0.999....=1

[Teeqoz]

JWeinCom said:

mjk45 said:

my answer to that is  yes a 1/3 and 2/3 = one whole but .333 recuring added to .666 recuring doesn't simply because it ignores the recuring part.

But... 1/3 is equal to .333... and 2/3 equals .666... they're interchangeable. They're different ways to denote the same amount.  

Replace the fractions in the equation with their decimal equivelants.

.333333333 (1/3)+ .666666666 (2/3)= .99999999 (1)

The thing is that you're not really dealing with infinity.  .33333333... is a specific finite amount that is equal to exactly one third. The numbers may go on and on in theory but the quantity they represent is specific amount.  It just so happens that in a base ten system, with the way division works, certain ratios (fractions) can only be represented in decimal form using repeating decimals.  The number itself is not infinite, the language we use to describe it is. It's just a quirk of our number system.  

For example, if we used a different number system, base 3 for example, we wouldn't have the problem of repeating numbers.  .333333333333... (1/3)in base 3 would simply be .1.  And in base 3, .66666666666... (2/3) would simply be .2.  If you added those two numbers, it would come out to a nice even 1.  And if 1/3+2/3=1 in a base 3 system, it has to also equal that in a base 10 system.  

(in base 3 .1+.2=.3  But, in a base 3, the only digits are one and two.  When you get to 3, you carry it over a place, like when you get to ten in base 10.  So, .3 would become 1.  1 in base 3 is the same as 1 in base 10)

Dunno if that helps, but the point is there is nothing inherently repeating about numbers like 1/3 or 2/3.  They don't *have* to be repeating numbers, it just so happens that due to the limitations of working with the particular number system we use, we can't describe these numbers in decimals without them repeating.

It's not just a flaw with base 10 though. Base 3 also has fractions that can only be represented by infinitely recurring decimals. The only thing that changes when you change the base is what fractions.

If there is a system that has a flaw, then it's the decimal system itself. However I like to think of it as just another way of writing the same thing. 2/2 (As a fraction), 10^0, 1, 0.999.... All just different ways to write the value (not the number) 1. The recurring 9 notation works for all finite sequences of numbers as well (ie 0.25=0.24999....)

It's like a synonym in language. It's just that people are a bit more stuck up when it comes to math.

It may be a flaw in the system, but the flaw isn't that this should be incorrect, because it is correct. The flaw is that notation like this is ever used/ever has to be used. However when you denounce something as "simply the result of a flaw in the system", people are often quick to think "aha, so it's not actually correct", so you have to be careful when proclaiming things as the result of a flaw in the system.