I feel many people miss the elegance of this video. Firstly, he motivates the reasons for expanding the set of numbers. Then, after a quick introduction to the term "root", he gives an informal proof of the Intermediate Value Theorem. This gives people an idea of how mathematicians prove theorems and informally introduces quite advanced concepts such as continuity. All of this leads smoothly into an extremely hard and intriguing question which allows him to introduce Dedekind Cuts (another advanced concept) without ever getting entangled with the formalities of mathematics. Although these topics probably seem like old news for many, this (should) make you realize that a first-year calculus course is not sufficient to understand mathematics. The underlying theorems are very complex and not easy at all. I challenge you to dive into the pure analysis branch and not get hopelessly stuck. Yes this video is elementary, but why people use it as criticism is beyond me.
And, if they can’t, they love writing the word: ”Conjecture”. Also; if the theorem is fancy enough, they love writing the words: ”The Fundamental Theorem of [x]”. 😅
There are times that I watch these video and they're so over my head or they're so 'deep' that I can't get interested at a point in time. This is one of them. When this came out over a year ago, I watched this, but it flew over my head and I didn't get it. I didn't find it interesting at the time. But now that I've watched this just now at work, and im revisiting the material when my mind, is at it's sharpest with numbers(at the office as opposed to laying in bed with the tablet), i find it very enjoyable. My mind is blown and interested. I'm pretty sure that I've watched all of the +Numberphile Videos. I love the replay value. Thanks for making these videos.
I found in school (I'm now 71 yr old) that I often didn't understand new concepts the first time they were presented. I also found that I usually understood what was going on after the second or third time I saw them. I also found that often I would be stumped with a problem at work, and while taking my shower at night I would figure out what was going on. That is one of the ways our brains work, they need time to make new connections.
So many know-it-alls in the comments in these videos. Not everybody knows everything about mathematics and you don't have to be a mathematician to watch numberphile. This video is very good at explaining equations
This is high school math. The average person should know this stuff. So, yes, I expect there to be a lot of "know-it-all's" on a video about mathematics.
Still, I think you come very close :) Dedekind and so on...speaking as a math hobbyist whose Knowledge of Everything has steadily decreased since high school, there are so many delicious rabbit holes.
You guys have an amazing talent for illustrating the concepts graphically =D whenever such an animation is used it really helps me get my head around the idea instead of mulling over text books. Well done, please keep the videos coming!
The most appealing thing about Numberphile, other than trying to condense a complicated idea (this is not one) into a simple discussion, is the overall geniality and goodwill of the folks who agree to be on these videos. Numberphile really does a nice job of showing not just how beautiful and interesting mathematical ideas are, but how human notions of beauty and interest can be shared through mathematics. Probably the purest expression of human spirit imaginable.
Some people in these comments are really up themselves, complaining about how the video wasn't intellectual enough for them, like math should be reserved for the academic elite. I see other people commenting that this is maths for 12-14 year olds, when it clearly isn't. This kind of stuff doesn't get taught until A-levels in the UK. Some people on here need to get off their high horse and stop kidding themselves.
But... I’m 14 and I’m learning this stuff this year, so... I do see your point, though. Some people are flat out rude. Honestly, I’ve just had to get used to it.
It doesn’t matter what age group this math is designed for. even if it was for 6 year olds, numberphile would make it interesting. Besides, the way they interact with the math could not be taught to the average 14 year old.
This stuff is actually really deep. People who thought this video was too simple are missing out on some important details. Questions you might want to ask: - how do you know that functions that start off negative and then become positive must pass the x axis? - how would you prove this? - why does the rationals have holes but the real line does not? - how does the definition of a dedekind cut fill all the holes in the real line? - is the dedekind cut the only way to "complete" the rational numbers? - now that we know odd polynomials always have a root, how would you prove even polynomials don't have a root?
First let’s define what they mean by “hole” in this case. Roughly speaking, a hole means: “An empty region with a begging (lower bound) value and an end (upper bound) value on the number line inside of which there are no numbers.” Imagine there is a hole in the real numbers between a and e with lower bound b greater than a but less than upper bound d that is less than e. That is to say that a
Even polynomials can have real roots ex x^2-4=0 has 2 real roots, x=-2 and x=2 And x^2=0 has one real root, x=0 BUT Unlike odd polynomials, even polynomials can also have no real roots. E.x. x^2=-1, since m the square root of -1 is an imaginary number, this polynomial has no real roots. He is showing proof that, unlike even polynomials, odd polynomials must always have AT LEAS ONE real root.
Okay I watched the rest of the video and now I understand what a dedekind cut is. I’ll try my best to explain it, but can’t make promises, since it’s my first time encountering it. So rational numbers repeat. For example 1/3=0.3333 and the 3s repeat. Or 1/6 is 0.166666 and the 6s repeat. Irrational numbers do not repeat. Suppose I have an irrational number R, then I can never find the closest rational number Qa that is bigger than R. I can always find a rational number Qa1 bigger than R and smaller than Qa, and I can find a rational number Qa2 bigger than R and smaller than Qa1, so far and so forth. The same applies to the closest rational number Qb smaller than R, it cannot be the closest, for the same reason that I can always find a closer number Qb1 bigger than Qb and smaller than R, so on so forth. So he is saying that he can define the irrational number R as the set of all rational numbers {Qb, Qb1, Qb2,...} less than R and all rational numbers {Qa, Qa1, Qa2,...}. As you can see, since Are does not repeat, we expect to find that the set of these rationals that define R has an infinite size (meaning there are an infinite number of these rationals to the left and to the right of R). I hope this helps. This is the best that I can explain what I understood of something that I learned for the first time.
This video was just perfect cuz i was literally studying about dedekind cuts in real analysis just right now, i never thought this video would veer into the path of defining real numbers
Reading the comments below, in which people have commented that they learned all this before 15 years old, I suddenly feel stupid. I didn't learn any of that until I was in my twenties. And I was still barely grasping it. Great video though!
I felt that he should have mentioned that his proof not only relies on f being continuous on that interval, but that it also appeals to the Intermediate Value Theorem. Those commenters that call this "basic algebra" are missing powerful mathematical concepts within this example that are learned in proof-based math classes.
nearly 110,000 in just 9 days! ... I'm really glad to see there are so many math-lovers out there. Thanks for your channel my friend, it's just like a breath of air to watch most of your videos. from Iran (and Sweden) with love A.M.G.
Well, I wasn't expecting to hear about Dedekind cuts when I started watching this video. There was quite a leap from polynomial roots to a way of defining the real numbers. I liked it.
Yoyoyo, hold up, can anyone explain to me how this dude, in 4 min, managed to explain what a teacher used a month to try and explain to my class at school (and failed miserably at that)._.
Mellow Tortoise - I was thinking the same thing... I think it has to do with *Prof. Eisenbud's* passion for the subject... the maths teachers I had never even came close to caring about even trying to get us excited, which is a real shame :(
Mellow Tortoise Probably something to do with him being *Prof* Eisenbud of the Mathematical Sciences Institute and not Mr. Doughnacare at Whatchamacalit High
I'm really happy that I actually could see and follow along were this was going after taking calc this past semester! It's a good feeling to be able to keep up during an episode!!
Why is it that when I hear something like: _an odd function must ALWAYS have at least 1 root_ I immediately try to think of a counterexample. Even after the proof has been shown, I'm still thinking "there's got to be an exception, there's just got to be!"
I wish Numberphile had a video that would help give a more intuitive understanding of what Curl and Divergence is. I recently finished my Calc III class and I understand the formulas and how to computer Curl and Divergence, but I often have a hard time understanding intuitively what they mean. They should get James to do the video! I love when he talks about advanced mathematics!
Every odd-degree function must cross the x-axis, because they can only have an even number of turning points (points where the function's slope changes signs). Since the number of turning points must be even, the function will always have a negative-to-positive end behaviour, or a positive-to-negative end behaviour. As a result, the function must cross the x-axis, in order to achieve that end behaviour.
To everyone complaining about the (lack of) complexity in this video: What Prof Eisenbud states starting 10:52 is by no means trivial! So even though many people are already well acquainted with polynomial equations (which by the way play a fundamental role in many areas such as Elliptic Curves), what he says at the end of the video is quite deep. Bear in mind, math is not just about theorems and proofs, *definitions* are crucial: why does one define certain mathematical object in a given way? This truly is an art.
not this way. it goes: If there is a hole, put something in. if there still holes, put more in.... in the end we have no holes. if there ar still holes, we are not at the end--> put somthing in the holes
For those don't understand **why** mathematicians "Dedekind-cut" real number: If you "define" a real number by a Dedekind-cut, then the property "A non-empty set of real numbers, say X, has a upper bound, will have a least upper bound" can be proved, without any advanced idea(i.e. Cauchy sequences), just set operation! And everything about real number can also be proved easily. i.e. Let [a,b) be such a set, then pick-up a point c, which is larger then every element is that set(in short, b < c), then you can pretty sure that the set has a least upper bound exists (it's b in this case)!
I for one enjoyed this video. I feel like we can get a little carried away with what we believe to be our vast understanding of numbers and mathematics. We (as individuals, not as the whole of humanity) think because we can do integrals and derivatives and have been doing basic functions for so long that it's no longer important to talk about them. But going back to a more basic area of mathematics and learning about it now, after having much more experience in the field really gives you a deeper understand of not only the more basic concepts, but the more advanced as well, and ultimately mathematics as a whole.
At 2:52, he says that f(x) = x^2+1= y but when you substitute in 2 for x, you receive f(2) = 2^2+1=y f(2) = 4+1=y f(2) = 5=y Therefore, when 2 is your x value, you will receive 5 as your y value. P.S. I'm only 14, but I enjoy mathematics a lot. :)
One of the delights of explaining maths is the endless variety of analogies and imagery that arise: e.g. the x-axis as the Mexican border. Thanks Numberphile - I always gain something from your videos - keep up the good work
I hate comments like this. Guess what? It's because you're not paying attention to your teacher. Maybe if you tried to engage in class you would actually learn something. I learned all the principles of this in school, and I sincerely doubt the education system has changed that much in 10 years.
More from Prof David Eisenbud! What he said was very clear to me, higher degrade odd x will rule the behavior of the function. "One x to rule them all"
This is the really cool part of math that I try to sell to people. Once you get advanced enough, you can start defining things to be whatever the heck you need it to be! So much fun :)
For me the key insight for rationals versus reals is that, it takes a finite number of steps to construct a rational, but an infinite number of steps to construct most reals. Dedekind builds a real number by stepping through an infinite sequence of rational cuts to close in on the real. Example: square root of 2: ...1|2... --> ...1|1.5... --> ...1.25|1.5... --> ...1.375|1.5... --> (infinite steps) --> all rationals less than root 2|all rationals greater than root 2.
Nice vid about a standard result - doesn't deeply need calculus, but a full proof needs the Intermediate Value theorem - which the good Prof. explains perfectly well anyway: Minor typo's/quibbles: At 3:25, Green "Solutions" overlap - ugh - that's ugly! At 6:54: "postive" should be "positive" Anyway, Brady, great stuff!!! Very, very welcome ...
Actually, the first construction of "Real Number set", is due to Méray based on Cauchy's sequences (I do not know if this is call this way in English :/) in 1869. Dedekind's construction was in 1872. The concept of Cauchy's sequences is a great way to understand what "real numbers set" is , because it's just a litteral way to "fill" the rational numbers set. However, this is a very interesting video and clearly explained :)
I wish I had had a maths teacher with [rational numbers less than x] x% [rational numbers greater than x] of the enthusiasm *Professor Eisenbud* has for the subject...
For those of you wondering about the negative one thing... Natural numbers (1, 2, 3,) Whole numbers (0,1,2,3) Integers (-2,-1,0,1,2) rational numbers (1/2, -3/5, ect) This is what is taught in the U.S. school system, not sure about elsewhere
I didn't thing I'd learn anything with this vid after 3 minutes but Dedekind cuts was very interesting as it basically means that in-between two irrational numbers there is always a rational number... A video about this would be interesting.
Tbh, last summer when I watched this channel, I didn't really understand most of it. Now that I am studying mathematics, this is actually pretty simple to comprehend :D
Just listened to his Numberphile Podcast interview.. I just love the guy. I just love maths a loooooooot more. I just wish my 7th grade teacher would have shown me that as clearly as he does!
Beauty of Maths is you have to know some basics and you can take part in any mathematical discussion. Extent of your contribution depends on how far your knowledge goes .
When you substitute x with -1 in the video, you're forgetting parentheses. So you end up writing -1^2 = 1 which is wrong. (-1)^2 = 1 is right. This is due to order of operations. Powers before subtraction. Hence -1^2 = -(1^2) = -1
don't want to sound annoying but when around 1:20 you say that f(x) is a function, it is not. f is a function. f(x) is a complex. So saying that f(x) has a root makes no sense. f has a root, but f(x) is a complex, let's say a real like 10 or 20. Saying that 10 or 20 has a root is non sense. Still, I just discovered the channel, and I love it ! Keep up with the good work.
What a horrible coincidence. I live in the UK and just had my Core 3 exam this morning, and this video comes out a couple hours after. It made it so easy to understand if only I knew for this morning :(
Well shit, you can't just leave us hanging there. You explained why the line must cross zero by depending on a proof you fully didn't explain! And it sounds even more fascinating than the question at hand...
You'd have to look at the proof of Bolzano's Theorem, which states that if a function is continuous on [a,b], and a and b have opposite signs, then there is a number c in (a,b) such that f(c)=0. This is a corollary of the Intermediate Value Theorem.
Easily shown using Lagrange's theorem... if you know that a function is "smooth" (It can be derived infinite times) then given 2 values of the function, you have to cross all the values in-between them.
At 2:54 he says that when x=1 then f(x)=2. That's correct. But when he says that when x=2 then f(x)=4, I believe that is wrong. 2^2+1=4+1=5. Correct me if I'm wrong. Not trying to devalue this video, just trying to help.
The counter example which everyone is looking for, which may have gone by a few times, is the case of f(x)=x^-1. The greatest power is odd. Negative integers are defined as odd or even. 9 cents debt split into two parts is as odd as 9 cents profit.
His proof specifically mentioned that any POLYNOMIAL function of odd degree has 1 root. and by definition a polynomial function cannot have a negative power. That would classify as a rational function.
Thank you genius, it is obvious that if the highest power of a polynom is odd, we can find at least one root ... Proof done in one phrase : If the highest power of P a polynom is odd, its +or-infinity limit is respectively +or- infinity. And as the polynom is continuous (even got the Cinfinity class), with the intermediary values theorem, it has at least one root ...
Something that can be added, as you go through more and more "exotic" numbers, by doing things to complicate your equations. When you get to where there are no real solutions anymore, you need to "invent" complex numbers. Then if you allow the polynomial's coefficients to be complex, you get no new kinds of numbers - all zeros of complex polynomials, are themselves found in the complex numbers. Fred
I am lost in Hilbert's Real Number Hotel where the rooms have rational AND irrational room numbers, and the hotel therefore includes an infinite uncountable collection of transcendental room numbers. Since the rooms are "uncountable," I am having the devil of a time finding my room. In fact, the number of digits in my room number is infinite. I suspect my room number is transcendental. Help! :0) The employees of the hotel are almost as lost as I am. The members of the Ethics Committee here all have transcendental room numbers, and I can't locate them either.
Want to know a really fast way of explaining this proof? Take the derivative of the odd power polynomial, the highest power is now even, this means the slope of this function for large values of x and -x will have the SAME SIGN. At, say, x=-1000 the slope is positive and at x=1000 the slope is also positive, try to visualize that, it means the graph is heading off to -infinity to the left and +infinity to the right, so it HAS to CROSS the x-axis somewhere. Hence, that is where the real root has to be.
Its funny that Im on complex analysis and I had a problem in witch an odd polynomial was given and 2 complex solutions too, than I had to prove that there must be a third one, and I was aware of the fundamental theorem of algebra but that wasnt enough, so I had this problen in my head for like one week until I had the class in witch the teacher just said ''rememmber Bolzano's theorem'' and suddently a light poped up in my head and I resolved the problem, one week later here I am watching a Numberphile video explaining that, thank you youtube
2:07 but any root in itself is both positive and negative. Sqrt of 2 and negative sqrt of 2 is the same thing there is no need for a negative in front of the root. (Sqrt2)^2 = 2 or -2 (-Sqrt2)^2 = -2 or 2 I think that shows a lack of understanding of mathematics to write it like that am I wrong? Please correct me if I’m wrong
Catch David on the Numberphile podcast: th-cam.com/video/9y1BGvnTyQA/w-d-xo.html
6:32 seems like the limit as x ➡️+/- ♾ plus proof of continuity proves the single root theorem of an Odd powered polynomial.
I feel many people miss the elegance of this video. Firstly, he motivates the reasons for expanding the set of numbers. Then, after a quick introduction to the term "root", he gives an informal proof of the Intermediate Value Theorem. This gives people an idea of how mathematicians prove theorems and informally introduces quite advanced concepts such as continuity. All of this leads smoothly into an extremely hard and intriguing question which allows him to introduce Dedekind Cuts (another advanced concept) without ever getting entangled with the formalities of mathematics.
Although these topics probably seem like old news for many, this (should) make you realize that a first-year calculus course is not sufficient to understand mathematics. The underlying theorems are very complex and not easy at all. I challenge you to dive into the pure analysis branch and not get hopelessly stuck.
Yes this video is elementary, but why people use it as criticism is beyond me.
thanks
i think it was the bolzano theorem (i think we are talking about the same theorem)
"Mathematicians love writing the word theorem, especially when they can prove what they're about to write" damn boy you got me there
And, if they can’t, they love writing the word: ”Conjecture”. Also; if the theorem is fancy enough, they love writing the words: ”The Fundamental Theorem of [x]”. 😅
The best joke. "You can't cross from Mexico to Canada, without crossing the U.S."
"… or you could dig a tunnel."
How about sailing around the U.S.?
@@josbertlonnee fly over it
@@chandlerlocklear3854 See above:
KepleroGT
2 months ago
Technically the land under the US is US soil as well
@@josbertlonnee yeah true I guess but couldnt you then go into space?
Round the globe in the Number space.
David Eisenbud is such a gift.
No matter the subject, it's always a delight hearing his explanation.
Saruman was a mathematician?
Damn that's made my day :D lol
Győri Sándor *mathimagician
There are times that I watch these video and they're so over my head or they're so 'deep' that I can't get interested at a point in time. This is one of them. When this came out over a year ago, I watched this, but it flew over my head and I didn't get it. I didn't find it interesting at the time. But now that I've watched this just now at work, and im revisiting the material when my mind, is at it's sharpest with numbers(at the office as opposed to laying in bed with the tablet), i find it very enjoyable. My mind is blown and interested. I'm pretty sure that I've watched all of the +Numberphile Videos. I love the replay value. Thanks for making these videos.
I found in school (I'm now 71 yr old) that I often didn't understand new concepts the first time they were presented. I also found that I usually understood what was going on after the second or third time I saw them. I also found that often I would be stumped with a problem at work, and while taking my shower at night I would figure out what was going on. That is one of the ways our brains work, they need time to make new connections.
So many know-it-alls in the comments in these videos. Not everybody knows everything about mathematics and you don't have to be a mathematician to watch numberphile. This video is very good at explaining equations
everyone thinks the TH-cam videos they watch are made just for them - it is hard to please all of them at once! :)
Yeah i'm no mathmatician i just seek better understanding of things everyone has to start somewhere.
This is high school math. The average person should know this stuff. So, yes, I expect there to be a lot of "know-it-all's" on a video about mathematics.
Still, I think you come very close :) Dedekind and so on...speaking as a math hobbyist whose Knowledge of Everything has steadily decreased since high school, there are so many delicious rabbit holes.
gcbound Having a mentality of knowing everything does lead to being dumber though. Modesty and thirst for knowledge shouldn't stop in ones life.
What a really nice guy this seems to be. I love his voice and his way of explaining. It's so calm and reserved! Great video!
You guys have an amazing talent for illustrating the concepts graphically =D
whenever such an animation is used it really helps me get my head around the idea instead of mulling over text books.
Well done, please keep the videos coming!
The most appealing thing about Numberphile, other than trying to condense a complicated idea (this is not one) into a simple discussion, is the overall geniality and goodwill of the folks who agree to be on these videos. Numberphile really does a nice job of showing not just how beautiful and interesting mathematical ideas are, but how human notions of beauty and interest can be shared through mathematics. Probably the purest expression of human spirit imaginable.
Some people in these comments are really up themselves, complaining about how the video wasn't intellectual enough for them, like math should be reserved for the academic elite. I see other people commenting that this is maths for 12-14 year olds, when it clearly isn't. This kind of stuff doesn't get taught until A-levels in the UK. Some people on here need to get off their high horse and stop kidding themselves.
Tom Watkins What are "A-levels". I'm from America and don't know anything about the U.K. academic system. I'm also fairly curious.
A Tr A-levels is a pre-university programme
Tom Watkins for me it was 12, but my class was the major exception, we have a kid doing college math.
But... I’m 14 and I’m learning this stuff this year, so...
I do see your point, though. Some people are flat out rude. Honestly, I’ve just had to get used to it.
It doesn’t matter what age group this math is designed for. even if it was for 6 year olds, numberphile would make it interesting. Besides, the way they interact with the math could not be taught to the average 14 year old.
It's incredible how intuitive David Eisenbud makes this. This is somebody who just wants to share what he has found with everybody.
That man has such a relaxing voice! I could listen to him explaining stuff all day long! Please do more videos with him, that'd be nice!
This stuff is actually really deep. People who thought this video was too simple are missing out on some important details.
Questions you might want to ask:
- how do you know that functions that start off negative and then become positive must pass the x axis?
- how would you prove this?
- why does the rationals have holes but the real line does not?
- how does the definition of a dedekind cut fill all the holes in the real line?
- is the dedekind cut the only way to "complete" the rational numbers?
- now that we know odd polynomials always have a root, how would you prove even polynomials don't have a root?
First let’s define what they mean by “hole” in this case. Roughly speaking, a hole means:
“An empty region with a begging (lower bound) value and an end (upper bound) value on the number line inside of which there are no numbers.”
Imagine there is a hole in the real numbers between a and e with lower bound b greater than a but less than upper bound d that is less than e.
That is to say that a
Even polynomials can have real roots ex x^2-4=0 has 2 real roots, x=-2 and x=2
And x^2=0 has one real root, x=0
BUT
Unlike odd polynomials, even polynomials can also have no real roots.
E.x. x^2=-1, since m the square root of -1 is an imaginary number, this polynomial has no real roots.
He is showing proof that, unlike even polynomials, odd polynomials must always have AT LEAS ONE real root.
Okay I watched the rest of the video and now I understand what a dedekind cut is. I’ll try my best to explain it, but can’t make promises, since it’s my first time encountering it.
So rational numbers repeat. For example 1/3=0.3333 and the 3s repeat. Or 1/6 is 0.166666 and the 6s repeat.
Irrational numbers do not repeat.
Suppose I have an irrational number R, then I can never find the closest rational number Qa that is bigger than R. I can always find a rational number Qa1 bigger than R and smaller than Qa, and I can find a rational number Qa2 bigger than R and smaller than Qa1, so far and so forth. The same applies to the closest rational number Qb smaller than R, it cannot be the closest, for the same reason that I can always find a closer number Qb1 bigger than Qb and smaller than R, so on so forth.
So he is saying that he can define the irrational number R as the set of all rational numbers {Qb, Qb1, Qb2,...} less than R and all rational numbers {Qa, Qa1, Qa2,...}. As you can see, since Are does not repeat, we expect to find that the set of these rationals that define R has an infinite size (meaning there are an infinite number of these rationals to the left and to the right of R).
I hope this helps. This is the best that I can explain what I understood of something that I learned for the first time.
Really good explanations and how he spins on what Brady say is really thought provoking. I would love to see more videos with prof David Eisenbud
That guy has a very soothing voice. Wish he taught my math classes :)
1:48 seeing someone write -1^2 without brackets just kills my soul.
-1²
@@screamsinrussian5773 No please
@@PatentedSugar9 you cannot escape it PatentedSugar9
@[screams in Russian] −1²
@@nousername5673 yes
its power grows
This video was just perfect cuz i was literally studying about dedekind cuts in real analysis just right now, i never thought this video would veer into the path of defining real numbers
This increased my intuïtion for imagining odd polynomials more than 20 math lessons I had in high school combined. Thanks :)
Scientists are the most interesting people in the world.
True dat
+Lau Bjerno Well,after Dos Equis pitchman they are.
And/because they are the most interested
Mathematicians to be exact, the term “scientists” are kinda broad in a sense.
@@x_gosie no. Mathematicians are no more interesting than chemists, physicists, zoologists, mycologists, historians etc.
Reading the comments below, in which people have commented that they learned all this before 15 years old, I suddenly feel stupid. I didn't learn any of that until I was in my twenties. And I was still barely grasping it. Great video though!
I felt that he should have mentioned that his proof not only relies on f being continuous on that interval, but that it also appeals to the Intermediate Value Theorem. Those commenters that call this "basic algebra" are missing powerful mathematical concepts within this example that are learned in proof-based math classes.
Nicholas Kent He mentioned that the proof relies on f being continuous--that's the "second point of the proof" he mentions at 9:00.
nearly 110,000 in just 9 days! ... I'm really glad to see there are so many math-lovers out there. Thanks for your channel my friend, it's just like a breath of air to watch most of your videos.
from Iran (and Sweden) with love
A.M.G.
Well, I wasn't expecting to hear about Dedekind cuts when I started watching this video. There was quite a leap from polynomial roots to a way of defining the real numbers. I liked it.
Yoyoyo, hold up, can anyone explain to me how this dude, in 4 min, managed to explain what a teacher used a month to try and explain to my class at school (and failed miserably at that)._.
The key might be in the audience.
Are you in high school by any chance?
Mellow Tortoise - I was thinking the same thing... I think it has to do with *Prof. Eisenbud's* passion for the subject... the maths teachers I had never even came close to caring about even trying to get us excited, which is a real shame :(
Mellow Tortoise I know!
Mellow Tortoise
Probably something to do with him being *Prof* Eisenbud of the Mathematical Sciences Institute and not Mr. Doughnacare at Whatchamacalit High
Thank you Brady and David! I really enjoyed that :)
you're welcome
I'm really happy that I actually could see and follow along were this was going after taking calc this past semester! It's a good feeling to be able to keep up during an episode!!
At 2:52 2^2+1 does not equal 4.
yes, you followed the principle he was explaining though, right?
Numberphile of course, I just love to nitpick. Great video
Numberphile hmm i don't get what you are saying there^^... he is right f(2)=5 with f(x)=x^2+1; in the viodeo it is 4. But only a minor flaw^^.
I wish I was just as smart as you are!
gotem
Why is it that when I hear something like:
_an odd function must ALWAYS have at least 1 root_
I immediately try to think of a counterexample.
Even after the proof has been shown, I'm still thinking "there's got to be an exception, there's just got to be!"
+AlanKey86 Comments not loading for me but has anyone said (√-x)^3 +1?
+Jim Giant ur equation is equal to (-x)^1.5 +1=0 because
(√-x)^3=((-x)^1/2)^3)=(-x)^3/2 and the power here is not an odd number
+AlanKey86 Actually all cubics have exactly 3 and there are no exceptions for this. Sorry bud. No 'at least'. EXACTLY 3. No more, no less
+Zachary Mitchell What? That's not true. There's an example of a cubic equation with only one real root in this video...
I never said the roots had to be real.
these videos with David are a joy
I wish Numberphile had a video that would help give a more intuitive understanding of what Curl and Divergence is. I recently finished my Calc III class and I understand the formulas and how to computer Curl and Divergence, but I often have a hard time understanding intuitively what they mean. They should get James to do the video! I love when he talks about advanced mathematics!
Sounds like something for 3B1B or a similar channel. Numberphile rarely gets into calculus beyond just dipping your toes into it.
Oh man that ending conclusion had me mind blown, everything until then I was like okay just another math class.
“It’s worse than that..” he said 😭😭😭
This man is the best mathematics teacher I have seen.
It's a pleasure to listen to professor Eisenbud talking
Always enjoying his voice and his explanations.
This takes me back to those good ol' days of calculus. Ahh the headaches and traumatic experiences I'll never forget(:
Every odd-degree function must cross the x-axis, because they can only have an even number of turning points (points where the function's slope changes signs). Since the number of turning points must be even, the function will always have a negative-to-positive end behaviour, or a positive-to-negative end behaviour. As a result, the function must cross the x-axis, in order to achieve that end behaviour.
By what theorem does function with an odd degree have to have an even number of turning points?
It could also have no turning points (0 can be even if you want), but it will then have a point of inflection instead.
That is using "higher" math, then the proof in the video...
Элемент Магии See, he can't tell.
what about 1/x ? thats an odd function that doesn't cross the x-axis
Finally an episode explaining Dedekind cuts. Came across the concept back in grade 11. Couldn't understand til now.
To everyone complaining about the (lack of) complexity in this video:
What Prof Eisenbud states starting 10:52 is by no means trivial! So even though many people are already well acquainted with polynomial equations (which by the way play a fundamental role in many areas such as Elliptic Curves), what he says at the end of the video is quite deep.
Bear in mind, math is not just about theorems and proofs, *definitions* are crucial: why does one define certain mathematical object in a given way? This truly is an art.
2:52 When x is 2, f(x) is actually 5. So the coordinates are (2,5) and *not* (2,4).
I really like this speaker Brady! He gives very good explanations while alluding to more advanced topics. Definitely interview him again!
I just love mathematics: "I define the hole does not exist. q.e.d" :D
not this way. it goes: If there is a hole, put something in. if there still holes, put more in.... in the end we have no holes. if there ar still holes, we are not at the end--> put somthing in the holes
@Tarsonis42
It doesn't go that way, though. But alright, I'll take the joke nonetheless...
i enjoyed reading your comment, tarsonis, don’t listen to da haters
For those don't understand **why** mathematicians "Dedekind-cut" real number: If you "define" a real number by a Dedekind-cut, then the property "A non-empty set of real numbers, say X, has a upper bound, will have a least upper bound" can be proved, without any advanced idea(i.e. Cauchy sequences), just set operation! And everything about real number can also be proved easily. i.e.
Let [a,b) be such a set, then pick-up a point c, which is larger then every element is that set(in short, b < c), then you can pretty sure that the set has a least upper bound exists (it's b in this case)!
I for one enjoyed this video. I feel like we can get a little carried away with what we believe to be our vast understanding of numbers and mathematics. We (as individuals, not as the whole of humanity) think because we can do integrals and derivatives and have been doing basic functions for so long that it's no longer important to talk about them. But going back to a more basic area of mathematics and learning about it now, after having much more experience in the field really gives you a deeper understand of not only the more basic concepts, but the more advanced as well, and ultimately mathematics as a whole.
At 2:52, he says that f(x) = x^2+1= y but when you substitute in 2 for x, you receive f(2) = 2^2+1=y
f(2) = 4+1=y
f(2) = 5=y
Therefore, when 2 is your x value, you will receive 5 as your y value.
P.S. I'm only 14, but I enjoy mathematics a lot. :)
One of the delights of explaining maths is the endless variety of analogies and imagery that arise: e.g. the x-axis as the Mexican border. Thanks Numberphile - I always gain something from your videos - keep up the good work
if it's "too easy" for you then just leave, this is great stuff i loved it
01:46 The second solution is (-1) because -1²=-1 because it equals -(1²)
And (-1)²=1 because it equals (-1)×(-1).
God bless you Brady. for bring us these gems.
A month of maths class in school to learn pretty much the same as in this 13 minute video. I applause education.
You should really be "applauding" it
I know your pain
Higgs Boson Sorry, XD
Joroba 3 I'm in a math class and we didnt even learn this..
I hate comments like this. Guess what? It's because you're not paying attention to your teacher. Maybe if you tried to engage in class you would actually learn something. I learned all the principles of this in school, and I sincerely doubt the education system has changed that much in 10 years.
Slight error at 2:52: if f(x) = x^2 + 1, then f(2) = 5, not 4.
Ruined the whole video for me /s
Thank y'all so much. I love your videos; especially the ones with Prof. Eisenbud.
2:38 Use complex numbers. For example square root -1i
More from Prof David Eisenbud!
What he said was very clear to me, higher degrade odd x will rule the behavior of the function. "One x to rule them all"
9:52 "...or you could dig a tunnel"
Genius way to counter a border wall!
This is the really cool part of math that I try to sell to people. Once you get advanced enough, you can start defining things to be whatever the heck you need it to be! So much fun :)
For me the key insight for rationals versus reals is that, it takes a finite number of steps to construct a rational, but an infinite number of steps to construct most reals. Dedekind builds a real number by stepping through an infinite sequence of rational cuts to close in on the real.
Example: square root of 2:
...1|2... --> ...1|1.5... --> ...1.25|1.5... --> ...1.375|1.5... -->
(infinite steps) --> all rationals less than root 2|all rationals greater than root 2.
As a freshman in college this material was the most mind-blowing stuff in the universe for me. That was a long time ago.
Nice vid about a standard result - doesn't deeply need calculus, but a full proof needs the Intermediate Value theorem - which the good Prof. explains perfectly well anyway:
Minor typo's/quibbles:
At 3:25, Green "Solutions" overlap - ugh - that's ugly!
At 6:54: "postive" should be "positive"
Anyway, Brady, great stuff!!! Very, very welcome ...
Actually, the first construction of "Real Number set", is due to Méray based on Cauchy's sequences (I do not know if this is call this way in English :/) in 1869. Dedekind's construction was in 1872.
The concept of Cauchy's sequences is a great way to understand what "real numbers set" is , because it's just a litteral way to "fill" the rational numbers set.
However, this is a very interesting video and clearly explained :)
I wish I had had a maths teacher with [rational numbers less than x] x% [rational numbers greater than x] of the enthusiasm *Professor Eisenbud* has for the subject...
I never expected to see something on Numberphile, that I already had in school. :)
For those of you wondering about the negative one thing... Natural numbers (1, 2, 3,) Whole numbers (0,1,2,3) Integers (-2,-1,0,1,2) rational numbers (1/2, -3/5, ect) This is what is taught in the U.S. school system, not sure about elsewhere
I didn't thing I'd learn anything with this vid after 3 minutes but Dedekind cuts was very interesting as it basically means that in-between two irrational numbers there is always a rational number... A video about this would be interesting.
I absolutely love this guy. His words drop into my mind like pebbles into a clear pool.
Tbh, last summer when I watched this channel, I didn't really understand most of it. Now that I am studying mathematics, this is actually pretty simple to comprehend :D
This is a beautiful display of function transformation.
Props on the thumbnail Brady. That shit was golden.
Just listened to his Numberphile Podcast interview.. I just love the guy. I just love maths a loooooooot more. I just wish my 7th grade teacher would have shown me that as clearly as he does!
Beauty of Maths is you have to know some basics and you can take part in any mathematical discussion. Extent of your contribution depends on how far your knowledge goes .
8:08 the correct way to write is x>>1, not zero, because we can assume any (pozitive non-zero) number is infinitely larger than zero
When you substitute x with -1 in the video, you're forgetting parentheses. So you end up writing -1^2 = 1 which is wrong. (-1)^2 = 1 is right.
This is due to order of operations. Powers before subtraction. Hence -1^2 = -(1^2) = -1
A minus ain't squared, unless it's been snared.
8:39 Astronomers would laugh, "You think that number is big?"
Jordan Fischer Then mathematicians will laugh because astronomers are confined to reality, whereas their numbers know no limits.
Jordan Fischer :)
A sect of Astronomy is cosmology.
Do you think cosmology of the Early Universe concerns reality?
Moregasm the Powerful Yes.
Have you studied Early Universe Cosmology?
don't want to sound annoying but when around 1:20 you say that f(x) is a function, it is not. f is a function. f(x) is a complex. So saying that f(x) has a root makes no sense. f has a root, but f(x) is a complex, let's say a real like 10 or 20. Saying that 10 or 20 has a root is non sense. Still, I just discovered the channel, and I love it ! Keep up with the good work.
I never thought about it like this before. Great episode.
"So let's start with some algebra."
A common nightmare for the average teenager.
I guess I'm no average teenager then.
Why a nightmare? We all know H*W=A easy :-) Height * Width = Area
I'm a tween and I take Algebra 1
I took algebra as a 12 year old. It was decently easy for me
I took it at 11
What a horrible coincidence. I live in the UK and just had my Core 3 exam this morning, and this video comes out a couple hours after. It made it so easy to understand if only I knew for this morning :(
I like this guy, please make more videos with him.
I'll try!
There is at least one more to come, where he expands on what we just discussed here and takes it to the next level!
This professor is very 'watchable', explains it well, nice video enjoyed it, looking forward to the next.
Thanks for the great video
Oh by the way at 1:46 i think you may forgot the parentheses
at 2:12, -sqrt(2)^2 does not equal 2, but (-sqrt(2))^2 does.
A 30-second rewind button on the TH-cam player would be a beautiful thing.
Well shit, you can't just leave us hanging there. You explained why the line must cross zero by depending on a proof you fully didn't explain! And it sounds even more fascinating than the question at hand...
You'd have to look at the proof of
Bolzano's Theorem, which states that if a function is continuous on [a,b], and a and b have opposite signs, then there is a number c in (a,b) such that f(c)=0. This is a corollary of the Intermediate Value Theorem.
Easily shown using Lagrange's theorem... if you know that a function is "smooth" (It can be derived infinite times) then given 2 values of the function, you have to cross all the values in-between them.
My favourite point on the video is at 10:18. "You have to decide what you think the real numbers R."
At 2:54 he says that when x=1 then f(x)=2. That's correct. But when he says that when x=2 then f(x)=4, I believe that is wrong. 2^2+1=4+1=5. Correct me if I'm wrong. Not trying to devalue this video, just trying to help.
Greetings from Mexico, we do watch your vids !
first 40 seconds is what we do in class..
the rest is what teachers ask in exams..
Take a shot every time he says "for the sake of argument"
The counter example which everyone is looking for, which may have gone by a few times, is the case of f(x)=x^-1. The greatest power is odd. Negative integers are defined as odd or even. 9 cents debt split into two parts is as odd as 9 cents profit.
His proof specifically mentioned that any POLYNOMIAL function of odd degree has 1 root. and by definition a polynomial function cannot have a negative power. That would classify as a rational function.
Thank you genius, it is obvious that if the highest power of a polynom is odd, we can find at least one root ...
Proof done in one phrase : If the highest power of P a polynom is odd, its +or-infinity limit is respectively +or- infinity. And as the polynom is continuous (even got the Cinfinity class), with the intermediary values theorem, it has at least one root ...
Something that can be added, as you go through more and more "exotic" numbers, by doing things to complicate your equations.
When you get to where there are no real solutions anymore, you need to "invent" complex numbers.
Then if you allow the polynomial's coefficients to be complex, you get no new kinds of numbers - all zeros of complex polynomials, are themselves found in the complex numbers.
Fred
I am lost in Hilbert's Real Number Hotel where the rooms have rational AND irrational room numbers, and the hotel therefore includes an infinite uncountable collection of transcendental room numbers. Since the rooms are "uncountable," I am having the devil of a time finding my room. In fact, the number of digits in my room number is infinite. I suspect my room number is transcendental. Help! :0) The employees of the hotel are almost as lost as I am. The members of the Ethics Committee here all have transcendental room numbers, and I can't locate them either.
Word Sailor Well, at least Hilbert's Real Number Hotel has Wi-Fi. :P
I find this so interesting because it makes my head hurt. I am terrible at math but I love this channel and trying to "get it "
Want to know a really fast way of explaining this proof? Take the derivative of the odd power polynomial, the highest power is now even, this means the slope of this function for large values of x and -x will have the SAME SIGN. At, say, x=-1000 the slope is positive and at x=1000 the slope is also positive, try to visualize that, it means the graph is heading off to -infinity to the left and +infinity to the right, so it HAS to CROSS the x-axis somewhere. Hence, that is where the real root has to be.
first time i hear about this dedekind definition. i know real numbers as equivalence classes of rational cauchy-sequences.
The two constructions are isomorphic :)
@@Nikifuj908 of course they are, you wouldnt want to have two definitions which yield different results.
Its funny that Im on complex analysis and I had a problem in witch an odd polynomial was given and 2 complex solutions too, than I had to prove that there must be a third one, and I was aware of the fundamental theorem of algebra but that wasnt enough, so I had this problen in my head for like one week until I had the class in witch the teacher just said ''rememmber Bolzano's theorem'' and suddently a light poped up in my head and I resolved the problem, one week later here I am watching a Numberphile video explaining that, thank you youtube
Complex solutions would be a brane (2D) and complex roots would be a line (1D) would they not?
2:07 but any root in itself is both positive and negative. Sqrt of 2 and negative sqrt of 2 is the same thing there is no need for a negative in front of the root.
(Sqrt2)^2 = 2 or -2
(-Sqrt2)^2 = -2 or 2
I think that shows a lack of understanding of mathematics to write it like that am I wrong? Please correct me if I’m wrong
You're wrong
Is the "More Polynomial Proofs" video still "Coming Soon"?