Where Did the Rest of the Cake Go?
You must’ve heard this joke.
A kind kid has a cake and he is kind enough to share it equally with his two other friends.
He cuts it into three equal parts.
Mathematically each part is: 1/3 = 0.333...
“But wait! 0.333... + 0.333... + 0.333... = 0.999...
Where did the rest of the cake go?”
Legend is:
“It’s on the knife.”
Or You Might Remember It from Your High School
Not the joke , but the fact that 0.999... = 1. It showed up in math textbook maybe in chapter on real numbers and you showed it using the algebra trick.
/In truth,Letting x = 0.999... Pre-Multiply both sides by 10: 10x = 9.999... Subtract the original: 10x - x = 9.99999..-0.9999...=9 => 9x = 9 ⇒ x = 1Boom.
You take it in blood now like an innocent obedient kid. Most of your classmates accepted it and moved on. Maybe some of you (the more curious ones) found it hard to digest.
To the latter folks , let me try Digene you today.
First off, i would introduce to you this idea called limit, lets say when we write:
0.9, 0.99, 0.999, 0.9999, ...
And every time, the number is just a little lesser than 1. It feels to you like 0.999... is forever “approaching” 1, but never quite getting there.
But in math, when we say:
0.999...
We don’t usually say “still going/processing". What we say is: the limit of that process, written as lim (of that thing)—.
And here in 0.999999..... although limit machinery is not written explicitly but observe these infinite dots throwing infinite 9s,no gpt can tell you this;it can be without any sorcery translated to: $$ \lim_{n \to \infty} 0.\underbrace{999\ldots9}_{n \text{ times}} $$ which is 1(fact fact).
that's how limits re defined, gettin rid of writing infinite dots,
and its ok if the above paragraph came off to you incoherent,its ok.)
The Infinite Series View
Another way to look at it is through a geometric series:
by writing0.999... as 0.9 + 0.09 + 0.009 + 0.0009 + ...ie 0.9999999....=0.9+0.09+0.009+0.0009+.............
This is easily a textbook defined geometric series with:
- First term: a = 0.9
- Common ratio: r = 0.1
The formula for such a sum is:
a / (1 - r)= 0.9 / (1 - 0.1) = 0.9 / 0.9 = 1
once again:
0.999... = 1
Different path,same conclusion.
And same old confusion!
NOw the real question to be asked— Is there any Number SANDWICHED b/w 0.99999... and 1(?)
This is where it gets real & its over to real analysis .
You might think:
Okay, cool. But surely there's a tiny number(s) secretly between 0.999... and 1?
oh if thats the case ,why not? try to find one.
You try putting more 9s:
0.9, 0.99, 0.9999999.....9
But you still end up at the same old ugly number 0.99999999...
Then maybe you try something like:
0.999...1 ,may be 0.99999999...2 or may be 0.999999......8
But that number doesn’t make sense. Firstly, there’s no place “after infinite 9s” where you can poke your nose and put 1 or any digit.
And even if you live long enough to put 1 after writing infinite 9s:
0.9999999999....... 1
That number is actually lesser than 0.9999....9... It doesn’t fit between 0.999... and 1 — it actually goes behind 0.999...
You must do something better with those years. Lol.
Completeness /No Gap property:
(area where you analyze real system), we say the real numbers have this niceness called completeness and we say real numbers 're complete unlike rational numbers etc .Which more or less translates to :
There are no holes ,no gaps,nothing whatsoever, in the number line. If two real numbers don’t have any number or anything between them, then they must be the same number, as no gaps when I say nothing I meant literally nothing not even emptyness to hold in between,thus essentially the same numbers.
This sounds trivial, but it holds great wisdom.
Since there’s no number , no decimal, no fraction that fits between 0.999... and 1,they overlap thus equal.
To keep it short, It’s okay if this still feels strange.
Even I, as someone who 've studied some pure math, stand in solidarity with you.
But then when you understand how real numbers and limits work, you realize:
The chase is over. The number has reached its destination with all its baggage.
And the cake? It’s whole.
0.999... = 1
And if you’re still wondering where the missing part went...
It’s on the knife. lol
If you pick up math someday, you’ll find much more rigorous and satisfying proofs for it , ones that hit all your senses. But for now, that’s too technical for this post.
Also If you're curious, feel free to write to me. I’d love try explaining the deeper side of it.
tata.
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