What do you think about "1 = .9999 repeating" ?

Question

so there is a debate about 1.00 being exactly equal to 0.999........ (the nine repeating itself infinite times)

so algebraicly it looks like this,

1\3 = .333..... lets call the number this represents with "x"

(x) * 3 = __

if we plug in either "1\3" or ".3333...." for "x" the result should algebreicly be exactly equivilent, alas......

(1/3) * 3 = 1

however

( .333...) * 3 = .999.....

so you see, 1 should equl .99... exactly but, well how can they they are two different numbers...

personaly, i belive (1/3) does not equal .333.... and there lies the flaw of this debate.

Answer 1

I thought exactly the same as soon as I started reading the question: 1/3 does not equal .333....

Edit: Now, after reading Blah's and She Nerd's answers, I am baffled.

Answer 2

Well, let's see here.

0.3333.... = 0.3 + 0.03 + 0.003 + 0.0003 + ...
= 3(0.1 + 0.01 + 0.001 + 0.0001 + ...)
= 3(0.1^1 + 0.1^2 + 0.1^3 + ...)
= 3*sum(i from 1 to infinity) 0.1^i
= 3(0.1/(1-0.1)) since for |r|<1, sum(i from 1 to infinity) r^i = r/(1-r)
= 3(1/9)
= 1/3

Look at what I just wrote; no inequalities, no tricks. 1/3 EQUALS 0.333...(where the 3's go on forever). Period. And also 0.999... EQUALS 1. Again, the 9's go on forever. That's the key point here; if the 9's ever stop, then it's not equal to 1; if they don't, then it is equal to 1.

Likewise 0.4999999.... = 0.5, 1.24999999...... = 1.25, and so on, as long as the 9's go on FOREVER. Every rational number has an infinite decimal expansion. That means that every number that is of the form p/q where p and q are integers can be written with a decimal that goes on forever.

There are many other ways to prove that 0.999... = 1. There is no number between 0.9999.... and 1. You cannot possibly find one. So they must be equal since any two different numbers have a number between them. You can also use the method I used above; instead of multiplying by 3 you would multiply by 9. But it is a fact that 0.999... equals 1.

edit: Also, you can determine that 0.3333.... = 1/3 using long division

3 goes into 1 0.3 times. Remainder is 0.1
3 goes into 0.1 0.03 times. Remainder is 0.01. So far 3 has gone into 1.0 0.33 times.
3 goes into 0.01 0.003 times. Remainder is 0.001. So far 3 has gone into 1.00 0.333 times.
And so on.

Just continue to carry this out forever. 1/3 = 0.3333.... where the 3's go on forever. You don't need any more than a third grade education to figure this one out.

Answer 3

In ordinary number bases, any irrational number has exactly one representation (which must be infinitely long).

A rational number whose denominator is not an exact divisor of any power of the base also has exactly one representation (which must be infinitely long). This is the case for 1/3 in base 10, contrary to your stated belief - there is no other possible representation except 0.333.... (But perhaps you disbelieve in ALL infinitely long representations. Do you? If so, you and Pythagoras have something in common. :-)

A rational number whose denominator IS an exact divisor of some power of the base has exactly two representations. One representation terminates, and the other is infinitely long, but they are simply two different names for the same number, just as "(1-1)" and "(pi-pi)" are different names for zero.

Answer 4

There's a few ways to do this.
One is by the usual way to change a repeating decimal into a fraction.
Another is to consider it as an infinite sum.
Those three dots really do mean infinitely repeating digits.
Just a minute and I'll get you some links that explain this better.
Here's how to transform a repeating decimal into a fraction...
http://mathforum.org/library/drmath/sets/select/dm_repeat_decimal.html
Here's some more theoretical explanations involving infinite sums...
http://mathforum.org/dr.math/faq/faq.0.9999.html

Answer 5

I think that eventually, the infinitely repeating decimal gets so close to 1 it doesn't really matter anymore. Why would you think that (1/3) doesn't equal .333 repeating? Well...I suppose I understand....I just think it's so close eventually it doesn't matter.

Answer 6

well....if you say that 1/9=.111...and 2/9=.222...

then 9/9 should equal .999.....but its equals to 1....it just doesnt make that much sense....but 9 divides 9 is 1....but following the pattern that 1/9=.111...3/9=1/3=.333...

hmmm...Indigenous!

Answer 7

Hmm, well, it's an infinite series.
The equality was proved by mathematicians about 200 years ago.

Answer 8

I think .999 is equal to 1 because .9 rounds that 0 to a 1.............
but .333 doesnt look right to me either

Answer 9

1/3 DOES equal 0.333 recurring.

Answer 10

I like to think that integer numbers like 1, 2, 3, etc. are ideals, while numbers like .99999... imply our limitations in reaching those ideals.