Bear in mind that the decimal .33333 etc. is an approximation of one third. If it were an exact representation of one third, then multiplying it by three would yield 1; but the actual product is .99999 etc.

Also, remember that a decimal represents parts of 10, or parts of 100, or parts of 1000, etc. (That is, 3 tenths, or 33 hundredths, or 333 thousandths, etc.) Rather than thinking about the repeating decimal as dividing something into infinitesimally smaller pieces, as we keep adding threes, we are just naming a larger number which is a part of an even larger number. So if 1 (one) was divided into a million parts, we would be separating out 333,333 of those parts.

It is really the limitations of language and our number system that makes your question perplexing. As a purely mathematical concept, repeating decimals have no end. And I agree with what Vesuvius and Wade Casaldi have said.

Hi Jaianniah,

Sorry to be the bearer of bad news. It does not end. :)

But: Your doubt is not trivial at all. Mathematicians required a long long long time to really clear up such difficulties.

Your confusion is similar to a paradox raised by an ancient Greek philosopher Zeno of Elea. It is the following and is called the paradox of Achilles and the Tortoise.

Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 metres. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 metres, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 metres. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise.

The point of view of mathematics, developed later to deal with apparent paradoxes such as the above, is that a sequence like 0, 0.3, 0.33, 0.333, 0.3333, 0.33333, 0.333333, 0.3333333, ........... converges to 1/3. Perhaps I could supply some more details in the following way. Of course it would not address your question; but it might convince you that this infinite chain of 3's after the decimal point really gives 1/3.

Surely you would agree that a sequence like 0.3, 0.03, 0.003, 0.0003, 0.00003, 0.000003, 0.0000003, 0.00000003, ............ ultimately goes to zero?

Ok, so please keep the above in mind. Now, I use some little bit of algebra. Let x denote your number 0.333333333333........

ie x = 0.3333333.... So, 10x = 3.33333333333....

subtracting, we have: 9x = 3, ie. x= 1/3. So by algebra we shown that your number must equal 1/3.

I have cheated a bit: In the above subtraction, since we do not know how to subtract an infinitely going on thing from another, what we do is actually to take a sequence of subtractions, like 3 -0, 3.3 - 0.3, 3.33 - 0.33, 3.333 - 0.333, 3.3333 - 0.3333, ....... and so on. These sequence of subtractions in a sense "converge" to the subtraction we actually want, ie. the subtraction of these infinite sequences.

Ok, why this type of thing happen? There what Wade Casaldi said applies. It is because you base our number system on 10. If our base were a multiple of 3, then the expansion of 1/3 would have terminated. Anyway regarding this kind of things you might like the song of Tom Lehrer.

But to say the complete truth, I should also mention that there are numbers which do not terminate in whatever base expansion you choose. One example of such a number is pi, the ratio of the circumference to the diameter for any circle. The precise expression for that number does not terminate in any base you choose for your representation. Moreover such numbers never have a repeating decimal expansion. A repeating decimal can be always expressed as a fraction of two integers, by the same argument given above for 0.3333333......,

Now, returning to the philosophy: Math is based on the real world; but it is not actually "real". For example, take the notion of "three". It is there in three apples, three oranges, three birds, three songs, three ideas..... But it is not any of these. In other words, this "three" is not a notion of the physical world, though it represents many things in the physical world. Just like that, the infinite sequence you have seen, is not actually a thing of the real world. Similarly, take the notion of a circle. You see circles in a lot of places. But an abstract circle is not any of those circles. Mathematics hovers uneasily between the real world and the world of thought, and so things which do not match with reality fully, are also allowed. Things are kind of "safe" in mathematics since you follow logical reasoning. It need not fit in completely with the prevailing worldview of the point. But then the advantage is that the worldviews might keep changing; but math is fixed.

Numbers do not have to have a corresponding physical counterpart to still be valid numbers. So yes, the number 1/3 really repeats indefinitely. This fact becomes important when dealing with certain mathematical proofs.

The number of atoms in the observable universe is estimated at 10⁸⁰ atoms, so a third of that would be three with seventy-nine threes after it. I suppose you could consider this the largest significant number with a physical counterpart, that contains all threes.

I read someplace the reason we have these anomalies is because our number system is based on 10, had our number system been based on 9 for example 1/3 would be perfect.

So it is not that reality is wrong but the systems we made up for measuring it, in other words the limits of our own creations, that makes a strange point change the system and everything believed and experienced also has to change.