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As penance for liberally beating a dead goat in the other Brainteaser thread, here is a fresh one that should be easy for many of you:

Does .9 repeated = 1?

Please prove mathematically and explain in English. SB for the first to do both.

• You can explain this without any fancy math, using the definition of numbers alone.

• just do a geometric series with a=.9 and r=.1

or could do something like this:
1/9=.1111111...
9*(1/9)=9*.1111111....
1=.999999...

• In reply to Cola Coca

Cola Coca wrote:
x = 0.9

10x = 9.9

10x - x = 9

9x = 9

x = 1

10*.9=9

not 9.9, so in your equation x does not equal .9 like you said.

the short answer is no it does not equal 1

• i think its because of the absence of nonzero infinitesimals in the real number system

• Yes - my simplest 'explain in english' is that .3 repeating is generally known to equal one third, and 3 times one third equals 1

• Let me also do it with math:

0.999 repeating equals x.
10*x equals 9.999 repeating equals 9 + x
Subtract x from both sides
9*x = 9
therefore x=1

• Oh Cola Coca got the answer first. Everyone who's disagreeing with him isn't taking the time to think about what it means for the decimal to be repeating indefinitely

JDimon wrote:
Yes - my simplest 'explain in english' is that .3 repeating is generally known to equal one third, and 3 times one third equals 1

That's a good way to explain it. Also if
1/3 = .3 repeating
2/3= .6 repeating

Add them together and you get

3/3=.9 repeating
1=.9 repeating

JDimon wrote:
Let me also do it with math:

10*x equals 9.999

Assuming you mean 10* .999 repeating = 9.999 repeating, this step is only true if the series converges (i.e it equals 1). In order to finish your proof you essentially assumed that what you were proving was true.

Are you retarded?

Yes. When you want a repeated decimal, you just take that number and divided it by 9.
0.7777.....=7/9
0.5555.....=5/9

Thus, 0.999999.....should theoretically equal 9/9, and 9/9 = 1.

Do I get the SB?

illiniPride wrote:
Cola Coca wrote:
x = 0.9

10x = 9.9

10x - x = 9

9x = 9

x = 1

That's the math, now why does it work?

Because an infinite string of 0.9 will always equal 1? There is no number between 0.9 repeating and 1; they are the same number, represented differently.

0.25 = 0.24(9) repeating

FreezePops wrote:
JDimon wrote:
Let me also do it with math:

10*x equals 9.999

Assuming you mean 10* .999 repeating = 9.999 repeating, this step is only true if the series converges (i.e it equals 1). In order to finish your proof you essentially assumed that what you were proving was true.

Are you retarded?

All I'm doing is using that fact that the 9s go on infinitely. If they go on forever, then multiplying everything by 10 doesn't reduce the amount of 9s after the decimal point....they still go on forever (not forever - 1, because that value still equals 'forever')

• Gave it to Dimon, wasnt thinking of it that way but it makes sense.

You can prove this another way using negation

• how can one number equal another number? even if a limit, it still doesn't technically equal it...

plz splain

JDimon wrote:
Yes - my simplest 'explain in english' is that .3 repeating is generally known to equal one third, and 3 times one third equals 1

Give it to this guy.

TechBanking wrote:
.9 repeated has a limit value of 1, but is never actually 1.

This statement makes no sense at it stands. If you mean to say constructing a sequence where each additional term is made by appending another 9, then yes, this sequence has a limit of 1. If you mean the number .9 repeating, well, it's a constant, so its "limit value" must be itself. By the way, that is most definitely 1.

JDimon wrote:
FreezePops wrote:
JDimon wrote:
Let me also do it with math:

10*x equals 9.999

Assuming you mean 10* .999 repeating = 9.999 repeating, this step is only true if the series converges (i.e it equals 1). In order to finish your proof you essentially assumed that what you were proving was true.

Are you retarded?

All I'm doing is using that fact that the 9s go on infinitely. If they go on forever, then multiplying everything by 10 doesn't reduce the amount of 9s after the decimal point....they still go on forever (not forever - 1, because that value still equals 'forever')

JDimon is right. This is a legitimate proof. He's not assuming anything about the series convergence- all he assumes is the presence of infinite 9s (follows by the assumption of repeated 9s). Multiplication by 10 is a well defined operation- even for infinite decimal places- as a "place shifter".

• This was how I thought of it:

If a b,
x exists where a x > b.
If not, a=b

• The answer is yes. There is a long mathematical proof that you learn in calc 2.

But the easy straightforward answer is...

1/3 = .3 Repeated
2/3 = .6 Repeated

3/3 = .9 Repeated

• Wikipedia has a pretty cool proof with (Dedekind) cuts too! I didn't know that one since my analysis class skipped the construction of real numbers, but it was nice. I liked it!

• Assume that 9.999.../=10

Then you can list a number that is greater than 9.999... and less than 10.

Since you can't, 9.999...=10