0.999....=1

[mjk45]

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.

Thank you very much , I really appreciate your explainations, I suppose most of my thinking isn't so much about the result , especially since you thoughtfully  explained it to me , more to do with how it has been put forward in some quarter's,  where it is seen in a very black and white way , that being it's a mathmatical fact end of story , while you point out that the reason has as much to do with us dealing with a systems limitation, something that i feel pokoko was alluding to in his replies.