0.3333… = 1 (in base 4)
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- Опубліковано 2 жов 2024
- This is a short, animated visual proof demonstrating the infinite sum of the powers of 1/4.
To buy me a coffee, head over to www.buymeacoff...
Thanks!
To see two other versions of this sum, check out
• Geometric series: sum ...
• A Hexagonal Geometric ...
To see the related base 10 fact, check out:
• 0.999999… = 1
And to see the related binary fact, check out:
• Adding powers of 1/2
To see related base-3 fact, check out:
• Geometric series: sum ...
or
• 0.2222…= 1 (in base 3)
To see a related base-5 fact, check out
• 0.4444... = 1 (in base 5)
This animation is based on a proof by Sunday A. Ajose from the June 1994 issue of Mathematics Magazine (www.jstor.org/... page 230).
#mathshorts #mathvideo #math #calculus #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #geometricseries #series #infinitesums #infiniteseries #shorts
To learn more about animating with manim, check out:
manim.community
If only these visuals were available during my college years. Hah!
This is high school maths
@@Devilhunter69 Visuals. Not the curriculum. Visuals of high order calculus was a luxury 15 years ago.
@@AbhishekBilkanAindwhat?
@@mihailmilev9909 Between 2008-12, internet was a different place. Mathematical or even Engineering visuals and CG were propriety materials of most Universities behind paywall. It wasn't this easy to simply get exact visuals or CG pertaining to your specific question.
For example, it was quite difficult visualize a complex plane with chalks on board. We learnt the approximated way. But now, you can get a comprehensive video just on youtube about intricacies of complex plane, derivative functions, integral functions, differential equations, and higher order polynomials.
We had pen, paper and C++.
if only these visual sums were helpful. visual proofs are not helpful for your logical reasoning and lead to false proofs.
This reminds of whenever the fact that 0.999..... =1 there's akways at least half a dozen people in the comments that just won't accept it
It's so simple actually.
let x = 0.999...
10x = 9.999..
10x = 9 + x
9x = 9
x = 1.
@@f.r.y5857 yeah I know, that's why I called it a fact as there are several proofs including the one you just cited, my point was that even when you present proofs like that or the ⅓ proof (⅓= 0.333.. 3 x ⅓ = 1 3 x 0.3333...=0.9999.. so 0=9999..= 1) there will still be at least one usually several people who'll still won't accept it and use arguments such as "it can't be" I think the problem they have is not fully appreciating what an infinite series decimal is... I can empathize to an extent as it is hard to think of 0.999... without thinking of the "last 9" even tho you or I know there is no "last 9"
It's not equal to 1.
@@mojaveclimber tell me you're joking ...please
It's me. I don't accept it. It is as close to one as possible without being one. It is also close enough for all practical applications...
But it's not one. I don't care what the "smart" people say.
0.nnn... is always 1 in base n+1
Bruh
@@haipingcao2212_. wdym bruh?
Wht do u mean
@@theiigotriangularround4880 for example 0.999... = 1 in base 10 and 0.111... = 1 in base 2
I was just wondering about the generalization into arbitrary base. Completely overlooked this fact.
"Sir, I asked you how many people there are in your dinner party."
Thank you for showing me this
Legend has it he is still cutting cubes into four equal pieces.
😀🤣
Squares actually
So what if you cut a cube into 8 equal parts, color in 7 of them, then repeat the process with the 8th piece?
Very Interesting ❤️ the animation 👌
"imagine the limiting process" -- we dont know what that means, and "we see that the entire square diagram will be shaded" -- no, we dont, if you just said we only shade 3/4 indefinitely... this is asymptotic, not "take an irrational number and round it to make it rational"
His content is literally awesome
Which apps you are using to make these type of animation?
I use manim
@@MathVisualProofsYooooo that's 🔥
Since |r| < 1 we can use infinite geometric series formula
Sum = a/(1-r)
Sum = (¼)/(1-¼)
Sum = (¼)/(¾)
Sum = ⅓
Thats why maths is awesome
It would also be nice to set a limit so that the expression really equals one.
Sir this is Wendy's
Same as how 0.9999…=1
can you add anything to it to make 0.9999... equal 1, is there a fraction that is equal to it, which there is, divide 0.9999... by 3 and you get 0.3333... which is equal to 1/3 times by 3 = 1 and is also equal to 0.9999...
@@purple-47 what do you mean "which there is"? ....there is NO fraction you can add to 0.999... (recurring) to make it 1 - or indeed no fraction you can subtract from 1 to make it 0.999.... that's just one more reason to realise they are equal
0.9999... = 1
@@izzabelladogalini "Is there a fraction that is equal to it, which there is." 0.9999... is equal to 3/3, I did not mention anything about adding.
@@purple-47 excuse me but your first comment literally started with the words "can you add anything ..."
I loved my maths (yes with an 's' ...I'm British) as a kid and was lucky enough to have a teach who got me through differential calculus, quadratics and the like by age 11 ....got my Bachelor's in applied mathematics age 19 and my Master's in number theory age 21 - after the death of my parents and a need to earn an income I had to drop my objective of a PhD but I've maintained a keen interest.... You're not alone many highly intelligent people have a problem with the concept but if you know maths you know 0.999... actually is equal to 1 even if initially it appears counterintuitive
@@izzabelladogalini I think I might try and learn Differential Calculus today, it might help with a few exams I've got in the next months.
0.3 repeating=1 in base 4 is the same as 0.9 repeating=1 in base 10 right?
This works for all bases
I wonder if this can be generalized to other bases?
See my channel. I have base 3,4,5 and 10 so far. Base 8 in the queue.
3*S for S = sum(1/4ⁿ)
Mipmap 😮
So that ((n-0.111111111111...)in base n)=1
Shared. With my wife and daughters. Because I know they'd appreciate this. #MathDad
:)
I wonder if there's a visual demonstration for every a_n = b/(c)^n where b , c and n are whole numbers. 🤔 (b and c being coprime of course)
This video shows one way to get a lot of them (though still some issues trying to use it to get all that you suggest): ua-cam.com/video/bxSCJR6RRxs/v-deo.htmlsi=TBoYBO50p67B0Xvv
Different ways to do the same thing
1/3
The problem is that this argument for disregarding the remaining unshaded corner point hinges on disregarding the unshaded corner point. It's a circular argument.
It's not circular, limits are already a well-defined construct in mathematics. This video is explaining that the limit approaches 1, and so given what we already understand about limits we can just say this value is equal to 1
In the same way we can prove 0.99999.....=1
Just use the infinite GP sum = a/(1-r) then u will get the answer
It is important to know the formula as degree of accuracy could effect weather or not you have a radiation leak .
3/4 + 3/4² +... = 1
1/4 + 1/4² +... = 1/3
Let n = 4
1/n + 1/n² +... = 1/(n-1)
Hence my theory is proven again, which is
1/n + 1/n² +.. = 1/(n-1)
where n = integer only, of course
Taylor theorem expansion
Look up the expansion for 1/(1-x) I'm sure you're going to love it
@@dougr.2398 doesn't it work for all real n such that |n| > 1 though?
I think your target audience needs a bit of an explanation about bases, and the "unit square". To the non-maths person, this is very confusing. Each digit after the decimal is also base 4, so thinking of each 3 as 3/4, and knowing that 4 is written as 10 in base 4 might clear things up.
❤
1
so -1/18 in base 1/2 is equal to 1?
Thank you,sir
Whats tha Base at which pi isnt irrational?
Base pi works.
@@MathVisualProofswell technically it's always irrational, but it doesn't always have an infinite decimal expansion :P
@@williammanning5066 yes. For sure. Pi is irrational. I was thinking the question was about pi’s representation but that wasn’t the question l :). Thanks!
Numbers worry me.
Classic. Totally.
So it's basically a geometric progression where summation(1/4)^n can be approximated to 1/3 using the formula a/1-r
Why in most cases when k>1 the infinite sum of (1/k)^n , when n=1 to infinity=1/(k-1) ??? Hmm???
Now I need an aspirin 😅
I need the extra strength dosage. I listen and watched this video three times and I can feel a really bad headache coming on. (Thought I must admit the video and the creator’s logic are brilliant.)
Интересно
Wouldn't that be equal to 0.4 not 1 because it's only 4 sections not 10
Leaving 0.99999... = 1?
That is how it is
@@Thesilentone88 its still approximately not precisely
0.999999.. is actually precisely 1
@@davedixon2068 If you did this for any real number of times, no matter how big, you'd be correct in saying it's an approximation. HOWEVER if you got to infinity the approximation becomes an equality.
A nice way to think of it is to ask: "what number falls between 1 and 0.99....?" there is none, atleast no real number (we'll ignore the hyper reals cus it's kinda na invented number)
@@mrfancyshmancy it is still not 1, it is some infinitesimal point off 1 so therefore not 1
What does he mean by "in the limit, the entire square will be shaded"?
It means that if you continue to the process infinitely, we can effectively treat the square as though it is fully shaded in.
It means precisely the opposite of that, the limit of a convergent series, is the closest value it approaches without ever being able to exceed (gregoire we saint Vincent - "The terminus of a progression is the end of the series, which none progression can reach, even not if she is continued in infinity, but which she can approach nearer than a given segment.")
Where does this fall apart for other fractions? Say 2/3
And 0.777777… = 1 in base 8
It’s in the queue! 👍😀
Interesting an i like the visual but i often what what application this has
The geometric series formula (of which this is one example only) has an amazing range of applications.
It's negative 1. Which doesn't work well with a visual proof lol
0.99999999999... = 1
Shouldn't it be:
0.999999...?
Oh, youtube flips my numbers because i have arabic as default... stupid
Or you could use infinite sum of gp
=a/1-r
my brain. ouch.
only reason this irks me is because “indefinite” is a dubious term. As soon as I ask how many times you’ve measured it, it’s definite, and that teeny tiny piece is still missing if I zoom all the way in.
That's why the square is only a representation of infinite sum. You can't show infinity in real life(or via a computer at least)
How does one get to the base 4 representation? I think it is 0.3 + 0.03 + 0.003 etc
Base 4 representation uses
Powers of 4 and not powers of 10
Question is: does nature have infinite zoom?
No, it doesn't. But we are doing maths in a math world.
Sum 41
Makes sense for every base tbh
Yep. I have base 3,4, 5 and 10 all in the channel so far :)
Really 0.xxxxxxxxx=x+1 in a x+1 base(x is a number from 0 to 8)
close, 0.xx... = 1 for base x+1
If , as many claim, 0.999999.... =1,
What is (1+1/0.99999......)^0.99999.......?
Surely it is equal to 2
Correct
@@FrostFlame75 So if I put 9,999$ into the bank I can take out 10000$ the following day because that's close enough, or am I missing something here, like reality?
@@davedixon2068 the difference is that 9.9 repeating goes on to infinity where you were 9,999 does not
You could add another number onto the end of your amount of money whereas with 9.9 repeating there is no end for you to add a number onto.
@@davedixon2068the difference between 9999 and 10,000 is 1. The difference between 0.99... and 1 is 0.
Odd way to make the bi flag
Approaching a value infinitely does not make it equal. It's like that delivery I ordered from China. It's only approaching. It was approaching six months ago, it was approaching a year and a half ago. It's still approaching. It's not here yet...
I understand that the limit will approach 1, but saying that it is equal to one is like saying that f(c)=limf(x) as x approaches c. Or am I wrong somehow? I know that in the AP exam for ap calculus they'll mark you as wrong if you do that because it isn't true. Also, no matter how far you take it, no matter what n you are at, there will always be a portion of the square that is not shaded, making it not 1. The limit will approach 1, but there is a difference between the limit approaching 1 and it being equal to 1. I'm not prideful tho, so if I'm wrong somehow, please tell me.
limits are fixed values. So the limit is 1. In this case, the sequence of partial sums will approach 1 (which is the limit of the sequence). But the limit itself is 1. The infinite sum of a series is defined to be the limit of the sequence of partial sums, so here when we say the infinite sum equals 1, we mean the limit equals 1. Limits don't approach things; things approach limits.
What the fuq is a base?
@@cyrus1586 so binary and decimal are bases? I (somehow) think that decimal was the universal accepted way of counting
@@inhnguyenminhkhoa4241it is for most things now but for computers it is base 2 and a lot of cultures have used different bases in the past
@@inhnguyenminhkhoa4241 Yes, binary, decimal, hexadecimal, etc. are notational bases for writing numerical quantities. Different cultures at various times in history have used base-12, base-20, and base-60 positional writing systems for numbers. In modern English speaking cultures, we are using base-10. You can still see some historical artifacts of other number bases, for example, the existence of the words "eleven" and "twelve" rather than "one-teen" and "two-teen". Also words like "dozen", "gross", and "score"... like "Four score and seven years ago", and how there are sixty seconds in a minute, and sixty minutes in an hour, and twenty four hours in a day -- twelve hours for day, and twelve hours for night.
@@inhnguyenminhkhoa4241 Yes, binary and decimal are both bases, being base two and base ten respectively. Hexadecimal is another base, base 16. You very rarely use other bases, but playing around with them can be fun.
Only near One but it will Never be one
It will at infinity, like how he does for the INFINITE sum
When do you colour in that last 1/4 of the square though?
Mathematicians cheat. That's the truth about it.
They do all of it in one go. Axiom of choice. Look it up.
You don't, but it's area is 0, so the area colored is still the full thing
I still say theres a difference, but you can never, never ever, tell the difference.
????????????????
I love how other number bases can help to solve infinite sums. I realized this thing one day while studying "decimals" in other number bases. Basically if m is the largest digit in base m+1, then the number 0.mmmmmmmmm...=1 in that base.
Like 0.999… =1; or for any base (n+1), 0.nnn… =1 (0.111… =1 in base 2; 0.FFF… =1 in base 16, etc.)
I'll admit, I looked it up before and there is a concept for an infinitely small value above or below zero but... Functionally it's just fluff to rewrite things as .9 repeating + something. Still for fiction purposes it's a great concept to draw inspiration from.
This should also allow us to solve for 0 by borrowing a bit from the future and the past.
This problem makes it harder than it actually is, and the visuals here should help to unmask it. You're basically taking the whole thing from the get-go, since you're continuing to take 3/4 of whatever remains after each step, which literally leaves nothing to be set aside. Whatever isn't taken in one step will be subdivided by the same amount in the subsequent step, and whatever's remaining there will be subdivided in the next step by the same amount, ad infinitum.
No such thing as one fourth. It's a quarter.
One fourth is a correct term.
@@officialteaincorporated243 Do you also use 'one twoth' instead of 'one half'.
@@howardg2010 No, because twoth isn't a real word. One second would be a valid term if it were not used for both time and rankings as well, making it extremely ambiguous. While I mainly say quarter, fourth is a completely valid and unambiguous term.
Then 4.444444.. = 27
When you think about it, its a very strage conception of "equals." How can these squares that never ultimately reach the complete whole somehow equal the whole?
What does equal mean if this is true?
Here you can find many ways of showing that 1 = 0.(9) from intuitive arguments to mathematically rigorous proofs. I like the one using fraction 1/3. en.m.wikipedia.org/wiki/0.999
He took the other surfaces and slid them on top of each other then asserted that it means 1/3. That's not how this works.
Was wondering if anyone else was gonna say that
It works in math, but if you need 1 full something to have something work (to travel to some point, to power something, to finish a task) then .99999999 repeating means you will always be just an nth away from finishing.
0.999999... is actually equal to 1
The limit will never reach what it's trying to approach, so they will never be equal to 1
that's not how limits work
Quite literally, the opposite of how limits are used. You'd be right in saying that it approaches 1 if we do this a finite number of times, no matter how big. HOWEVER once you do it infinitely you will get an equality.
But the last square is never shaded
The concept of infinity leaves us with an unshaded square with zero area
which is a nonsense... If the 1/4th has zero area, then the 3/4 also has zero area, and so on and so forth back down the line until you're further and further from 1, rendering your proof false and invalid... That's why this is a LIMIT of the geometric series, and not the VALUE, the value at n is 1-1/4^n, no matter how infinitesimal it is, there is always another potential digit that could be added at n+1 that separates the geometric series from the number 1, numbers do not have a planck length, they have no conceivable minimum size.
Except it is not fully shaded, there will always be an infinitesimal area that is unshaded.
That's where and why the talk of Limits came into it.
unless you find any point inside of the square's area that is not shaded (which is impossible) its is exactly 1!
Not if we go to infinity like in this video (or if we use hyper reals which we don't cus that would've been made clear)
I don't agree. Why? Because there is ALWAYS one point that is not filled. Even after infinite amount of these you can still fit infinitely more 3/4
Dude, it is ∞.
And it is really 1, not smaller/bigger.
You don't have to agree, also unless you find any point inside of the square's area that is not filled (which is impossible) its is exactly 1!
Any single point you can possibly give, it will be filled after a finite amount of steps, so when saying "after infinite steps" those will all be filled
Math does not care about you agreeing or not.
@@badorni69The very corner, the point directly adjacent to two sides of the perimeter, will never be filled. Each step shades 3 of the squares and always leaves behind the corner, meaning that no matter how many steps take place the point where two sides of the square meet will always be left unshaded.
Ah I see, 1/4 = 1/3
You’re falsely assuming there is this thing called; “At Infinity” - which contradicts The premise of “Infinity” ( ? )
clever language tricks.. but not true ..
100% true.
If you do that infinitely, you always have an infinitesimally small piece that is missing. At some point, the piece becomes of a scale so small, it becomes insignificant, as the measuring tool is larger than the thing needed to measure. It approaches 4, though, as it is always 3+x filled in, but never fills in the fourth square. You may say the compared unit is one square, but it is made of 2 left and 1 right or 2 bottom and 1 top. The final is 1, as the limit approaches 1 on the lower scale, but by comparison, the overall (on the larger scale) approaches 4.
Wrong
Ehm no. It's very true. And math doesn't really care about you believing it or nor
However I don’t like what’s written at the bottom of the vid. Cause it’s not true!😊
I mean yeah, but also no surely? Why is there a limit? It's infinite. Very very very very very very nearly 1 doesn't equal 1 😂
Don't get me wrong, I get that in _practical terms_ sure yeah it's 1. But surely if we assume some make believe infinitely scaling mathematical world, it's not 1
No, every single point you can possibly give, will be filled after some finite amount of steps, therefore after "infinite" steps all those points will be filled
But surely any finite is smaller than any infinite? So if there's infinitely small space you can't fill it with a finite number of steps
@@tomjardine-smith2793 I mean if you want to look at it that way, that infinitely small space also has an infinitely small area, so the area covered is still the same size as the full area
I believe it's not. It will be infinitely close to 1, but not 1.
That is just false
Incorrect . It's the same logic as 0.999999... = 1
Math is dumb
🥲