0.8888… = 1 (in base 9)

This is a short, animated visual proof showing the sum of the infinite geometric series with first term 8/9 and ratio 1/9, which in turn allows us to compute the sum of the series of powers of 1/9 and determine an interesting base 9 representation of 1.

For a longer, wordless version (more dramatic) of this animation see

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To learn more about animating with manim, check out:

[UserYouTube]

Math in school: 🥱
Math in internet: 🤩

[anujthakur]

This is basically geometric progression in which the common multiple r has value less than 1 so show the formula to get sum of infinity a/(1-r) can be used, its the same thing. Btw a is the first term in the above formula

[aireyroblox]

So basically for any n, 0.nnn repeating will be equal to 1 in base n+1

[pieceofwaterofficial]

1. Let 0.(n)=x
2. Multiply both sides by 10
n.(n)=10x
3. Substract x from both sides
n.(n)-0.(n)=10x-x
n=9x
x=n/9
4. Recall 0.(n)=x
0.(n)=n/9

[horriblememes]

I mean yeah you can do this with any fraction

[FaerieDragonZook]

The areas may be equal in measure, but there is a point in the center of the square that is never shaded. One way you can think of this is that the limiting difference between 0.888..._9 and 1.0 is an infinitesimal.

[Azuro-pjxj]

Isn't x-1 over x + x-1 over x² + x-1 over x³ ... always 1?

[Jff_Kng]

I've just realised that this videos of proving that something is 1 in base X is not as impressive since it's just like 0.999... is 1 in base 10 lol

[mitsunam]

How many series are left with these proofs?

[brianhale]

Or we can be perfectly happy knowing that the area of the square is one square and go about the rest of our days without obsessing over tic tac toe boards. 😅

[scottabroughton]

You could also do 0.eee = 1 in hexadecimal.

[FelanLP]

Good example how is equal to 1 (in base 10).

[BennoRob]

Asymptoticism should be a word because this would be Antiasymtoticism

[jasonhilliker]

Wouldn't it be more accurate to say it approaches 1 instead of equals 1?