# .333 does not equal 1/3?

#### Algr

In another thread it was stated that an infinite number of zeros is undefined - different equations could make this value be any number you wanted. It was also stated that multiplying a number by infinity was meaningless, and inf/inf is NOT one, but also undefined.

How then is it possible for an infinite number of .3's to equal 1/3, or indeed any real number? Wouldn't an infinite number of positive values invariably equal infinity?

Also, by definition, adding an extra 3 to a finite string of 3s would never result in 1/3, so why would repeating a failed process an infinite number of times result in a success?

#### ice109

a number with infinite decimals is almost the definition of a real number.

the sequence of partial sums of this:

$$\sum^{\infty}_{i=1}\frac{3}{(10)^i}$$

is definition of 1/3 in the reals.

#### Hurkyl

Staff Emeritus
Gold Member
In another thread it was stated that an infinite number of zeros is undefined
I imagine what was actually said is a slight variation of this -- such a thing would be perfectly reasonable in certain contexts. (e.g. there are strings of characters that consist of infinitely many zeroes)

It was also stated that multiplying a number by infinity was meaningless,
Meaningless in the real numbers (and the naturals, complexes, integers, etc) -- however other structures may have an element named "infinity", and those structures may have a multiplication operation for which some products involving infinity are meaningful.

(Two of the most common examples are the projective real numbers and the extended real numbers)

How then is it possible for an infinite number of .3's to equal 1/3
I assume you didn't mean an "infinite number of .3's", but instead "the decimial number consisting of infinitely many 3's to the right of the decimal place (and 0's to the left)"? It's fairly simple, and follows directly from the definition of decimal multiplication, decimal equality, and division: 3 * 0.333... = 1.

Alternatively, you can apply the formula to compute the real number denoted by the decimal string 0.333..., and observe that it results in the real number 1/3. (As ice109 did)

Also, by definition, adding an extra 3 to a finite string of 3s would never result in 1/3, so why would repeating a failed process an infinite number of times result in a success?
Not by definition. And there are flaws in your logic:

. You cannot form 0.333... by (ordinarily) iterating that process -- you would have to invoke some sort of transfinite iteration, which also includes "limit" steps.

. That form of induction only works for ordinary iteration -- it does not pass through a limit, and so cannot be used to generalize from the finite to the infinite.

#### Werg22

A number with infinite number of decimals should be understood to represent a limit; in the case of rationals, this limit can be expressed as a ratio of two integers.

#### DavidWhitbeck

Wouldn't an infinite number of positive values invariably equal infinity? Also, by definition, adding an extra 3 to a finite string of 3s would never result in 1/3, so why would repeating a failed process an infinite number of times result in a success?
? You are doing nothing but playing with words.

The set $$S=\{.3,.33,.333,\hdots\}$$ is bounded above, and thus must must have a least upper bound, either by axiom, or proven by construction of the real number line using Dedekind cuts. It shouldn't be difficult for you to figure out what sup S is. Since infinity, last time I checked, is not a real number I think we're done now.

Notice that the least upper bound is not an element of S, nor does it need to be. Jumping to it must be infinity if it doesn't below does not make logical sense.

#### HallsofIvy

Homework Helper
In another thread it was stated that an infinite number of zeros is undefined?
I suspect what was said was that an infinite number of zeros (after the decimal point), followed by a 1 (or other nonzero digit) is undefined. An infinite number of zeros, after the decimal point, is, of course, just 0.

There have been people who tried to argue "The difference between 1.0 and 0.999... (repeating) is 0. an infinite number of 0s followed by a 1" and so they were NOT the same. The only way such a construction could make sense would be as the limit of the sequence 0.1, 0.01, 0.001, 0.0001, 0.00001, ... and it is easy to show that that limit is 0, not "an infinite number of zeros followed by 1".

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Also, by definition, adding an extra 3 to a finite string of 3s would never result in 1/3, so why would repeating a failed process an infinite number of times result in a success?
Because every time you repeat it, you get a little bit closer to 1/3.

#### Hurkyl

Staff Emeritus
Gold Member
Because every time you repeat it, you get a little bit closer to 1/3.
You get a little bit closer to 1/2 too.

Fair enough:

Because every time you repeat it, you get a little bit closer to 1/3, and the size of the next step you take is proportional to the remaining distance to 1/3.

#### Algr

What I meant about an infinite number of zeros is 0 * infinity, or {0+0+0+0+...}. These values are undefined.

For example, start with a value of 5. Divide it by x. As x becomes larger, 5/x becomes smaller. So it stands to reason that when x becomes infinite, 5/inf = 0. It also seems logical that if you take your divided parts and put them back together, you'd get your original 5 back. But if you describe that mathematically, you get 0 * inf = 5. This can't be right, since you could have started with any number instead of 5, and gotten that same number back. But what went wrong? It can only be that you can't multiply the equation by infinity to put the parts back together.

Now .333... means {.3+.03+.003+.0003+...}. How could that be smaller then {0+0+0+0+...}, when every value within is larger? If infinity isn't a real number, then how can a value defined by infinity ever be real?

What I understand about the calculus concept of "limits" is that it seems to be based on the _assumption_ that .333...=1/3 and that other infinite sequences add up this way. But because of that, citing limits as the answer is just circular logic. Limits sound useful for real world calculations, but as far as I can tell, they simply assume this result without ever really justifying it on theoretical grounds.

Because every time you repeat it, you get a little bit closer to 1/3, and the size of the next step you take is proportional to the remaining distance to 1/3.
What you are describing here is Zeno's paradox, which most philosophers do NOT consider solved. Again calculus simply assumes and declares a solution without ever really stating one. If I'm cutting wood or working out the volume of some shape, I'd be happy to assume that .333...=1/3, but in pure mathmatics I'm sure that this is going to jump out and bite someone someday.

For example, start with a value of 5. Divide it by x. As x becomes larger, 5/x becomes smaller. So it stands to reason that when x becomes infinite, 5/inf = 0. It also seems logical that if you take your divided parts and put them back together, you'd get your original 5 back. But if you describe that mathematically, you get 0 * inf = 5. This can't be right, since you could have started with any number instead of 5, and gotten that same number back. But what went wrong? It can only be that you can't multiply the equation by infinity to put the parts back together.
Right, which implies that "infinity" is not a sufficient mathematical description of the dividing process you have in mind. If you instead come up with a description that DOES contain all the pertinent information, then it's no problem to "reverse" the division process. I.e., you observe that x*5/x = 5, no matter how big x becomes, and so conclude that you divide by an infinite number and then paste it back into the original.

You can also use hyperreal numbers for this kind of thing, if you're so inclined. The resulting notation may be closer to what you seem to want.

Now .333... means {.3+.03+.003+.0003+...}. How could that be smaller then {0+0+0+0+...}, when every value within is larger?
Why is it a problem that 0.333... is not smaller than 0?

What you are describing here is Zeno's paradox, which most philosophers do NOT consider solved.
I don't share your impression that "most" philosophers consider this problem to be open. But, anyway, math is not the same thing as philosophy; mathematical systems are simply the outcome of certain sets of assumptions, and there is no question that 0.333... = 1/3, under the usual definitions of these things. Whether or not these definitions properly correspond to the real world is a seprate issue, and that's what people who debate Zeno worry about. But, really, do you actually think that Achilles will never catch the hare?

but in pure mathmatics I'm sure that this is going to jump out and bite someone someday.
No, for the reasons mentioned above, it will not be a problem. It WOULD be a problem if you assumed 0.3... not equal to 1/3, as this is demonstrably inconsistent with the axioms of number theory.

#### CRGreathouse

Homework Helper
It sounds to me like your problem is with real numbers in general, not just with 1/3. Why don't you google for "finitism". I also recall a paper with a title like 'What's real about real numbers' although I don't suspect you'll find it at an appropriate level.

#### DavidWhitbeck

What I understand about the calculus concept of "limits" is that it seems to be based on the _assumption_ that .333...=1/3 and that other infinite sequences add up this way. But because of that, citing limits as the answer is just circular logic. Limits sound useful for real world calculations, but as far as I can tell, they simply assume this result without ever really justifying it on theoretical grounds.
Well it's a good thing then that my post did not use limits, maybe you missed it? And you're confusing series with sequences. But saying that limits have no theoretical justification is nothing more than grasping for straws.

#### ice109

do you have any formal mathematical training? some of your ideas are good but you can't play this game unless you know the rules

#### Algr

Formal Training: I've gotten A's in Trigonometry and symbolic logic.

David: Not useful posts.

#### CRGreathouse

Homework Helper
Formal Training: I've gotten A's in Trigonometry and symbolic logic.
So, no.

If you have an understanding of what "1/3" means, then it's not hard to see that 1/3 has decimal expansion 0.333.... Without set theory (for Dedekind cuts) I'm not sure I could convincingly show that the decimal expansion 0.333... corresponds to a unique real number -- but this is mainly because any formal treatment of real numbers is not possible without such tools.

The completeness property of the real numbers is the reason that we can say that the limit of the sequence 0.3, 0.33, 0.333, 0.3333, ... exists in the real numbers. (The property actually works for all Cauchy sequences, but for you it's enough to know that it works for all decimal expansions.) If you don't like the completeness property, then what you're saying is that you don't accept that real numbers exist. (This in turn requires removing at least one standard axiom, but I take it you're not using any particular axiomatic formulation like ZFC?)

#### JonF

What you are describing here is Zeno's paradox, which most philosophers do NOT consider solved.
You're not serious right?

I strongly suggest you listen to what these people are saying, most of them actually do have a formal background in what they are talking about, and aren't making completely false claims, such as the one you posted above.

#### ice109

why don't you just divide 1 into 3 for a little while with your favorite division algorithm and see what you get? that should be sufficient proof.

#### HallsofIvy

Homework Helper
And since 33333333333333333333331/1000000000000000000 is equal to 0.33333333333333333333 (for the first 20 digits anyway). you would consider that sufficient proof that it is equal to 1/3?

#### ice109

And since 33333333333333333333331/1000000000000000000 is equal to 0.33333333333333333333 (for the first 20 digits anyway). you would consider that sufficient proof that it is equal to 1/3?
umm that's not the same thing. i'm sure i can prove by induction on the division algorith that 1/3 = .3333...

#### mathwonk

Homework Helper
you should read xeno's paradox. roughly, if i try to walk from here to the front door, first i have to walk half way, and then i have to walk half the remaining distance, and then half the remaining quarter of the way,....... and so on.

thus i have to do an infinite number of things before i get to the door, and each takes a finite amount of time, so why does it not take altogether an infinite amount of time to reach the door?

i.e. is it possible for the sum of an infinite number of positive numbers to be finite?

e.g. here maybe it takes 3 seconds to walk 9/10 of the way, then .3 more seconds to walk 9/10 the remaining distance, then .03 seconds to walk 9/10 the remaining distance after that,.... and so on,
then all told it takes 3.3333333...... or 3 and 1/3 seconds to walk to the door.

more briefly, if a finite quantity can be divided infinitely often, then it must be possible to reassemble the infinitely many parts into a finite whole.

this paradox occurs in galileo's dialogues as well.

#### CRGreathouse

Homework Helper
i.e. is it possible for the sum of an infinite number of positive numbers to be finite?
That is my interpretation of Zeno's paradox of Achilles and the tortoise, as well.

#### HallsofIvy

Homework Helper
umm that's not the same thing. i'm sure i can prove by induction on the division algorith that 1/3 = .3333...
No doubt. But that wasn't what you said before: you said "just divide 1 into 3 for a little while":

why don't you just divide 1 into 3 for a little while with your favorite division algorithm and see what you get? that should be sufficient proof.

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#### ice109

No doubt. But that wasn't what you said before: you siad "just divide 1 into 3 for a little while":
the suggestion of induction was implicit. but you're right i should've been clearer though not like this thread is very clear to begin with.

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#### Pere Callahan

thus i have to do an infinite number of things before i get to the door, and each takes a finite amount of time, so why does it not take altogether an infinite amount of time to reach the door?

i.e. is it possible for the sum of an infinite number of positive numbers to be finite?
One could however argue- and people do - that it's not so much about the time you need to accomplish those infintely many tasks, but if it makes sense to do an infinite number of things in the first place, regardless of how long if it would take you if you actually could ....well apparently you can, walking to the door is not a big deal. 