Rusty
2007-03-19 14:17:50 |
This one is an easy one but stumping me!
X = 3.9999 (repeating)
Make X = 4, You can not use addition however. |
Vernon Lewis
2007-03-19 19:06:40 |
Re: This one is an easy one but stumping
Hi Rusty.
I have a spare 0.0001 I am not using. You can borrow it if you like.
Perplexus asks that you refrain from posting problems on the forums, since that upsets the overall structure of the site. |
George
2007-03-20 18:43:14 |
Re: This one is an easy one but stumping
It's more of a question really.
Rusty,
0.9~ = 1
The wikipedia article should tell you all you need to know. |
JayDeeKay
2007-06-12 16:13:51 |
Re: This one is an easy one but stumping
Not sure what you're asking, Rusty, because 3.9~ *is* equal to 4. This is easily proved as follows:
3.9~ = 3 + 9/10 + 9/100 + 9/1000 + ... [definition of a decimal fraction]
= 3 + 9/10 * (1 + 1/10 + 1/100 + 1/1000 + ...)
= 3 + 9/10 * lim{n -> inf} (1 - (9/10)^n) / (1 - 1/10) [sum of geometric series
= 3 + 9/10 * 1/(9/10)
= 3 + 1
= 4.
(No need to borrow any fractions, at all!) |
Federico Kereki
2007-06-13 14:31:29 |
Re: This one is an easy one but stumping
An even easier way: let's try to find out what X is if X=3.99999... Multiply by 10, to get 10.X=39.99999.... Subtract one equality from the other to get 9.X=36. Therefore, X=4. |
JayDeeKay
2007-06-14 17:28:26 |
Re: This one is an easy one but stumping me!
Dear FK: That is a common "proof" which gives the correct answer but which harbors a subtle error: multiplying the series by 10 assumes that it converges to a finite, real number, but that is actually what you are trying to prove!
If the series were, in fact, divergent, you could get a crazy answer, such as 1=2 or similar by performing "simple" arithmetic on the series.
For example, consider the series 1 - 1 + 1 - 1 + ...
If we sum it as (1 - 1) + (1 - 1) + ..., the sum is obviously 0.
However, if we sum it as 1 - (1 - 1) - (1 - 1) + ..., the sum is 1.
Which would imply that 1 = 0. |
Sunil
2007-06-16 08:54:07 |
Re: This one is an easy one but stumping
Hey JayDeeKay,
the way you you used limits, doesnt it simply mean : x{lim x-> 4} = 4 |
