Do you think .9 repeating is 1?

Question

.99999999999999999999999999999...
Do you think this number is really 1? Of course it is not, but when you use the "9 trick" (putting a repeating decimal over 9 to express it as a fraction) it will be 9 over 9, which is actually equal to 1. Of course, this number gets so close to 1, but it never truly reaches 1. What do you say?

Answer 1

It is 1, just like 0.3333... *is* exactly 1/3 and 0.6666... *is* exactly 2/3 and 0.9999.... *is* exactly 3/3 = 1.

It's not *close* it just *is*, exactly equivalent.

There is no trick to it... ask any mathematician and they will confirm that 0.9999.... repeating is just another way of representing 1.0.

1/9 = 0.1111...
2/9 = 0.2222...
3/9 = 0.3333... (1/3)
4/9 = 0.4444...
5/9 = 0.5555...
6/9 = 0.6666... (2/3)
7/9 = 0.7777...
8/9 = 0.8888...
9/9 = 0.9999... (1)

or use this proof:

x = 0.9999...
10x = 9.9999...
10x - x = 9.9999... - 0.9999...
9x = 9
x = 1

To say that 0.9999.... doesn't equal 1 because it might eventually end and be different than one is like saying 0.0 repeating might not equal zero.

Check Wikipedia or any of the math forums to see that 0.9999... is confirmed to be 1, despite what our intuitive logic might say otherwise.

Answer 2

Geez, some of you make this sound like it's an act of faith. It just a geometric series. If you "believe" in the formula, then a/n + a/n^2 + a/n^3 + a/n^4 + .... = a/(n-1). So, 9/10 + 9/100 + 9/1000 + 9/10000 + ... = 9/9 = 1.

Answer 3

Interesting question.

1) No,in general if you repeat .9 ( n times) with n = 1000000000000000 by
example your answer will be very close to 1 but not equal to 1.


Example 2: f(x) = 1/x f(x) = mathematical function

f(1000) = 1/1000 = 0.001

f(1 x 10^6) = 1/(1 x 10^6) = 0.000001

f(1 x 10^20) = 1/(1 x10^20) = 0.0000000000000000
0001

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-
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f(1 x 10^ 1000000) = 0. 000...0001
repeat ''0'' 999 999 times ! before 1

but the final answer is very very close of 0 but it's not 0.


2) Yes, in particular for calculus. The reason it is more practical for calculations to put .999999999... if we take limit of certain number with n => infinity and suppose .99999.... (9 repeat infinity
time).So, it' s more practice to take ''1'' for answer to complete
or finish the reasoning.(depends of the problem)

Answer 4

From the answers, I think the phrase ".9 repeating" is ambiguous.

.9 repeating infinitely is 1, which should be the default understanding when 'infinitely' is not specified.

.9 repeating for n times, where n is any positive integer, is not equal to 1.

Answer 5

.9 repeating is so close to 1 that most mathematicians believe it to be 1. I believe that it is 1 as well.

Answer 6

Well it all comes down to where you want to use this number. For things like space travel, you have to be dead accurate, so you'd probably need 0.9999 to 12 decimal places!!

Answer 7

It is close to 1, but it isn't. Of course, making it into a fraction is really estimating it. It should be like 999999999999/10000000000000! LOL ROFLMAO actually that isnt funny at all...

Answer 8

nope it's just .9 bar