses lots of notebook paper and can film closer. You supposedly get more larger type using brown paper, plus its cheaper.
I hate the way he writes x.
When a mathematician throws out graph paper and notebook paper, replacing it with paper from a 1 meter wide roll, you know that you will soon be confused.
So...from a simple dream I had...it evolved into a lot more. I came up with the equation n*(xn-(n-1)/2) If n=5, x=1, y = 40. :) 40 also equals the nth triangle number at base n=5 plus n squared. Apparently this also means that (n + square root of -y at x=1) * (n - square root of -y at x=1) is equivalent to n*(xn-(n-1)/2)...which is 40. 15 is the nth triangle number at 5, 25 is n squared at 5, and 40 is the second y= value in the base n=5 chain where the first value is 15 the nth triangular number, the second is 40 which is 1n squared plus the nth triangular number, and each proceeding number in the chain adds an additional n squared to the nth triangular number to form a proceeding y= value in the chain. I've taken the y= values of n=1 though n=10 out to y= 500 and noticed some interesting things. I plan on charting them all, as well as organizing a paper around them. I am beyond an amateur mathematician who has nearly hated math all my life, but I've unfortunately become obsessed by this one particular equation...which has branched off into many different ideas in basic algebra and geometry... :) Im sure all of it is already known, but its still awesome none the less...