# .333333333333*3 = 1?

1. Dec 5, 2004

### ldzcableguy

How can 1/3 multiplied by 3 give us 1 if 1/3 is the repeating decimal .3 to an infinite number of decimal places? If you multiplied 3 * .3 with an infinite number of repeating 3's wouldnt you get .9 with an infinite number of 9's repeating? Why do we say that (1/3) = .3 repeating and when you multiply that times 3 you get 1 instead of .9 repeating?

2. Dec 5, 2004

Yikes!

3. Dec 5, 2004

### Tom McCurdy

because .999999999999999999999999999999999 repated equals one when you apply calculus

look up zeno's paradoxs it has similar ideas

4. Dec 5, 2004

### Math Is Hard

Staff Emeritus
heh heh heh! here we go again! ... poor Hurkyl... poor Matt.. poor Tom...
I feel sorry for them already.

5. Dec 5, 2004

### Tom McCurdy

well at least people are thinking

6. Dec 5, 2004

### Gokul43201

Staff Emeritus
Because the real number represented by the repeating decimal 0.999... is defined to be identical to the number represented by 1.

7. Dec 5, 2004

### Zurtex

Just so you guys know I actually managed to convince some people on another forum that 0.9 recurring = 1 who were arguing otherwise.

8. Dec 5, 2004

### Tide

Can you find any number between $0.\bar 3$ and $\frac{1}{3}$ ? If not, then they must be the same! :-)

9. Dec 5, 2004

Haha, my thoughts exactly!

I will say it can be an indicator someone has been using their coconut to try and make sense of things but, well, not always (often turns ugly for no good reason).

10. Dec 5, 2004

### Janitor

Where has Organic been of late?

11. Dec 6, 2004

### ldzcableguy

What is the calculus behind it?

12. Dec 6, 2004

### Gokul43201

Staff Emeritus
I believe what Tom had in mind was the limit of the infinite sum "0.9 + 0.09 + 0.009 + ..." which is just a geometric progression, and its sum to an infinite number of terms is simply 0.9/(1- 0.1) = 1.

13. Dec 6, 2004

### ReyChiquito

i like this one

$$0.33333\bar{3}=a$$

$$3.33333\bar{3}=10a$$

$$10a-a=9a=3$$

$$a=\frac{3}{9}=\frac{1}{3}$$

$$0.333\bar{3}=\frac{1}{3}$$

14. Dec 6, 2004

### matt grime

One thing I've never understood is why people argue against the fact that it is even remotely possible for two decimals to represent the same real number yet are perfectly happy to accept there are an infinite number of rational representations for some element in Q {1/2, 2/4. 3/6, ...} Surely two different ones for only a few numbers must be a fantastic improvement over infinitely many for all.

15. Dec 6, 2004

### Hurkyl

Staff Emeritus
My best guess is that they don't internalize 1/2 as being a number, but rather an arithmetic expression. The thing that bugs me is how many think of 0.999... as some strange sort of varying number.

16. Dec 6, 2004

### Staff: Mentor

Because of this, I don't understand why this should be that hard of an issue:
Some people have tried to get around it by pulling new numbers out of their a--air (0.000...1), but if you can't answer the question using the real number system, game over. I'm currently in page 9 of a similar thread at BadAstronomy, where the answer to that question was "an infinitessimally small number." It makes me wonder whether this is an honest argument.

The one (sorta) legitimate concern I've seen is from engineers who think its a matter of precision: you have to stop somewhere to round it off and thats how you get 1.

Last edited: Dec 6, 2004
17. Dec 6, 2004

### matt grime

It cannot be an infinitesimal number, and non-zero, in the sense of non-standard analysis, since 1/3 - 0.33... is real, and an infinitesimal, hence zero - even in nonstandard analysis 0.9.. and 1 are the same.

18. Dec 6, 2004

### omicron

We should have a sticky for this. Or maybe we should have a faq forum for the different topics. Btw you can try here

19. Dec 6, 2004

### check

Nah, people would still continue to post this 5 times a day. I recommend just changing the url to 0.9repeating=1.end_of.story_so.please.stop_asking.com and replace the PF banner with something similar.

Last edited: Dec 6, 2004
20. Dec 6, 2004

### nnnnnnnn

9 = 9.9... - .9... = 10*(.9...) - 1(.9...) = (10 - 1)*(.9...) = 9*(.9...)

-> 9 = 9*(.9...)
-> 1 = .9...

Using this same method can be used to prove the geometric series (I think its the geometric series) which is where it ties into calculus...

21. Dec 10, 2004

### okidream

Hi. If you don't mind, I like to share my opinion on this.

This type of thing is new, as before the decimal system was invented, there was not such a problem. This is a problem with the decimal system and not a problem with fractions, as in ancient times, I guess.

As an illustration: 3 is odd and hence dividing it into itself i.e to 3 components, (1/3 each) it should never be perfect in reality, except in your own imagination, (which is the fraction 1/3...which reveals the beauty of maths). Such are the case with all the other odds, except for the special odd - 5 and its selective multiples. (I'm still figuring out what is so special about this, like the fact we're born with 5 fingers and toes and 4 limbs+head).

To be honest, I have not seen any usefulness of decimal point system in number theory - it could be very useful to precision engineering, etc, elsewhere though.

My conclusion is: 1/3 is never 0.333..., but 0.3333... is 1/3.
(Just as a square is a rectangle but a rectangle is not a square, kind of arg)

22. Dec 10, 2004

### matt grime

"My conclusion is: 1/3 is never 0.333..., but 0.3333... is 1/3.
(Just as a square is a rectangle but a rectangle is not a square, kind of arg)"

Words almost fail me.

If you can't figure out why 1/5 has a nice decimal base 10, then try understandin why 1/3 = 0.1 base 3.

23. Dec 10, 2004

### Cosmo16

Here's a proof that I think is sound. Tell me if I'm wrong

x=.9 repeating
10x=9.9 repeating
10x-x=9.9 repeating - .9 repeating
9x=9
x=1

24. Dec 10, 2004

### matt grime

Presumes that the arithmetic operations have been defined on infinte strings.

25. Dec 10, 2004

### StatusX

its all limits. in that proof, using a finite string of 9s, the error gets smaller and smaller as the length grows, so in the infinite limit, it is exact. Every proof depends on the idea of a limit and the fact that all these symbols are intended to represent is the real number limit of a certain sequence. each decimal corresponds to exactly one real number. none of them correspond to a process that never ends, as some would tend to believe about 0.999...