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0.66666… = 1 (in base 7)
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- Опубликовано: 16 июн 2024
- This is a short, animated visual proof showing the sum of the infinite geometric series with first term 6/7 and ratio 1/7, which in turn allows us to compute the sum of the series of powers of 1/7 and determine an interesting base 7 representation of 1.
If you like this video, consider subscribing to the channel or consider buying me a coffee: www.buymeacoffee.com/VisualPr.... Thanks!
For a longer, wordless version of this animation with two other proofs, see • Three Geometric Series...
Also, check out my playlist on geometric sums/series: • Geometric Sums
This animation is based on a proof by Stephan Berendonk (2020) from the November 2020 issue of The College Mathematics Journal, (doi.org/10.1080/07468342.2020... p. 385)
#mathshorts #mathvideo #math #calculus #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #geometricsums #series #infinitesums #infiniteseries #geometric #geometricseries #equilateraltriangle
To learn more about animating with manim, check out:
manim.community
...At this rate, you should be using a hexagon...
Wouldn't a heptagon (because it has 7 sides) be even better?
@@wyattstevens8574 No, it's like using a triangle to figure out a sum of 1/4^n (Which will become 1/3)
Why ? You'll get 6 sides and a central piece that is a smaller version of the entire shape that is 1/7 of the area, making this visual proof possible
In a heptagon, what visual representation can be made about such a thing ?
It is the bestagon
Best reference@@gabekrieck6772
@@gabekrieck6772 true
0.nnnnn... = 1 in base (n+1)
is this always true?
@@RandyKing314 yes, always true
@@cccexestartedwhat about base 1
Oh yeah I forgot about that rule
@@taylormarinescu805 also true
I thought it was Pink Floyd at first
Usually we use the formula
S = a_1 / (1 - r) but the visual proof makes it much easier to understand with the eye and I'm glad you can help others understand it more easily
Love this! I only wish RUclips didn't cover the bottom of the video
Longer version on my channel (wordless) doesn’t have as much covering it
@@MathVisualProofs I'll have to give it a watch!
If you go into comments and hold the bar near the bottom you can see it without obstruction.
I have never been as much satisfied by a visual proof than this. My god did it look beautiful when the 6 triangles folded over to reveal themselves.
😀
By reading the title, I was gobsmacked, collaterally dumbfounded and rendering my prime knowledge of math into shreds.
Sigma means sum? If that’s all I learned today…
Yes
You mean that your teacher didn't mention sigma meaning sum in passing in your 7th grade math class?
That's why they chose Σ. It's the Greek equivalent of S, and S stands for sum. Furthermore, Π is the Greek equivalent of P, and P stands for product, so Π is used for products.
Omg I love these. I always was a visual learner. Numbers just are language I don’t speak. Awesome. I’ll maybe learn a bit too.
New Roblox logo just got leaked
This is why i hate supertasks
Best representation
It's Not Art
It's Not Math
It's Both
this is pretty easy regarding 0.999... = 1 in base 10, so in base 7 0.666... is the same as theres no integer between 0.666... and 1
What software do you use for animation? Thank you
Pretty sure it's manim
manim
So, polydivide (my preferred way of saying divide multiple times) by 2 and 3, then connect opposing trisectors and vertices, then connect the remaining trisectors with the midpoints, and then repeat the process with the middle triangle indefinitely, which will *precisely* give you [ *1* ]
Love listening to these videos. I don’t learn or retain any of it but it seems pretty interesting.
One can appreciate why the ancient Greeks adored geometry!
The infinite sum of a single simple fraction is the one below. 1/3+1/9+1/27... = 1/2
And 5/6+5/36+5/6³ = 1 ?
In all configurations.
So cool.
1/n < 1/n-1. Therefore, it's not one below.
@alex_ramjiawan I mean that the sum of thirds gives 1/2, the number as denominator gets smaller, making the portion get larger.
1/5+1/5²+1/5³.....=1/4
wrong the 1st one's ans. is 3/2 And 2nd one's ans. is 6
@@ketanchavda5511 That's incorrect.
Even more interesting than 0.nnn... = 1 in base (n+1)
I find it fascinating the way the triangle is divided into 7 similar pieces, smaller equilateral triangles (after transformation). Dividing to 3n(as long as n is equal or greater than 3) and n^2 number of pieces is fairly easy. But I have been obsessing over this for over a week now.
Is there a way to divide an equilateral triangle that the resulting sections are 1/2 of the original, and to be transformed into equilateral triangles as well? (similar to when you rotate the pieces to match)
What about 5, 6, and 8?
Why is it that 7 works so much more "easily"?
Hi, can you please recommend any books you found useful in your journey of math? Ty❤
It’s always the black magic of the concept of infinity….
I lost you at “equilateral triangle”
Oh no, fractal
its the same thing as 0.99999...=1
Bruh, I'm in holidays rn stop overloading my mind
For all sums of the type m*(1/n+1/n*n+...), the sum is equal to m*1/(n-1)
I always wondered how to prove that 1/(x-1) can be represented by an infinite sum of 1/(x^n)
S=1+x+x²+x³+...
Sx=x+x²+x³+...=S-1
Sx=S-1
Sx-S=1
S(x-1)=1
S=1/(x-1)
This video has me all sixes and sevens
YOOOO MY COMMENT WAS RIGHT DIDN'T KNOW THERE WAS A VISUALIZATION FOR THIS ONE AS WELL
n being a digit
0.nnnnnnn... = 1 in base (n+1)
Well, meaning 0.99999999999… = 1 (in base 10) is true?
Well, for bases higher than 10, maybe HEXAdecimal might be 0.FFFFFFFFFFF…….. = 1 (in hexadecimal)
@@chrisrodriguezm13 yeah that works too
I'm confused but I feel enlightened
7 is very weard
Base 7 minus base 1 equals base 6. Therefore, can I see the base 6 triangle? Thank you.
My brain is just saying: So triangle is triangle cuz triangle gets smaller meaning triangle. My mind is discombobulated 😭
Or for short
6/7 + 6/49 + 6/343... = 1
0,66666 periodic in base 7 equal 1, just like 0,999 periodic in base 10 = 1
no summation of (1)/(7^i) upto infinitity is 7/6
routh's theorem?
0.k repeating in base (k+1) is 1
Fractangle. :)
Wow. He did it again. The infinite sum of 1/n is always equal to 1/(n-1)
sum of 1/n from n to infinity always equals 1/n-1
Easy maths proof using
telescopic sum
I think it's "tends to 1"
…and so on and so forth…
Does that mean 0.000000000000 ... is = 1 in base 1?
Depends on how accurate you really wanna be.
I mean, if youre buying a car and youre a dime short, chances are you can still buy the car, but if youre trying to buy a soda and youre a dime short, chances are you wont be able to buy the soda.
Does that mean that (price of a car-a dime)=price of a car?
No, but in most cases, the dime is not as important as the rest of the price.
Unless a dime=0, or at least a dime
@xinpingdonohoe3978 has to do with percentages. Car is 40k, soda is 1, dime is 0.1. You do the math.
@@blu-birb right, so that's not applicable. Our percentage is 0. We want a change of 0.
@xinpingdonohoe3978 the example is sound. This video claims that X is equivalent to 1, where X=/=1. So that's where I am saying, "depends on accuracy"
In the real world, 1-0.1= no soda, but 40k-0.1= yes car.
Being that it would be 1/10 of a soda and 1/400,000 of a car.
@@blu-birb but your claim that x≠1 is not correct, so you've either lost soundness or validity.
Nice !
does 0.99999... equal 1 in base 10?
yes
Well, two minutes into the video and already the objectively incorrect take of "Heavy Blade is really bad" has appeared. Not giving me confidence in the rest of this video.
Idk seems like some type of fractal to me.
Now do 0.111111111111111111111111111111111111111111111111111111111... = 1 (in base 2)
It’s on the channel… 😀
0.NNNNN eventually equals to M
Depends on what you mean. 0.3333... won't equal 6.
So basically if it's hexadecimal, 0.FFFFF... is 1 right?
Maybe
Yes.
Great
I allways thought u would have 1/ inf not shaded but i think it means the same thing
You do. But then if 1/∞ is a number, the only number it can be is 0. If 1/∞ exists, 1/∞=0. Otherwise, 1/∞ doesn't exist, and so you can't have 1/∞ not shaded.
@@xinpingdonohoe3978 huh i thought that 1/ inf would be like how 1/10 would be 0.1 but 1/inf would be 0. Then infinite zeros then one so idk how to describe what I'm thinking but like above 0 but it's infinitely small
@@JaphethThomas-um9gw I get what you mean. Let me explain why it doesn't work.
First, there isn't such a thing as an "∞th" decimal place. There's a 1st, and a 2nd, and a 3rd, and so on. There's an nth for every positive integer n. There are infinitely many of these things, because there are infinitely many natural numbers, but every single one is in a finite position. Just like the numbers themselves. 1,2,3,4,5,6,7,... There are ∞ of these, because they never end, but each individual number can be reached by travelling a finite distance. Same with the decimals. Every single bit of the decimal can only be finitely far along, so we can't have a 1 in the ∞th place because there isn't an ∞th place.
Also, let's assume it does exist. 0.00...1, so that's 0 point, then ∞ many 0s, then a 1. Call it x.
Then 10x=0.00...1 as well. It has ∞-1 0s, but we know ∞-1=∞. Nothing changes. That's the special thing about ∞. For regular numbers, n-1≠n, but for infinity ∞-1=∞. This means we have the same thing. So 10x=x. But then if we subtract x on both sides, 9x=0. Two numbers multiply to 0 if and only if one or both are 0. 9≠0, which means x=0, so 0.00...1=0
All I understood was 1 7th
in base 7.... 😡
\sum_{n=1}^\infty\frac{6}{7^n}
Niceee sir
By using infinity you can prove any number equal to any another number
You don't need infinity. Just let 1=0, then you can prove all numbers are equal.
Only if you use it improperly. Limiting processes are well-defined
Huh
I’m nodding my head in agreement as I have no clue what’s going on
No there won't, there will alsways he a missing 1/7!
1/7 is pretty bit. That gets eliminated quickly.
6/7⁶⁹⁴²⁰=1
What
Surely there will always be an empty black triangle in the middle. It will never be completely filled.
Seems like black hole energy: the drawing in of energy or expelling energy out like that hawkings energy thing. 🤔
If you stop after a finite time yes, but in the limit no
☠️☠️☠️
Bro I understood nothing 😬😬😬
Bro I understood nothing 😬😬
😢
All the triangle is coloured except the point in the center! No matter how many steps you do, this point will never be colored.
A non-mathematician could argue that this point is the difference between 0.66666 and 1.
Basicly its like saying 0.9999999999999999_ = 1
Not even, this is an asymptote, no matter how many parts you fill in unless you fill it in fully it will not be equal to one, also 6/7 isn't .666, that's for the 1/3
What's the area *not* shaded after infinitely many iterations? If it's not 0, what number is it?
@@xinpingdonohoe3978 it's infinitely small but still never zero, it can't be zero because it never fully filing in that part, so it is constantly striving to become zero but it cant
@@hyoominoid01 what does it mean to be an infinitely small number that's not 0?
Is it smaller than 10¹⁰⁰? What about 10¹⁰⁰⁰? Or 10⁹⁹⁹⁹⁹⁹⁹⁹⁹? Or 10^n for *ANY* number n? Because there is no positive number that can satisfy all that.
@@xinpingdonohoe3978it's always smaller than 1/7 of the last iteration of the process show in video
@@xinpingdonohoe3978I suppose I could've been more specific when I said infinitely small
It seems more like the infinite sum is like a sub-atomic rounding up principal
Isn’t 6/7 more than .6
Base 7. We can only use the digits 0 to 6.
@@xinpingdonohoe3978 thanks chief
Technically no. While if you continue that process, you will always be moving closer to 1, you will never quite reach it. Even if it continues infinitely, it will never quite reach 1.
It doesnt fill in the triangle completely do yk what a fraction is and how it works?
What's the limit of the area that isn't shaded?
How come
The centroid will never be shaded. There may be no difference in measure, but it is false to say that the entire triangle will be shaded.
In the limit the entire triangle will be shaded
@@InsertPi not every point. Just an amount of the triangle that has the same area measure as the entire triangle.
If it’s 0.5 or greater, it’ll always be 1.
What do you mean?
@@xinpingdonohoe3978 i was thinking of something else my bad
but i was thinking if you get a half and add a half to the space left forever it'll be 1
I through shade on 6/7th of your mom's 😂
There is something ambiguous about number
Too trivial and put into a too long context. Waste of time.
Not quite. But sounded very convincing. Nice try.
Given that it's right, you seem to have an illogical grievance.
But it will never be all shaded. Even if you reach the Planck's limit.
In real life not, but in math it will...
Planck's limit? Explain what you mean by that.
I have heard this many times from very well educated sources but it is false, infinite means it goes on for ever and therefore consists of a constant repetition forever of the same process. Each process always leaves a small part unaccounted for, therefore an infinite process APPROACHES 1 but never reaches it. It does not equal 1, it approximately equals 1, that may sound small but its a very big difference when this incorrect proof is used in certain ways to state that it equals 1.
Convert this same thing into base 10 with an infinity repeating fraction that approaches 1 (0.999...) is equal to 1. Convert it into a fraction and you get 3/3=0.999...=1.
@@TurtleBoi-lg4rw No this proof is what people use to justify the statement that 0.9 recurring equals 1. You cannot use a false proof to make a statement and then claim the statement proves the false proof.
Just call it 1-
Problem solved
@paulosullivan3472 then prove that my statement is wrong, do 1-0.999... and tell me the answer
@paulosullivan3472 and if I am wrong that means 3/3 =/= 1