posted on Sep, 14 2009 @ 01:33 AM
... ohhhhhh dear...
Something that needs to be made clear here is that just because something is comprised of infinitely many pieces does NOT mean that it is infinite.
I'm calling on Zeno's Arrow Paradox here. Just because a number has the mystical "..." attached to the end of it does not render it unusable in
standard mathematics - it simply means that it has an infinite decimal expansion.
Furthermore...
ALL NUMBERS ARE INFINITE DECIMAL EXPANSIONS.
This is true and has been since the day you were born, and most likely well before that. "1" is shortened notation for "1.00000...", 4/5 is
shortened notation for "0.80000...", and "pi" is shortened notation for "3.14159...". These are three different numbers answering to three
differing number sets - whole, rational, and real, and all THREE of them follow this rule. No matter what set your number belongs to, there are
several ways to write it - in convenient shorthand (such as 1, 2/2, .5*2,) or with its infinite decimal notation (good luck writing out something
comprised of infinitely many parts, like 1.00000000000... [and by 'write it out, I mean write out an infinite number of zeroes. Good luck.])
Now, to business...
.999... = 1.
It's true, I'm sorry if it disturbs or disappoints any of you. .999... is another way of writing the shorthand form of 1, and there are many
mathematical proofs that prove this.
Take the Additive Property, for example. It states that any real number "a" plus the number zero will equal "a." Therefore, [a+0=a]. By
rearranging this equation, we get what I like to call the Subtractive property - any number minus itself equals zero [a-a=0]. NOW, remember that "0"
is actually the shorthand of 0.000... SO, when we plug the two different forms of writing the number "1" into the equation (1 and .999...), we get
this:
1-.999... = 0
0.000... = 0
Therefore .999... = 1.
Imagine that.
But there should be a "...001" at the end of that "0.000..."!
The entire point of the "..." is that the expansion goes on forever, without end. There IS no end to tack the "...001" onto.
The Limit Argument
A sequence can have ONE and ONLY ONE limit.
Let us observe the sequence:
.9
.99
.999
.9999
...
therefore, the sequence gets closer and closer to .999... infinitely close.
It also gets infinitely close to 1. Therefore,
.999... = 1.
I cannot find for the life of me why THIS algebra is being disputed:
let x = .999...
Therefore,
10x = 9.999...
Subtracting x from both sides (legal) gives us
9x = 9.999...-x
Substituting ".999..." for 'x' on the right side (which is completely legal) gives us:
9x = 9.999... - .999...
9x = 9
x=1.
Above is pure, completely correct algebra.
But we already said that x=.999... therefore 0.999... equals one.
FOR THOSE OF YOU THAT DISPUTE THE ALGEBRA:
Assume that .999... is less than or equal to 1.
Let us next assume that .999... does NOT equal 1. (INCORRECT ASSUMPTION)
Therefore, .999... < 1.
Therefore, there should be some positive number N such that
.999... + N = 1.
HOWEVER, for ANY positive number N,
.999... + N > 1.
Therefore, one of our assumptions is wrong. I'll give you a hint: It's the second one.
".999... = 1" is INDISPUTABLE. There are NO PROOFS ANYWHERE THAT OPERATE UNDER STANDARD MATHEMATICS that can make this equation false. I've
checked, but if you'd like to yourself, go ahead. Religious reverence of the concept of infinity or plain old disbelief does not make you correct -
neither does the fact that 'so and so' agrees with you. Proof is everything in mathematics - and there are no proofs to the contrary. I rest my
case.
Sam Hughes has an excellent article on this subject at his website "Things of Interest." [qntm.org]. I advise anyone who disputes this truth to look
up the arguments posted there.
[edit on 14-9-2009 by ARandomGuy]
[edit on 14-9-2009 by ARandomGuy]
[edit on 14-9-2009 by ARandomGuy]