# .9999...=1

Discussion in 'The Lounge' started by Josepho1997, Nov 25, 2012.

1. ### Josepho1997 Guest Thread Starter

I've been told that .999...=1 (.999... means .9999 repeating)
What do you think? Does. 9999=1?

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2. ### Mikestony ~30% Carbon Black ± Moderator 3,118

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It's close enough for me, but .999.... is repeating to me, so no, it does not equal "1". "1" = 1.0000000.... My opinion anyhoo

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3. ### jhawkkw Chinchillin' 1,333

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.99999... does in fact equal 1 and can be easily mathematically proved.

Let x=.9999999......
Therefore 10x=9.99999.......
10x-x =9x. == 9.99999.....-.999999....=9
So 9x=9
9x

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5. ### jhawkkw Chinchillin' 1,333

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Following the same set if steps, you would end up with 9x =2. Thus x =2\9, which is indeed what. 2222222..... is equal to.

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6. ### olbriar   Moderator 1,973

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7. ### scary alien not really so scary Moderator 2,138

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8. ### Mikestony ~30% Carbon Black ± Moderator 3,118

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Tell me if I'm wrong here:
x=.22222222......
10x= 2.22222222......
10x-x=9x 2.22222222.....-.2222222222= 2
2=9x
divide each side by 9
2/9=x
x=.2222222222222
back to where we started

ah, I think I get it...
.99999999.....would be the only number equal to 1?

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9. ### jhawkkw Chinchillin' 1,333

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Correct, it is an odd situation that comes up in mathematics. There others out there too, but I will spare everyone the messy details because the proofs involve higher mathematics and aren't easy to type. This method shown above can be used to figure out the fraction for any repeating decimal. Pretty nifty I'd say.

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10. ### Mikestony ~30% Carbon Black ± Moderator 3,118

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I knew I should've taken math a little further in school

Thanks jhawk

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11. ### Mikestony ~30% Carbon Black ± Moderator 3,118

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Anything after the "Analytic Proof" section in that wiki link, just fried my brain:help:

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12. ### Prinny Resident Linux Nutcase 433

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I like how one of the tags is "trolls." o.o

This was an awesome thread. Great read. It makes total sense.

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13. ### h4x0rj3ff Chemist 323

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It depends on the setting really. If your manufacturing gears with a wire edm then .9999 is still .0001 underside from 1.00000. now if you have a geometric tolerance of lets say +- .00001 than that dimention being at .9999 would bring you out of tolerance by .00009 rendering the part non conforming and scrap.

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14. ### jhawkkw Chinchillin' 1,333

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Partially true. .9999 itself is not equal to 1. But the idea behind the question and the proof is for .9999...., meaning repeating infinitely forever and is absolutely 100% true.

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15. ### h4x0rj3ff Chemist 323

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Also its not repeating unless it uses ~ above the last number

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16. ### jhawkkw Chinchillin' 1,333

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That is one way to show repeating decimals, but the ellipses "..." is also widely accepted in mathematics to signify a repeating sequence.

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17. ### DaSchmarotzer Blame it on me 313

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It's been said already, but it's true.

You can also see it even more easily like this:

(1/3) + (1/3) + (1/3) = 1

0.3333... + 0.333333... + 0.333... = 1

Therefore

0.999999..... = 1

It also works with other numbers that end with a decimal periodic 9. Such as:
23.56999999.... = 23.57

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2+2=3

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19. ### psionandy Well-Known Member 763

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thats only true if you assume 0.3333... = 1/3
which (depending on context, may or may not be true

There are 10 types of people in the world.. those that count in binary, and those that don't.

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20. ### jhawkkw Chinchillin' 1,333

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1/3 is exactly = .333... repeated for ever and you can easily discover that for yourself by dividing 1 by 3. If you do it manually using the standard you will constantly get repeating 3's forever until you decide you've had enough punishment.

I wanted to avoid the use of higher mathematics, but the reason behind the equality is stated right in the wiki article:

Furthermore:
Whether anyone chooses to accept it or not is their choice, but nonetheless, it is 100% true. I often run into non-believers when I teach this topic in the Infinite Series part of Calculus II, granted the proof for that class is different giving the context of the class.

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21. ### scary alien not really so scary Moderator 2,138

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One can look at all of the Notes, References, Further Reading, and External links listed in the wiki page.

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22. ### jhawkkw Chinchillin' 1,333

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I could give you the real reason, but would require the use of real analysis which is usually a 4th year mathematics course. As I stated earlier, it has to deal with the fact that there is no non-zero infinitesimal between .999... and 1.0. In other words, there is no number between .999... and 1.
For the purpose of a proof, lets assume they are distinct numbers. For any number not ending in a infinite sequence of 9's, you can always find another number in between two different numbers by simply finding the midpoint (x1+x2)/2. This doesn't work with numbers ending in the infinite sequence of 9's because the midpoint between .9 and 1 is .95. so if you could take the midpoint of the two, following that idea, the last number would have to be a 5, which contradicts the whole notion because .999 is an infinite sequence of 9's and would actually be greater than our midpoints. This creates an absurdity, which many mathematicians call a proof by contradiction. So since our original idea about them being two different numbers let to a contradiction, that means that assumption must be false, and thus means they have to be the same number.

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23. ### Liamo_210 Well-Known Member 233

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Without a doubt .999 =1 it can be mathematicaly proved as shown above. .222 does not =1 nor does .3333 .444 etc. Again all this was said above but i thought i would verify it.

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24. ### psionandy Well-Known Member 763

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well.. in that case pie is 3.14

or whatever number you get before you decide you've had enough punishment.
or 3.2 if you're a stupid person from indiana
Indiana Pi Bill - Wikipedia, the free encyclopedia

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25. ### scary alien not really so scary Moderator 2,138

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Hey, I'm from Indiana!

Oh, wait--that means....nooooooooooo!

Glad that all happened way back 1897 .

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