Adding powers of 1/2

Рет қаралды 120,052

Жыл бұрын

This is a short, animated visual proof demonstrating the infinite sum of the powers of 1/2 using an isosceles triangle. #mathshorts #mathvideo #math #calculus #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #geometricseries #series #infinitesums #infiniteseries
Here is a bit longer and slower version (with dramatic music and no words): • Geometric series: sum ...
This animation is based on a proof by Roger B. Nelsen from the February 2006 issue of Mathematics Magazine page 60. (doi.org/10.2307/27642904).
To learn more about animating with manim, check out:
manim.community

Пікірлер: 158

@ScenicFlyer4 Жыл бұрын

Half it and give it to the next person

@MathVisualProofs Жыл бұрын

Great idea for a trend ;)

@Faer_Zizh Жыл бұрын

Halve it and take it from the previous person

@mathman15 Жыл бұрын

New approach, it was normally done by a square but you tried new that is amazing

@MathVisualProofs Жыл бұрын

Thanks! I like the triangle better than the square because each shape is similar at each step (whereas with the square, you have to alternate between squares and rectangles).

@mathman15 Жыл бұрын

@@MathVisualProofs makes sense i guess i will from now use right angled triangles to do my mathematical research because it's more accurate, stable and always it's type

@CorporateZombi Жыл бұрын

@@MathVisualProofs well if you split the initial square across the diagonal. It'd basically look exactly the same as the video you have here, but with one extra step before, to make the square.

@ritorapid Жыл бұрын

my Life is now complete following these shorts from 1/8 hehe

@MathVisualProofs Жыл бұрын

But have you seen this : kzfaq.info/get/bejne/gNqVhKh_lr60qZc.html

@carultch Жыл бұрын

An infinite number of mathematicians walk in to a bar. First orders a pint, the next orders a half pint, followed by a quarter pint order, and so forth. All of the same kind of beer. The bartender serves them two pints, and says, "you guys don't know your limits, do you?"

@ThePeterDislikeShow Жыл бұрын

How many orders before you have less than one molecule of beer?

@hamzehalomari7520 Жыл бұрын

Very convincing and interesting

@jackstonebaby Жыл бұрын

I love these geographic representations of infinite sums.

@itzmrinyy7484 Жыл бұрын

*geometric This has nothing to do with geography

@AkariLimano Жыл бұрын

Thanks, I love your geometric interpretations. I am a visual person and infinite series drove me nuts in university. Now I can see the beauty behind those formulas

@MathVisualProofs Жыл бұрын

Wonderful!

@Logarithmic-Spiral Жыл бұрын

I really like your channel

@fire17102 Жыл бұрын

So sum of 1/N^k = 1/(N-1) Seems consistent across your videos

@SteveMathematician-th3co Жыл бұрын

This can be solved just by Geometric progression formula, but this is better to understand it

@thechocolatemonster3392 Жыл бұрын

Superb channel

@MathVisualProofs Жыл бұрын

Thank you so much!

@anatolykarpov2956 Жыл бұрын

Man , you're great

@MathVisualProofs Жыл бұрын

Appreciate this!

@Sloop_Jonz_B 11 ай бұрын

These videos are great. I’m a project cost estimator who was never allowed to take difficult math courses in school (they didn’t believe someone with autism can do math I guess). Now I’m in my late 20s and math is easier than I ever imagined. Thank you for helping me change my life for the better.

@XanderAnimations 11 ай бұрын

Sorry if this sounds rude (I've been trying to find a way to say this that doesn't sound rude) but what is it like living with autism? My sister is autistic and my family is always trying to find ways to help her strive above her diagnosis

@NabiruBogdan Жыл бұрын

I love these videos

@jogeshjoshi2535 10 ай бұрын

Very Nice

@MarySmith-ko9qv Жыл бұрын

Good stuff.

@MathVisualProofs Жыл бұрын

Thanks!

@dvargart Жыл бұрын

My brain is ripped apart between choosing to believe this lol

@ummoco8940 Жыл бұрын

The infite sum of 1/x^n = 1/(x-1)

@Xbrh4499 6 ай бұрын

One must imagine the isosceles triangle happy...

@cipherxen2 Жыл бұрын

Now he'll do the same series with all other shapes

@MathVisualProofs Жыл бұрын

Already done : kzfaq.info/get/bejne/gNqVhKh_lr60qZc.html

@MohamedouMoctar Жыл бұрын

Amazing🎉🎉🎉

@p.g.8796 Жыл бұрын

Some comments here say it is actually=1. That is actually not true. By giving the infinite sum a number = x, you are already made the assumption that is in a fact a finite number. It is not. It is converging to 1. The math behind that is not complicated, it is the geometric sequence. In fact, here it is s= (1-q^(n+1)/ 1-q) - 1 In this case q = 1/2 and n would be infinity. So you have to find the lim of this function. For any q < 1 , q^(n+1) will always converge to 0. But this will ultimately cause that you cannot use the equal sign. You have to use ➡️

@MhdAliAlashkar Жыл бұрын

شرح رائع أحسنت

@AngryEgg6942 Жыл бұрын

Here is the mathematical solution very simple: a1 = 1/2 q = 1/2 Sum of an infinite geometric sequence: a1/(1-q) => (1/2)/(1-(1/2)) = (1/2)/(1/2) = 1

@sumitsarmah4142 Жыл бұрын

This can be also done with the help of a Square

@AdriBlt Жыл бұрын

Even though since the first video I demonstrated that sum(1/n^i) = 1/(n-1), I cannot miss all the videos !

@MathVisualProofs Жыл бұрын

Check this one : kzfaq.info/get/bejne/gNqVhKh_lr60qZc.html

@DrRiq Жыл бұрын

I prefer the square tbh. The triangle looks like half a square the way it's placed

@JackReacher110 9 ай бұрын

We can do it with square. 😊

@anjalidwivedi2057 Жыл бұрын

the actual answer is that the sum will keep on approaching 1 but will never reach 1....but for explaining this will work too👍👍

@MauritsWilke Жыл бұрын

Incorrect, it converges absolutely to 1. en.wikipedia.org/wiki/1/2_%2B_1/4_%2B_1/8_%2B_1/16_%2B_%E2%8B%AF

@anjalidwivedi2057 Жыл бұрын

@@MauritsWilke Oh.. I'm sorry.. My bad. I saw this series on some video and it showed like approaching 1 so I was confused.. Thanks😀😀

@lostsoul2455 Жыл бұрын

Here is an algebraic solution to show that it does definetely become 1. Let the sum equal x 1/2+1/4+1/8+1/16...=x Multiply both sides of the equation by 2 2(1/2+1/4+1/8+1/16...)=2x We see that the first term becomes 1 plus the same infite series. 1+1/2+1/4+1/8...=2x So we can sub in x, since we let x equal the same infinite series. 1+x=2x Using simple algebra, we can solve for x 1+x=2x 1=1x 1/1=x 1=x

@sickowhale6861 Жыл бұрын

@@MauritsWilkeYeah, just in mathematics.

@bagochips1208 Жыл бұрын

Lol i remember drawing something like this in elementary because i was bored in class

@amitavadasgupta6985 Жыл бұрын

Sum=1/2/(1-1/2)=1 ans. By G.P.

@eastonbrunet9996 8 ай бұрын

MISTAKE 1/1024 is actually the answer, not 1/1028.

@MathVisualProofs 8 ай бұрын

Oh good catch on the typo! Sorry about that.

@honkshire4139 Жыл бұрын

series of (1/2)^n from n=1 to ♾️ then use formula (1/(1-(a)) a=(1/2) to get 2

@MathVisualProofs Жыл бұрын

That’s not quite right. The formula you gave is for the series that starts with 1. This one starts with 1/2.

@zacharymogel9500 Жыл бұрын

“Start with an isosceles triangle” *draws a right triangle*

@LohStorm Жыл бұрын

it's an isosceles right triangle... the requirement for a triangle to be isosceles is having 2 sides with equal length

@MathVisualProofs Жыл бұрын

And I should have said isosceles right triangle :)

@zacharymogel9500 Жыл бұрын

@@LohStorm the requirement for a right triangle is one angle having to be 90 degrees exactly

@zacharymogel9500 Жыл бұрын

@@MathVisualProofs fair

@briancase6180 Жыл бұрын

Cool.

@BearJoyner00 Жыл бұрын

This reminds me of the joke about the bartender and the infinite number of physicists.

@JK-ph8pv Жыл бұрын

You are adding

@cherylchui4510 Жыл бұрын

@martymcfly5068 Жыл бұрын

Can someone answer this for me? If you solved it using the geometric sim Theorem you would get r=1/2 because it is the summation of (1/2)^n. Then when you plug it in to solve for the sum, you get 1/(1-1/2) which equals 1/(1/2) which would give you 2 right? So why didn’t that work right?

@MathVisualProofs Жыл бұрын

Remember that the formula needs to also account for the first term. This one starts at 1/2 and ratio is 1/2 so the formula is (1/2)/(1-1/2) = 1.

@sumdumbmick Жыл бұрын

no, you never reach the corner point. so it's 1 -L(0), where L(0) is the inverse of positive infinity, and the point adjacent to 0 on the numberline such that lim x->0+ 1/x = 1/L(0)

@100neshama Жыл бұрын

This is how you reach first base

@guleryazkan5702 Жыл бұрын

So mandelbrot will work with sums?

@jaimintankariya6955 Жыл бұрын

I used to understand this with square and circle also. Square was the easiest one.

@Unknown10446 10 ай бұрын

Let, 1/2+1/4+1/8+...=S Now, 1/2+1/4+1/8+...=S => 1/2(1+1/2+1/4+1/8+...)=S =>1/2(1+S)=S =>(1+S)/2=S =>1+S=2S =>1=2S-S =>S=1

@rogofos Жыл бұрын

its not equal merely approaching but it will never be exactly one

@MathVisualProofs Жыл бұрын

Infinite sum is the limit. It is exactly 1

@jackmills7706 Жыл бұрын

Can you please do one of these for sum from n=1 to infinity of (1/n)?

@beerilevinger6886 Жыл бұрын

It actually divarges so its not really possible, unless you mean the proof to it converging.

@darioprats1541 Жыл бұрын

OMG !!!

@guillermobuenohernandez847 Жыл бұрын

e=2.71or 1or 360° ,,, 1/2 = 0.5, which is equal to one hole..● of the other ○,,

@rodrigoodonsalcedocisneros9266 Жыл бұрын

What if the remaining unshaded area was the size of a quantum? Does that even matter in math?

@fighterinarras0998 8 ай бұрын

You realise that no matter how many times you divide by 2 it will never be 0 right

@Adhamadous 10 ай бұрын

Let's make a chain I say 1 The next say 1/2 Then other guy is gonna say 1/4 Let's do it

@Adhamadous 10 ай бұрын

@MathVisualProofs 10 ай бұрын

1/2

@letranky8a109 Жыл бұрын

fun fact (but i don't think it's fun): the sum of the positive powers of 1/sqrt(2) is equal to sqrt(2) +1

@matheustomatti843 Жыл бұрын

this works with any number x/2î

@zander513 Жыл бұрын

To the people who think it’s 0.9999999999999999…infinitesimals are not on the number line

@mastervibes2296 Жыл бұрын

Wow this guy must be a super geek, but it's very cool stuff

@MathVisualProofs Жыл бұрын

Professional geek

@stuartm7009 Жыл бұрын

So... Is the infinite sum of the powers of any fraction (1/n) equal to 1/n-1?

@bradyschultz808 Жыл бұрын

You sound like a certain Presh Talwalkar

@TanjoGalbi Жыл бұрын

You could just cut all these videos about each fraction and just say that the infinite sum of the powers of a fraction with a numerator of 1 equals the fraction with the denominator minus 1. E.G. the infinite sum of 1/3 is 1/2, the infinite sum of 1/4 is 1/3, etc. It would have saved you a lot of production time making all the videos.... But then I guess that's the point, more videos = more watches = more ad sense income!

@MathVisualProofs Жыл бұрын

Shorts don’t bring in a lot of ad revenue. I do these because I love math and I love the geometric series :)

@TanjoGalbi Жыл бұрын

@@MathVisualProofs Well then, carry on 😉

@carlkarl Жыл бұрын

Sir can u do the 1+2+3+4...... = -1/12 ? I learned it in 9th grade from my olampiad teacher once

@MathVisualProofs Жыл бұрын

It's not really true in the usual sense, so it won't have a visual proof. There are plenty of videos online that work through the process of using Cesaro summation to get there.

@arthurauillans5256 Жыл бұрын

Why does the summ of the powers of 1/n always equal to 1/(n+1). This pattern repeat itself in all your video. I would love to see à more in depht explanation

@arthurauillans5256 Жыл бұрын

I made à mistake Its not 1/(n+1) but 1/(n-1)

@emiel-io Жыл бұрын

Is the last triangle green or magenta?

@MathVisualProofs Жыл бұрын

What last triangle? ;)

@BAMBAMBAMBAMBAMval 9 ай бұрын

But how is it ever going to reach 1? It will always be 0.9999999... (Because theres always 1/2 of the previous number missing) ((and if this ever would reach 1 you would have to add 1/2 of N, then it would reach infinity, which it does not))

@ihavenoideawhattoputheresowhat 3 ай бұрын

1028 is not a power of 2

@MathVisualProofs 3 ай бұрын

Yep. Definitely a typo from me :)

@5k_Sub_challenge_with_no_video Жыл бұрын

1/N + 1/N² + 1/N³ ......... = 1/(N-1) Simple 😊

@alex1lane Жыл бұрын

how do you make these videos?

@quay6292 Жыл бұрын

Pretty sure he used manim

@MathVisualProofs Жыл бұрын

Yes. I use manim.

@KaliFissure Жыл бұрын

Why didn't you start with a square?

@MathVisualProofs Жыл бұрын

Square alternates between square and rectangular areas. I like that the triangles are similar all the way down

@KaliFissure Жыл бұрын

@Mathematical Visual Proofs no, you can cut a square in half into triangle and triangles all the way down I prefer a it shows unit square composed of scaling identical parts

@KaliFissure Жыл бұрын

@@MathVisualProofs I uploaded a short, garage format 🙂

@MathVisualProofs Жыл бұрын

@@KaliFissure oh yes! Nice one! I was thinking of a diff one

@scottwesner9362 Жыл бұрын

Wouldn't actually be equivalent to 1 and not exactly equal to 1

@nekogod Жыл бұрын

Nope it is exactly 1

@jorypeters5235 Жыл бұрын

So infinity means 1

@carultch Жыл бұрын

No. This is an infinite series of a decreasing sequence, that converges to equal 1.

@DerAndersdenker 10 ай бұрын

This i so unsatisfying! 😅

@MathVisualProofs 10 ай бұрын

Unsatisfying? Why? What would make it more satisfying?

@DerAndersdenker 10 ай бұрын

@@MathVisualProofs It's unsatisfying, because it's never complete. You can add as many parts as you want and it will sustain incomplete.

@nekto4658 Жыл бұрын

It's never gonna be 1

@nekogod Жыл бұрын

Yes it is

@dalwindersingh5114 Жыл бұрын

That doesn’t make sense 😢

@kavithavelu8282 Жыл бұрын

This could be also done in a square of side length 1 .😊😊 But it is a great work👋👋👋

@howhello354 Жыл бұрын

Yah 😅

@MathVisualProofs Жыл бұрын

Yes, definitely! But I like the triangle better than the square because each shape is similar at each step (whereas with the square, you have to alternate between squares and rectangles). :) Thanks!

@joyli9893 Жыл бұрын

Answer: 0.9999999999999999999

@nekogod Жыл бұрын

0.9999999... actually which is = to 1

@joyli9893 Жыл бұрын

@@nekogod wrong But if 0.9999999999 is not equal to one then the square root of 5 isn’t equal to the square root of 5

@KabalFromMK9 Жыл бұрын

@@joyli9893 0.9999... (Unending 9) is not the same as 0.9999999999 (the lack of three dots means that the decimal stops there, and that does not equal to one. 0.999... on the other hand, also written as 0.9 with a bar over the number nine, does equal to one.

@JeroHitsukami Жыл бұрын

Actually it's 0.9 repeating which does not equal 1 no matter what anyone tells me

@charlie-gamingshortsandmor9324 Жыл бұрын

Nope

@brainergaming4554 Жыл бұрын

Even if we do it infinitely lastly we'll end up half portion on last triangle unshaded. Do not give wrong info or If there is any justification please let me know

@ruthvikas Жыл бұрын

He said it's converging towards 1 not precisely 1. I will be nearly 0.99999999999999 which we can tell as 1

@cheeseburgermonkey7104 Жыл бұрын

Yes, you can think of the unshaded region as being an infinitesimal, which would make the shaded region equal to 0.999999999... but most people say this is equal to 1

@KabalFromMK9 Жыл бұрын

@@ruthvikas did you mean converging?

@ccbgaming6994 Жыл бұрын

No, it has to be 1. This is true because even though it will never pass 1, it is AT LEAST infinitely close. But even if it’s that close, you can still get closer because you are always adding a half. Therefore if the area is between infinitely close and not passing 1, then it has to be 1

@ruthvikas Жыл бұрын

@@KabalFromMK9 oh yea autocorrect, i hate it sometimes

@AMEL-gr4yj Жыл бұрын

Nah its more like ≈1

@nekogod Жыл бұрын

Nope it equals exactly 1

@AMEL-gr4yj Жыл бұрын

But man, it never gonna reach the end if triangles will always shrink, so its 0,984375... So its *≈* 1

@brainergaming4554 Жыл бұрын

No it's wrong

@aug3842 Жыл бұрын

he isn’t wrong - he’s saying as the size of the sum _approaches_ infinity, the result _converges_ to 1; he isn’t saying that if u get a bunch of powers and add them u get 1 but rather if u observe what happens as u approach a never ending sum

@beerilevinger6886 Жыл бұрын

@@aug3842 and to add to that, with infinite sums, "approach" is the same as "equal".

@lime.person Жыл бұрын

The "brainer" in your username is surprisingly ironic.

@aug3842 Жыл бұрын

@@lime.person lol i was gonna say

@lime.person Жыл бұрын

@@aug3842 lmao sorry