# Why is 1/3 considered a number line point when .333... isn't

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1. Oct 21, 2015

### Pjpic

Why is 1/3 considered a number line point when the decimal expansion never seems to stop at a point (if this is indeed the case)?

2. Oct 21, 2015

### Krylov

$\frac{1}{3}$ and $0.333\ldots$ are exactly the same real number. If you are bothered by the infinite decimal expansion, you can always write it in base $3$, as $0.1_3$.

3. Oct 21, 2015

### Staff: Mentor

The sequence of numbers {.3, .33, .333, ...} converges to 1/3. Each number in this sequence represents a point on the number line.

4. Oct 21, 2015

### jbriggs444

The real number line is considered to also contain a point at the limit to which the sequence converges. This is one of the ways in which the real number line can be defined -- as the set of all limit points to "convergent" sequences. To be more technically correct it can be defined as a set of equivalence classes of Cauchy sequences of rational numbers.

A Cauchy sequence is one where all of the terms eventually get arbitrarily close to each other. Two Cauchy sequences are considered "equivalent" if both of their terms get arbitrarily close to each other. So you can interleave their terms and still have a Cauchy sequence.

5. Oct 21, 2015

### Krylov

To be honest, I don't really understand the difficulty that the OP seems to encounter? I agree with what was written above, but $\frac{1}{3}$ is just an innocent rational number. Yes, it happens to have an infinite decimal expansion, but that's just an artifact of our common choice of base.

6. Oct 21, 2015

### Pjpic

I wonder why a sequence is considered to be the same as its limit.

Last edited by a moderator: Oct 21, 2015
7. Oct 21, 2015

### jbriggs444

A sequence is not considered to be the same as its limit. What we are saying is that the limit exists.

The way the real number line is constructed ensures that for every sequence whose terms approach one another arbitrarily closely, there is a limit that those terms approach.

8. Oct 21, 2015

### Staff: Mentor

I don't know why you would say that. For the number in this thread, 1/3, the sequence is {.3, .33, .333, ...}. 1/3 is not an element of this sequence, but the farther you go in the sequence, the closer to 1/3 that element is.

9. Oct 21, 2015

### WWGD

Because the number line models a continuum of points.

10. Oct 21, 2015

### WWGD

And this also puts still another nail in the coffin of why 0.9999..=1 , seeing them as equivalent under this relation.

11. Oct 21, 2015

### HallsofIvy

Staff Emeritus
The sequence is not the same as the limit but that was not the point. the sequence {.3, .33, .333, ...} has limit 0.3333... and 1/3. 0.3333... and 1/3 are just different ways of writing the same number. The answer to your original question, "Why is 1/3 considered a number line point when the decimal expansion never seems to stop at a point (if this is indeed the case)?" is that "This is NOT the case. 1/3 and 0.3333... are both the same number line point."

12. Oct 22, 2015

### symbolipoint

You learned intermediate algebra and the Pjpic did not.

13. Oct 22, 2015

### micromass

Staff Emeritus
I wouldn't be so harsh, since this is really a much more difficult problem than at first sight. Sure, people learn in intermediate algebra that $1/3 = 0.33333...$, but they never learn why. They don't even learn what $0.33333...$ is supposed to mean. Of course, when we are just kids, we accept it and we continue to accept it to this day. But I find it a sign of intelligence to start questioning this thing.
Sadly, the answer that would satisfy Pjpic is not an easy one. He will need to get familiar with calculus, and in particular the rigorous treatment of limits and series. That is the only way you can really come to a full understanding of decimal notation, without relying on statements of authority.

14. Oct 22, 2015

### Krylov

Because in order to understand the definition of $0.33333\ldots$ one needs to understand convergence of a geometric series? In my opinion, this does not immediately quality it as a "much more difficult problem", but it does probably explain why here it would appear in a calculus course, rather than in "intermediate algebra".

15. Oct 22, 2015

### infinite.curve

You should not be scared by the infinite numbers. As Krylov said, they are the same. Can you please elaborate in what you are thinking? Also, why are you asking the question. The answer lies in a not so basic proof. Here is something that hints on what you are searching for: http://www.mathswrap.co.uk/how-to-change-repeating-decimals-to-fractions/. What yu should take is real analysis.

16. Oct 22, 2015

### aikismos

On a fundamental level, $1/3 = 0.33333...$ is taught as repeated division. Three goes into 10 3 times and leaves a remainder of 1. Three goes into 10 3 times and leaves a remainder of 1. Three goes into 10 3 times... it may not be rigorous enough for real analysis, but it certainly answers the question why. What's a shame is we don't teach that all numbers ultimately result from one or more operations even if they are as simple as counting.