i remember seeing a whole site with proofs on it
2006-09-01 12:23:36 · 6 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
wikipedia:
http://en.wikipedia.org/wiki/Proof_that_0.999..._equals_1
2006-09-01 12:26:26 · answer #1 · answered by Puzzling 7 · 0⤊ 1⤋
When Einstein formulated his greatest equation, it indicated that mass and energy were pretty much interchangeable - he first thought it was just a mathematical fluke. After exhaustive analysis, he concluded it wasn't a mathematical illusion, but it was true in reality. This implies, to me, that mathematics isn't always the bedrock of precision it is thought to be when we start inserting numbers that are real only in our imagination.
No one - but NO ONE - can conceptionalize or truly define an infinite number in such a way as to be able to logically argue why, if you add any number to it, it must become a larger infinite number? - and does that mean the original infinite number is no longer an infinite number? Just how big is it if you square an infinite number? As you can tell, I'm not a big fan of the term "infinite" since it simply doesn't occur in reality.
That being said, how can anyone possibly "prove" that .9999~ is exactly equal to 1 if you know the reality of the concept exists only in the mind of the mathematician?
Note: The EXACT value of 1/3 cannot be expressed as a decimal - in the same sense that the EXACT value of Pi cannot be expressed.
2006-09-01 13:28:42 · answer #2 · answered by LeAnne 7 · 0⤊ 1⤋
Let x=0.99999....
Since 1 is greater than or equal to x, 1-x must be greater than or equal to zero.
Now let's guess that number (1-x). I say, hey, I think it's 0.0001. Well, 1-.99999 is less than that, so 1-x, which is less than 0.99999, must surely be less than .0001. If I say, hey, I think 1-x is equal to some other number y, I can just subtract 0.999999... (with a sufficient number of 9's) from 1 and the result will be less then y. So, no matter what positive number I choose for y, 1-x is less than that number. Therefore, 1-x is not equal to any number greater than zero. So, it must be zero. Therefore, 0.99999...... = x = 1.
2006-09-01 13:05:17 · answer #3 · answered by john 3 · 1⤊ 0⤋
1/9 = .111...
Therefore:
9(1/9) = 9(.111...) = .999... = 1
Don't be fooled by notation, 9(1/9) and .999..., are just represented differently. However, they mean the same thing.
2006-09-01 17:46:29 · answer #4 · answered by Jerry M 3 · 0⤊ 0⤋
try
1/3 + 1/3 + 1/3 = 1
.333333 + .3333333 + .3333333 = .999999
2006-09-01 12:50:51 · answer #5 · answered by Brian D 5 · 0⤊ 0⤋
Let x = 0.9999...
10x = 9.9999...
9x = 10x - x = 9.9999... - 0.9999... = 9
dividing by 9:
x = 1
2006-09-01 13:57:13 · answer #6 · answered by h2 2 · 2⤊ 0⤋