I think you summed up the problem in the first couple of seconds: "We all know what comes next, just from memory" For most people, these things need to be taught by memorization, rather than by understanding. The steps to reproduce (a+b)^2 = a^2 + 2ab + b^2 can feel daunting, and 'counterproductive' vs just remembering it. Think also of the quadraric formulat as a good example of this. This is mostly what highschool maths is based on. Knowing the steps is not required for most people to know, and remembering the steps is not something people are familiar with, so the emphasize is on memorization. One cannot really blame the system for this. A big part of studying maths is thinking differently about these concept, and realizing that it is often easer to remember the process that created the formula. Since the process is logicall, and follows simple rules (like the distrivutive property), you don't have to remember that much anymore. I hope this video has shown this for at least some people. Keep it up!
Another thing I feel is incredibly useful for many levels of math in general is Pascal's triangle. Honestly I can't be bothered to remember expansions beyond a cubic expansion and a few times when it's really come down to using the triangle, it actually has been useful! I find small quirks in math so interesting. It's odd how it just ends up the way it is.
I digress somewhat. I do not believe people *need* to be taught by memorization. Scientific studies have demonstrated repeatedly that memorization is a poor way of learning, and that it tends to only hinder the performance of people. The reason we seem inclined to think we do and learn better via memorization than by logical and critical thinking is due to societal conditioning, not due to a matter of fact. Historically, education was also memory-centered because in the past, civilizations believed memory was the only important aspect to knowledge. Now we know this is not true, but breaking old societal habits takes centuries, and we only just learned this not longer than two or three decades ago. This is part of the reason why the education system is the way it is, and why it needs significantly overhauling and updating. Mathematics need to be taught by understanding logical deduction, and by way of analogy with algorithms.
To be honest, it is much easier, if not necessarily to remember common integrals, series, theorems, definitions and proofs. Of course you could rederive everything and it would be really cool if you would not have to do it. However the short time you will have in the exam doesn't allow for this method. You have to know the answer instantly.
@@Caspar__ once you re-derive it enough times, you start remembering it automatically. After re-deriving the basic stuff a few times, memorization becomes easier. For instance, I re-derived the formula for integral of 1/(a²+x²) a few times, then I decided to remember the result for the next time and spent less than a minute on the formula. That way the amount of time dedicated to memorization was nil. Spend almost all of your time to understand and to solve. The time you spend on memorization should be extremely small.
What a unique way of deriving this formula!! I never thought of it in terms of n-length strings. It's a very combinatorial way of looking at it. It's always nice when you can prove/derive a formula in more than one way. Also, video editing is ON POINT, as always! 👍
@@l.1244 Yes, but most students are not taught what binomial coefficients or, more importantly, why they appear in a binomial expansions when they are supposed to be combinatorial objects, assuming that they ever learn their combinatorial significance to begin with, which as I said, is not taught to most students.
@@l.1244 Well, Europe isn't the whole world. IDK what to tell you. I've visited enough schools in the U.S.A to know that the education system here, particularly for S.T.E.M purposes, is only slightly better than trash. The only reason I don't say it's absolute trash is because many countries are even worse.
Very nice! This is (roughly) how I explained it to my students earlier this term in my elementary number theory class. Well, I gave them the chance to try to figure out a proof in groups - all wanting to use induction and running into a roadblock - and then I showed them a counting argument. I enjoyed that lesson :)
Just as an addendum to the video. The summand in the equation has the term a^k·b^(n - k), which can be rewritten as b^n·(a/b)^k. Meanwhile, (a + b)^n = [b·(1 + a/b)]^n = b^n·(1 + a/b)^n. Therefore, if 0 < |b|, then both parts of the equation can be divided by b^n, resulting in (1 + a/b)^n = Σ{0 =< k =< n; binomial(n, k)·(a/b)^k}. If x := a/b, then (1 + x)^n = Σ{0 =< k =< n; binomial(n, k)·x^k}. If n < k, then n - k < 0, so (n - k)! is undefined, as the analytic continuation has poles at the negative integers. However, this implies 1/(n - k)! = 0 via the analytic continuation, and since binomial(n, k) = n!/[(n - k)!·k!], it follows that for n < k, binomial(n, k) = 0. Therefore, (1 + x)^n = Σ{0 =< k =< n; binomial(n, k)·x^k} = Σ{0 =< k =< n; binomial(n, k)·x^k} + 0 = Σ{0 =< k =< n; binomial(n, k)·x^k} + Σ{n < k; binomial(n, k)·x^k} = Σ{binomial(n, k)·x^k} = (1 + x)^n, and this is just Newton's binomial theorem for natural numbers! In other words, the binomial theorem is a special case of Newton's theorem, where a = n from the set of natural numbers.
Another easy way to remember this is that the coefficients follow Pascal triangle and you always start from a^n, (a^(n-1))(b^1), (a^(n-2))(b^2), ..., (a^1)(b^(n-1)), b^n
@ゴゴ Joji Joestar ゴゴ Really? I've never had much intuition for that other than that you can prove that the choose function follows Pascal's addition property and symmetries. Could you enlighten me please?
@@EpiCuber7 Take look at the recursive definition of binomial coefficients, and this makes it more clear how fundamentally related they are to recursive binomial expansions. The fact that binomial coefficients are not associated with such a fundamental arithmetic connection and instead to only the Pascal triangle is very unfortunate and of the billions of failures of the education system.
What cute nicknames was he talking about for the distributive property? When I went to school here in Germany we sometimes jokingly called it the "Pinkelgesetz" which means something like the "piss rule" since you can imagine a small stickman standing on one side and pissing on the two terms on the other side but I always thought that was a regional joke. Is that also a thing in other countries or languages? Or is he talking about something else?
@@chrislombardi3968 Yeah, that's what comes to mind for me. (not that there actually is an order at all 😂) Those acronyms are useful for teaching and practical purposes, but I don't like how its underlying mechanism is kind of ignored.
Holy fuck the (a+b)^2 part blew my mind even though I’ve been doing maths for ages. Goes to show what blind memorization from a young age does to your head.
Trust me, you aren't alone. It's like we learn that it's a^2 + 2ab + b^2 before we'd even be able to recognize it as "all a,b combinations". By the time we would be able to recognize it as such, we've been using that basic math for so long that it's something you don't think about. Some of the most charming things I've seen in math came from looking at basic results with a more inquisitive mindset. There is a lot to appreciate in the routinely applied stuff like this.
@@Invalid571 exactly, permutations Then in the video, why do we take combinations of a and b, while in reality, ab and ba are the same Combination but different permutation
Thank you for Not using FOIL! I like to tell my students it is double distribution. I also warn them that FOIL fails once you have a trinomial or greater.
I actually had the very same iterpretation of the binomial theorem. I feel good now that even he thinks the same way. It means my thinking was definitely correct :DDD
"Of course, everything is obvious in Math" -Typical Math Professor. BTW, I remembered that I've read similar logical derivation in Ross. S. A First Course in Probability...
You could write (a + b)^2= (a + b)(a + b) = (a + b)a + (a + b)b = aa + ab + ba + bb = a^2 + 2ab + b^2 which i think is a little bit clearer than going from (a + b)(a + b) to aa + ab +ba +bb.
If my school teachers teach me Maths in this way, then I should not opt for Physics in college( Which I opted only because of my love for Knowing stuffs about Universe and atoms ).
Despite this seeming "obvious" at first, rigorously speaking you need to prove that, indeed it's just summing up the number of permutations. Please show this for rigour.
If you have already known a theorem with proof then it is fine to just state as choose. After all by definition the choose function is that. It involves factorials because it's simpler to write it that way
Do you prove the Pythagorean theorem every single time you use it in your exams? Do you also prove that 2 + 2 = 4 every single time you make usage of this in a calculation? No, I doubt it.
@@angelmendez-rivera351 , naive of you to doubt me. I know the proof of the Pythagorean theorem, and when my calculus instructor didn’t provide formulas on exams, I derived the necessary formulas on the exams and used those formulas to solve the problems.
@@Reivivus You completely missed the point of my argument. I guarantee you that, whether you believe it or not, you do not prove every single statement you use when you take an exam. You do not prove every geometric statement in an exam from only the 5 euclidean axioms in your geometry exams or trigonometry exams. I know you do not. Neither do you prove basic arithemetic facts from scratch, such as using the Peano axioms to prove 2 < 5.
"The binomial formula...it's not a bad formula." - Jon's glowing review of the binomial formula
😂
No you didn't!
Finally understood what
I think you summed up the problem in the first couple of seconds:
"We all know what comes next, just from memory"
For most people, these things need to be taught by memorization, rather than by understanding. The steps to reproduce (a+b)^2 = a^2 + 2ab + b^2 can feel daunting, and 'counterproductive' vs just remembering it. Think also of the quadraric formulat as a good example of this.
This is mostly what highschool maths is based on. Knowing the steps is not required for most people to know, and remembering the steps is not something people are familiar with, so the emphasize is on memorization. One cannot really blame the system for this.
A big part of studying maths is thinking differently about these concept, and realizing that it is often easer to remember the process that created the formula. Since the process is logicall, and follows simple rules (like the distrivutive property), you don't have to remember that much anymore. I hope this video has shown this for at least some people.
Keep it up!
Another thing I feel is incredibly useful for many levels of math in general is Pascal's triangle. Honestly I can't be bothered to remember expansions beyond a cubic expansion and a few times when it's really come down to using the triangle, it actually has been useful! I find small quirks in math so interesting. It's odd how it just ends up the way it is.
I digress somewhat. I do not believe people *need* to be taught by memorization. Scientific studies have demonstrated repeatedly that memorization is a poor way of learning, and that it tends to only hinder the performance of people. The reason we seem inclined to think we do and learn better via memorization than by logical and critical thinking is due to societal conditioning, not due to a matter of fact. Historically, education was also memory-centered because in the past, civilizations believed memory was the only important aspect to knowledge. Now we know this is not true, but breaking old societal habits takes centuries, and we only just learned this not longer than two or three decades ago. This is part of the reason why the education system is the way it is, and why it needs significantly overhauling and updating. Mathematics need to be taught by understanding logical deduction, and by way of analogy with algorithms.
To be honest, it is much easier, if not necessarily to remember common integrals, series, theorems, definitions and proofs. Of course you could rederive everything and it would be really cool if you would not have to do it. However the short time you will have in the exam doesn't allow for this method. You have to know the answer instantly.
@@Caspar__ once you re-derive it enough times, you start remembering it automatically. After re-deriving the basic stuff a few times, memorization becomes easier. For instance, I re-derived the formula for integral of 1/(a²+x²) a few times, then I decided to remember the result for the next time and spent less than a minute on the formula. That way the amount of time dedicated to memorization was nil. Spend almost all of your time to understand and to solve. The time you spend on memorization should be extremely small.
What a unique way of deriving this formula!! I never thought of it in terms of n-length strings. It's a very combinatorial way of looking at it. It's always nice when you can prove/derive a formula in more than one way. Also, video editing is ON POINT, as always! 👍
In schools, they give us Pascal's triangle and that's that...
Did the combinatorial thing surprise you? There is literally a binomial coefficient in the formula...
@@l.1244 Yes, but most students are not taught what binomial coefficients or, more importantly, why they appear in a binomial expansions when they are supposed to be combinatorial objects, assuming that they ever learn their combinatorial significance to begin with, which as I said, is not taught to most students.
@@angelmendez-rivera351 wtf, it's pretty much taught at every high school in Europe at least.
@@l.1244 Well, Europe isn't the whole world. IDK what to tell you. I've visited enough schools in the U.S.A to know that the education system here, particularly for S.T.E.M purposes, is only slightly better than trash. The only reason I don't say it's absolute trash is because many countries are even worse.
No contest, this is the best explanation!
a video i understand more then half of? thats the most mind blowing part of all of this.
Andrew dotson
A physics boi sent me
Very nice! This is (roughly) how I explained it to my students earlier this term in my elementary number theory class. Well, I gave them the chance to try to figure out a proof in groups - all wanting to use induction and running into a roadblock - and then I showed them a counting argument. I enjoyed that lesson :)
Love the Tool background music
This is so much more intuitive!!!! Thanks for this.
I've always loved combinatorics so much but never learned this method. This is gonna be helpful!
Just as an addendum to the video. The summand in the equation has the term a^k·b^(n - k), which can be rewritten as b^n·(a/b)^k. Meanwhile, (a + b)^n = [b·(1 + a/b)]^n = b^n·(1 + a/b)^n. Therefore, if 0 < |b|, then both parts of the equation can be divided by b^n, resulting in (1 + a/b)^n = Σ{0 =< k =< n; binomial(n, k)·(a/b)^k}. If x := a/b, then (1 + x)^n = Σ{0 =< k =< n; binomial(n, k)·x^k}.
If n < k, then n - k < 0, so (n - k)! is undefined, as the analytic continuation has poles at the negative integers. However, this implies 1/(n - k)! = 0 via the analytic continuation, and since binomial(n, k) = n!/[(n - k)!·k!], it follows that for n < k, binomial(n, k) = 0. Therefore, (1 + x)^n = Σ{0 =< k =< n; binomial(n, k)·x^k} = Σ{0 =< k =< n; binomial(n, k)·x^k} + 0 = Σ{0 =< k =< n; binomial(n, k)·x^k} + Σ{n < k; binomial(n, k)·x^k} = Σ{binomial(n, k)·x^k} = (1 + x)^n, and this is just Newton's binomial theorem for natural numbers! In other words, the binomial theorem is a special case of Newton's theorem, where a = n from the set of natural numbers.
Awesome derivation! Thanks for sharing! :)
Beautiful explanation of a fundamental result.
Reflectioooon, omg! This channel just got even better
Hey, thanks! But I just have good taste. Check out Sakis's channel (in description) for glorious Tool covers!
@@EpicMathTime Haha it's always very nice to find people who like Tool!
l already did! They're amazing, thanks for sharing
That's an awesome way to look at the binomial
Nice, i came up with the same explanation when first stumbling above n choose k
These videos are EPIC!
It looks like you’re flipping us off in the thumbnail
Whoops! 😂
wait this is literally so much more intuitive than what I learnt in lecture
that's so... obvious, that's so cool fr
Awesome video
Another easy way to remember this is that the coefficients follow Pascal triangle and you always start from a^n, (a^(n-1))(b^1), (a^(n-2))(b^2), ..., (a^1)(b^(n-1)), b^n
Yeah but this video was about explaining why that formula works, this just helps you remember those binomial coefficients
@@EpiCuber7 Yeah, pascal triangle is for those who don't know binomial formula yet.
@ゴゴ Joji Joestar ゴゴ Really? I've never had much intuition for that other than that you can prove that the choose function follows Pascal's addition property and symmetries. Could you enlighten me please?
@@EpiCuber7 Take look at the recursive definition of binomial coefficients, and this makes it more clear how fundamentally related they are to recursive binomial expansions. The fact that binomial coefficients are not associated with such a fundamental arithmetic connection and instead to only the Pascal triangle is very unfortunate and of the billions of failures of the education system.
@@Invalid571 There is no reason to not learn the binomial formula before learning about Pascal's triangle.
What cute nicknames was he talking about for the distributive property? When I went to school here in Germany we sometimes jokingly called it the "Pinkelgesetz" which means something like the "piss rule" since you can imagine a small stickman standing on one side and pissing on the two terms on the other side but I always thought that was a regional joke. Is that also a thing in other countries or languages? Or is he talking about something else?
Probably FOIL, First Outside Inside Last, an acronym remembering the order of multiplications.
@@chrislombardi3968 Yeah, that's what comes to mind for me. (not that there actually is an order at all 😂)
Those acronyms are useful for teaching and practical purposes, but I don't like how its underlying mechanism is kind of ignored.
"Pinkelgesetz", hahaha.
That was neat. Well done, sir.
Holy fuck the (a+b)^2 part blew my mind even though I’ve been doing maths for ages. Goes to show what blind memorization from a young age does to your head.
Trust me, you aren't alone. It's like we learn that it's a^2 + 2ab + b^2 before we'd even be able to recognize it as "all a,b combinations". By the time we would be able to recognize it as such, we've been using that basic math for so long that it's something you don't think about.
Some of the most charming things I've seen in math came from looking at basic results with a more inquisitive mindset. There is a lot to appreciate in the routinely applied stuff like this.
0:25 if you look at the pen, it shows what he is going to write before he writes it.
I'd love to say that's intentional...
Nice video
Great content bro! keep up with the good work ^_^
Nice! never thought of it that way...
would be really glad to see a video on the expansion of (a+b+c)^n . BTW video was great
It's the same, you'll have to write down all the permutations of length n of three characters.
@@Invalid571 exactly, permutations
Then in the video, why do we take combinations of a and b, while in reality, ab and ba are the same Combination but different permutation
@@Invalid571 plx explain this to me:)
@@johubify What Math & Coding stated is incorrect. You only need to work with combinations of 3 characters, not permutations.
@@angelmendez-rivera351 (a+b)(a+b)
= aa+ab+ba+bb
ab and ba are the same combination
What's going on here??????????
EXPLAIN OR I'M GONNA SUICIDE
How do you make your videos? do you actually write on a mirror or glass or something? I just cant wrap my head around how you do it
Yeah that's what I was thinking throughout the video lmao
This is so good
Thank you for Not using FOIL! I like to tell my students it is double distribution. I also warn them that FOIL fails once you have a trinomial or greater.
Wow i never understood this formula before today!!
bro ur a fricking mathematical chad
I actually had the very same iterpretation of the binomial theorem. I feel good now that even he thinks the same way. It means my thinking was definitely correct :DDD
big fan of the tool song in the background
Well, its more obvious than the proof of Fermat's Last Theorem for example 😃
Thanks!
I HAD forgotten this. How did you now?
Thanks lad
Funny and informativ.
"Of course, everything is obvious in Math" -Typical Math Professor. BTW, I remembered that I've read similar logical derivation in Ross. S. A First Course in Probability...
Awesome ♥️
Yup this is how I learned it
Holy. Shit.
This editing is absolutely nuts. Keep up the great work. 420/69.
You could write (a + b)^2= (a + b)(a + b) = (a + b)a + (a + b)b = aa + ab + ba + bb = a^2 + 2ab + b^2 which i think is a little bit clearer than going from (a + b)(a + b) to aa + ab +ba +bb.
Yes. The binomial theorem is just a special case of the distributive property.
Awesome
Can I pay you to tutor me in analysis
If my school teachers teach me Maths in this way, then I should not opt for Physics in college( Which I opted only because of my love for Knowing stuffs about Universe and atoms ).
(a+b)^n = a^n + b^n quick mafs
or should I say quick meth?
😂
Despite this seeming "obvious" at first, rigorously speaking you need to prove that, indeed it's just summing up the number of permutations. Please show this for rigour.
I think you're confusing this video with some kind of homework assignment being submitted to you.
Where is the proof that n choose k involves factorials?
in every good high school math textbook
If you have already known a theorem with proof then it is fine to just state as choose. After all by definition the choose function is that. It involves factorials because it's simpler to write it that way
Do you prove the Pythagorean theorem every single time you use it in your exams? Do you also prove that 2 + 2 = 4 every single time you make usage of this in a calculation? No, I doubt it.
@@angelmendez-rivera351 , naive of you to doubt me. I know the proof of the Pythagorean theorem, and when my calculus instructor didn’t provide formulas on exams, I derived the necessary formulas on the exams and used those formulas to solve the problems.
@@Reivivus You completely missed the point of my argument. I guarantee you that, whether you believe it or not, you do not prove every single statement you use when you take an exam. You do not prove every geometric statement in an exam from only the 5 euclidean axioms in your geometry exams or trigonometry exams. I know you do not. Neither do you prove basic arithemetic facts from scratch, such as using the Peano axioms to prove 2 < 5.
*to be read in an indian accent and imagined as an instagram comment*
Show me your Frobenius, dear.
Awesome, now do Newton's infinite version
This would require using Taylor's theorem.
(a+b)^n is harder to prove than you think
Me an intellectual: (a+b)^n =a^n+b^n mod n. QED. Gg wp
Uhh... n should be prime. Number Theory 😂
TMW you try to make a pompous joke and you fail miserably.
S H T R I N G S
Noice!
...quick maffs
He didnt do the funny (a+b)2=a2 + b2
** I am extremely disappointed and my day is ruined **
Ah yes, trivial smh, nice video!
It is obvious. After about 15 secs i proved it in my head using permitations.
Combinations, not permutations.
@@angelmendez-rivera351 same thing.
@@nevokrien95 Not the same thing.