3b1b viewer here. This video is awesome! If anyone wants a more mathematical proof, here it is (by induction): Firstly, check a base case (n = 1) 1^2 = 1^3, so the base case works Next, choose some case (n = k). By the inductive hypothesis, (1 + 2 + 3 + ... + k)^2 = 1^3 + 2^3 + 3^3 + ... + k^3 let x = (1 + 2 + 3 + ... + k) for simplicity. Note that also, by the triangle-number formula, x = k(k + 1)/2 So x^2 = 1^3 + 2^3 + 3^3 + ... + k^3 Lets check if it works for n = k + 1 (x + (k + 1))^2 is the new expression on the left hand side, expanding out, we get x^2 + 2x(k + 1) + (k + 1)^2 = x^2 + 2kx + 2x + k^2 + 2k + 1 substituting for x (leaving the x^2), we get: x^2 + 2k*(k(k + 1)/2) + 2*(k(k + 1)/2) + k^2 + 2k + 1 simplify (cancel out 2s): x^2 + k*k*(k + 1) + k * (k + 1) + k^2 + 2k + 1 x^2 + k^3 + k^2 + k^2 + k + k^2 + 2k + 1 collect like terms: x^2 + k^3 + 3k^2 + 3k + 1 factorise: x^2 + (k + 1)^3 note that by the inductive hypothesis, x^2 = 1^3 + 2^3 + 3^3 + ... + k^3 so: x^2 + (k + 1)^3 = 1^3 + 2^3 + 3^3 + ... + k^3 + (k + 1)^3 This is the sum of the first k + 1 cubes, so, by induction, if it works for k, it works for k + 1 However, we know it works for k = 1 (base case), so it must work for all natural numbers Ta da! :D P.S. Please provide critical feedback on this proof presentation if you have time! I'm practicing for the Olympiad! Thanks in advance! Edit: Oh, and happy tau day! :D
The way you 'explain' this using geometry is simply beautiful. If you had shown that equality up front, I would have waited for the punchline. Aside from what you showed, is there some other way to come to this conclusion intuitively? In any case, you got another fan.
I found maths distasteful last year but this year after reading Alex's Adventure in the numberland I feel like I have some connection with it. No one cares anyway still solving a math problem was boaring for me but after watching videos from channel's like mathologer,3blue1brown,Thinktwice and my favoutlrite Numberphile kept me motivated and insightful.
I now support on Patreon 3blue1brown, Veritasium, Eugene Khutoryansky, the Science Asylum and you! Your videos are awesome! You deserve more support. How old are you?
Great video! Just one tiny error. In the last sigma notation, you should've replaced the places for k and n. Because above you represented the sum from 1 to n, not from 1 to k.
I think it would nicer if for cases like 2^3 and 4^3(top layer comes from both sides), the one side was rotated and then moved, instead of shuffling cubes like you did.
Beautiful, I had just done this with a set of original Dienes wooden blocks and was making a Scratch version. Your animation is great. Do you know of a visual proof for the sum of powers of 4?
When I first found the sum of cubes formula, I realised it was the square of the sum of naturals, so I went into wolfram alpha to see if any other sum of power formulas where the same as other sum of power formulas raised to a power, but I couldn’t find any. Pretty disappointing. I would have loved it if the sum of power 5s was the sum of naturals cubed
Are there any other examples of this where the sum of powers, is equal to another power of sums? I.e. where the 1^a + 2^a + 3^a + ... + n^a = (1 + 2 + 3 + ... + n)^b?
Why haven’t I seen this equation before?? I’m sad I’ve missed such a beautiful equation. This is why i love math everything is somehow interconnected though it wasn’t mean to be when math was invented
Came from 3b1b and definitely loving this channel
the same!
same!
your videos are literally some of the most beautiful things I have ever seen.
3b1b viewer here. This video is awesome!
If anyone wants a more mathematical proof, here it is (by induction):
Firstly, check a base case (n = 1) 1^2 = 1^3, so the base case works
Next, choose some case (n = k). By the inductive hypothesis, (1 + 2 + 3 + ... + k)^2 = 1^3 + 2^3 + 3^3 + ... + k^3
let x = (1 + 2 + 3 + ... + k) for simplicity. Note that also, by the triangle-number formula, x = k(k + 1)/2
So x^2 = 1^3 + 2^3 + 3^3 + ... + k^3
Lets check if it works for n = k + 1
(x + (k + 1))^2 is the new expression on the left hand side, expanding out, we get
x^2 + 2x(k + 1) + (k + 1)^2 = x^2 + 2kx + 2x + k^2 + 2k + 1
substituting for x (leaving the x^2), we get:
x^2 + 2k*(k(k + 1)/2) + 2*(k(k + 1)/2) + k^2 + 2k + 1
simplify (cancel out 2s):
x^2 + k*k*(k + 1) + k * (k + 1) + k^2 + 2k + 1
x^2 + k^3 + k^2 + k^2 + k + k^2 + 2k + 1
collect like terms:
x^2 + k^3 + 3k^2 + 3k + 1
factorise:
x^2 + (k + 1)^3
note that by the inductive hypothesis, x^2 = 1^3 + 2^3 + 3^3 + ... + k^3
so:
x^2 + (k + 1)^3 = 1^3 + 2^3 + 3^3 + ... + k^3 + (k + 1)^3
This is the sum of the first k + 1 cubes, so, by induction, if it works for k, it works for k + 1
However, we know it works for k = 1 (base case), so it must work for all natural numbers
Ta da! :D
P.S. Please provide critical feedback on this proof presentation if you have time! I'm practicing for the Olympiad! Thanks in advance!
Edit: Oh, and happy tau day! :D
So the sum of the first n integers all squared is equal to the sum of the first n cube numbers... I love you.
James Wise I love you too
:(
Ashley Lee no one loves you though
naa
Think Twice 😂
why haven't I heard of this before? it's awesome
:)
Yusril Atfan he’s talking about this equation
It must begin from 1, and must be continuous.
Awesome, I'm loving this channel!
Nuno Mateus comments like that make me want to make more videos:)
And people like me want you to make more videos!!!
Nuno Mateus I'll upload as soon as i can, however I have to work a lot these days, so I might upload at slower rate.
Think Twice now worry just keep up the great quality :) very good work.
*No worry
As soon as it hit 1:01, I had to pause, lean back in my chair, and verify the absolute brilliance I just saw.
So True
The way you 'explain' this using geometry is simply beautiful. If you had shown that equality up front, I would have waited for the punchline. Aside from what you showed, is there some other way to come to this conclusion intuitively? In any case, you got another fan.
Your content keeps helping everyone, I just love that
Awesome
The RUclips algorithms seem to be favoring this channel as of late. Good to hear.
Thank you, reaaaaally thank you!!! That's beautifull
Thank you~
Excellent video, very well and nicely done, love the music as well. Thank you!
oh my god
everything in math being so different since i see this channel
oh...my...god
how couldnt i seen thí channel before?
I'm so glad I've found this channel. God bless you guys! >
This is one of the most beautiful theorem's i have ever seen. why am i only now learning of it?
Beautiful explanation !!!
I've been looking for a long time to find a video like this. This proof is so unknown I thought it didn't exist at all for years.
I found maths distasteful last year but this year after reading Alex's Adventure in the numberland I feel like I have some connection with it. No one cares anyway still solving a math problem was boaring for me but after watching videos from channel's like mathologer,3blue1brown,Thinktwice and my favoutlrite Numberphile kept me motivated and insightful.
Another cool one: The sum of the first n odd numbers is n^2
This one is so cool
This is really nice! I always wondered if there was a visual way of understanding this formula :)
This is sooo cool!!!
estuardoremi it is, indeed:)
Very nice. With these videos, education of mathematics is far more fun and enjoyable
Wonderful
Excellent channel! Your visualizations are great!
Came from 3b1b, awesome channel!!
I actually figured this fact out myself in junior high. Cool to see a video on it!
I can't know why this channel has only 35 thousands of subscribers.
This one gave me the chills
That’s beautiful
I love your process of intuition sir thanks a lot
The beauty of Math in full display!
Thanks for watching:)
Beautiful!
Subbed for sorcery
Awesome 👍
Excellent 💕
holy hell I didn't expect this (never heard of the theorem and when it came I was surprised)
unbelievable
AMAZING
Love your animations!
BTW: Song ?
it's sooooooooooooo beatifufllll
Brillante
Thank you
I now support on Patreon 3blue1brown, Veritasium, Eugene Khutoryansky, the Science Asylum and you!
Your videos are awesome! You deserve more support.
How old are you?
Wow
mind explosion!
Лектор доказал эту теорему на занятии по математическому анализу, но не назвал её. Спасибо!
100
1+2+3+4=10
(Inspired by numberblocks)
10²=100
Excellent. It's so satisfying... I'll recommend this channel for all my math's friend. Btw, what software do you use to make this animation?. Please 😊
Thanks a lot:) I used cinema 4d
Great video! Just one tiny error. In the last sigma notation, you should've replaced the places for k and n. Because above you represented the sum from 1 to n, not from 1 to k.
I didn't even know this formula =O
Woah
Wow.
Your goal of getting 1M subs gets closer with 3b1b.
What is the music?
I think it would nicer if for cases like 2^3 and 4^3(top layer comes from both sides), the one side was rotated and then moved, instead of shuffling cubes like you did.
Holy--
i luv vids
Wow I really wanna see the proof for this
3blue1brown brought me here. You got one more sub now!
Beautiful, I had just done this with a set of original Dienes wooden blocks and was making a Scratch version. Your animation is great. Do you know of a visual proof for the sum of powers of 4?
This really be called a theorem after someone when you can literally look at the formulae for the sums of first n natural numbers and n first cubes.
When I first found the sum of cubes formula, I realised it was the square of the sum of naturals, so I went into wolfram alpha to see if any other sum of power formulas where the same as other sum of power formulas raised to a power, but I couldn’t find any. Pretty disappointing. I would have loved it if the sum of power 5s was the sum of naturals cubed
Are there any other examples of this where the sum of powers, is equal to another power of sums?
I.e. where the 1^a + 2^a + 3^a + ... + n^a = (1 + 2 + 3 + ... + n)^b?
Bruhhhhhhh!
I dont see how i can visualize the general case from this.
Can anyone please tell me the title of the background music?
Can this be applied to higher exponents?
John Apawan i dont think so, since you cant make "nice" quadratic patterns for example 1^4+2^4= 17 and the square root of 17 isnt that nice
yeah, but isn't ONE square missing from every next square?
What's the sum of 1³+2³+3³+...+n³ though? Is it n⁴?
Matt GSM no 1^3+2^3 = 9 and 2^4= 16
ilPrincipe so what is it then?
Matt GSM watch the video xD
wow.. how to. Make these visuals
nice try dude.
why i'm saying this simply because 1^3+2^3+...+n^3 = (n(n+1)/2)^2 which is sum of natural numbers who square
How do you know that each colour will perfectly make up a cube every time? You didn't show that part
They were arranged into n groups of nxn, for instance the 4s were arranged into 4 blocks of 4x4 (one of which was cut in half).
but... but how... and why... and... AAAAAAAAAAAA
Nice, but doesn't work where n > 4. Seems too trivial to be a theorem.
Thor's Hammer it works for n=5, and all other n
What about the rasengan doe
2
But why?
Why haven’t I seen this equation before?? I’m sad I’ve missed such a beautiful equation. This is why i love math everything is somehow interconnected though it wasn’t mean to be when math was invented
I dont inderstand why earea of square transform to cube i dont agree hhhh
1.1k likes : 4 dislikes. Wow
I knew this when I was 8, or 9
Probably the least interesting think twice video I've watched.