# Can 1/3 really be equal to 0.333

Can 1/3 really be "equal" to 0.333...

Can 1/3 really be "equal" to 0.3333333333...........?

Seems like no matter what you do, when you add three of these together you will get 0.9999999999999.....

As close as that number is, it is NOT equal to 1.

Thoughts?

Can 1/3 really be "equal" to 0.3333333333...........?

Seems like no matter what you do, when you add three of these together you will get 0.9999999999999.....

As close as that number is, it is NOT equal to 1.

Thoughts?

Actually, yes it is equal to 1. Do a search of the forums, this topic has been covered numerous times before.

Gib Z
Homework Helper

There has been a lot of debate on whether 0.999999... equals 1 or not.

Gib Z
Homework Helper
Yes I've realised that, and there are many ROCK-SOLID proofs on why it does, I can think of at least 7 different ways of proving it at the top of my head. I can manage that because I've seen thread after thread of experienced mathematicians providing us the proof, and some ignorant people who absolutely REFUSE to believe them.

I'm not saying this poster, Holocene, is one of those ignorant people, but just search up the forums, theres many threads on this.

Math Is Hard
Staff Emeritus
Gold Member
There has been a lot of debate on whether 0.999999... equals 1 or not.

There is no debate. This thing keeps turning up like a bad penny.

It's a common question, but the answer is not obvious.

I seem to recall that in Conway's surreal numbers, there is indeed a difference between 0.99999... and 1.0. (Don't shoot me if I'm wrong)

Also- computer arithmetic does admit a distinction. 0.999999... is always less than 1.0 on a computer, because computers always work to finite precision on decimals.

Also- it is debatable whether real numbers such as 0.99999... can be ever realized in our physical universe, whereas 'countable' integers can. (It's easy to have 1 electron, whereas it's not clear that any number in physics can have an infinite decimal expansion- or whether it becomes fuzzy after so many decimal places.)

So - in my opinion it's true that 0.99999... is the same as 1 in the real number system, but we have to be aware that several important physical systems do not use the real number system.

Gib Z
Homework Helper
"Also- it is debatable whether real numbers such as 0.99999... can be ever realized in our physical universe, whereas 'countable' integers can. (It's easy to have 1 electron, whereas it's not clear that any number in physics can have an infinite decimal expansion- or whether it becomes fuzzy after so many decimal places.)"

Yes, it is possible to have 0.9999... electrons, because 0.9999... is EXACTLY 1, so its the same as 1 electron.

"Also- it is debatable whether real numbers such as 0.99999... can be ever realized in our physical universe, whereas 'countable' integers can. (It's easy to have 1 electron, whereas it's not clear that any number in physics can have an infinite decimal expansion- or whether it becomes fuzzy after so many decimal places.)"

Yes, it is possible to have 0.9999... electrons, because 0.9999... is EXACTLY 1, so its the same as 1 electron.

But is it possible for any number in physics to have an infinite decimal expansion? 0.9999... electrons may be a bad example. You have to show that physics obeys the real number system, which is a matter for experiment perhaps, rather than math. It may be that inherent quantum fuzziness prevents us from saying that a charge is exactly 0.333333... of an electron. I don't know- but I don't think the answer is so obvious.

It's not even clear that there is such a thing as 1 electron. There is a non-zero amplitude that the electron could annihilate with its antiparticle for example.

The very fact that it's possible to dream up number systems in which 0.99999... is never equal to 1 should give you pause for thought. Maybe electrons are better described by surreal numbers than by the real number system. I doubt it- but you can't say for sure without experiment.

Gib Z
Homework Helper
The surreal numbers are a field containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number.

But since there are many proofs 0.999..is indeed exactly 1 and therefore a real number, it doesn't matter if electrons are better described by surreal numbers, because the number is still a real number and equally well represented by the reals.

The surreal numbers are a field containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number.

But since there are many proofs 0.999..is indeed exactly 1 and therefore a real number, it doesn't matter if electrons are better described by surreal numbers, because the number is still a real number and equally well represented by the reals.

No. It's only true that 0.9999... is equal to 1 in certain number systems, including the important real number system.

Computers are a good example of a system that can't fully implement the true real number system. It is not true on a computer that 0.99999... = 1, because it is impossible to implement an infinite decimal expansion in a finite memory.

It's possible (though not likely) that the universe could be a big simulation run on an alien computer. If that were the case then I would not agree that 0.9999... = 1 in external reality, for the simple reason that such a universe does not admit infinite decimal expansions of any number, and that includes 0.99999....

Again, it is a matter for experiment whether 0.99999... = 1 in the external universe- though we are all agreed that it works in the real number system.

From http://www.maths.nottingham.ac.uk/personal/anw/Research/Hack/ [Broken]

It should not be thought that the Hackenstrings approach in any way shows that conventional mathematics is wrong'. Rather, the axioms of Hackenstrings arithmetic are different from those of the real numbers, one consequence being that in Hackenstrings 0.999... < 1. So any proof that 0.999... = 1 must fail when applied to Hackenstrings, which in turn must mean that one of the different' axioms has been used. In particular the `optical' proofs are making an indirect use of the Archimedean axiom; but this use is rarely made explicit, which makes the proofs misleadingly simple or even inadequate.

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I'm surprised some say that it is equal to 1.

The way I see it:

If you subtract 0.9999999... from 1, you are left with 0.0000000...1.

So, if you have something left over, they couldn't have been equal in the first place, no?

I'm surprised some say that it is equal to 1.

The way I see it:

If you subtract 0.9999999... from 1, you are left with 0.0000000...1.

So, if you have something left over, they couldn't have been equal in the first place, no?

In the limit- that little bit vanishes to zero and the two numbers are identical.

The real number system admits limits like that, but I'll leave it up to people more knowledgeable than I to show you why that is.

Also- try looking at the Wikipedia entry

http://en.wikipedia.org/wiki/Proof_that_0.999..._equals_1

I think for example if you take the equation 1/3*9 you get 2.999... & will continue on forever. However, if you rearrange the equation as is possible with the commutative property you get 9*1/3=3. Which is rather perplexing. Anyway, I'll have to check out that wiki link then.

Gib Z
Homework Helper