Does 0.333333... equal 1/3 ?

Yes.

I guess you slept through 4th grade math.

OK, genius, explain it to me then.

0.333…=3/10+3/100+3/1000+…
is a converging geometric series with a=3/10 and r=1/10

converging geometric series are a/(1-r) so the answer is (3/10)/(9/10)=1/3

It's real close.

So you are asking if 1/3 is equal to .33333333…

So if you divide 1 by 3 you get .333333333…

Or if you take .333333333… x 3 you get .999999999999999 which is 1.

That's not a proof, you just re-stated the question as a statement

X = 0.33333... = 3/10 + 3/100 + 3/1000 + ... = 3/10^1 + 3/10^2 + 3/10^3 + ...

10X = 10 * (3/10^1 + 3/10^2 + 3/10^3 + ... ) = 3 / 10^0 + 3/10^1 + 3/10^2 + ...

10X - X = (3 / 10^0 + 3/10^1 + 3/10^2 + ...) - (3/10^1 + 3/10^2 + 3/10^3 + ...)

10X - X = 3/10^0 = 3

9X = 3

X = 1/3

Eq #1: x = 0.333333...

Eq #2: 10x = 3.333333...

Subtract Eq #1 from Eq #2.

9x = 3 Divide both sides by 9.

x = 3/9 ----> x = 1/3

Works for finding the fractional equivalent of any repeating decimal.

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The key word is "equal." No, it does not equal .3333...

The reason is that "infinitely close" is the same same as "exactly."

It has, as usual, to do with definitions.

If you were an ant marching down a number-line towards the number 1 and you were at .999, you'd never "get there" if you could only add more 9s to the end of the decimal.

That is what 1/3 is, so sorry, it is not the "exact same thing."

But you (or your teacher) might define "equal" differently than I do.

He didn’t ask for a proof. He asked for an explanation. You failed.

I love this proof. I don't really agree with it, but it is right, nonetheless.

Not exactly, but roughly.

It's how it's most commonly depicted as a decimal.

OK, then look at the number ...333333333.0 (infinitely many 3s on the left of the decimal point rather than the right).

Eq #1: x = ...33333333.0

Eq #2: 10x = ...333333330.0

Subtract Eq #1 from Eq #2.

9x = -3. Divide both sides by 9.

x = -3/9 ----> x = -1/3

That seems odd.

Love this insane asylum of a forum. On the same page we have a thread where the discussion touches on a bit of number theory, and in another thread we have Rich calling everyone f4g5 and asking us to defend ourselves while flipping us off lol

For the people who are down-voting me, I agree on a mathematical level that the Wolfram proof is correct.

Where I disagree is that linguistically "very close to 1" is NOT the same as "exactly 1." Calculus is its own world though, so those guys can do what ever they think is best...

That is where math and linguistics will just have to disagree.

Instead of N 3's, choose a finite number of 3's. For 0.33333..., the limit as N approaches infinity is 1/3. For your 33333... number, the limit as N approaches infinity is infinity.

It's not "very close to 1/3." It's "equal to 1/3."

I agree with you mathematically. I don't agree linguistically. There is a difference.

No there's not.

Not trying to sound snarky but you obviously never took calculus because under your logic you would "disagree" with all of it.