Thursday, October 27, 2005
To Infinity and Beyond?
In July of 2001, Cecil Adams takes on a question that has been fought over for many years. It is a mathematical question that has to deal with infinity, a very difficult subject for the finite human mind to comprehend. The question is if 1/3 = .333 repeating, and 2/3 = .666 repeating, then why doesn’t 3/3 (= .999 repeating) truly equal 1? He begins by explaining that repeating decimals never terminate and therefore go on infinitely. Then he says that .999 repeating does actually equal 1 even though it seems that it may never get there. It seems that no matter what, if the repeating decimal is truncated at some digit, there will still be a difference between 1 and the decimal, however miniscule it is. However, that contradicts the idea of infinity. If the decimal itself is infinite, it can’t be truncated, that isn’t allowed by mathematical principles. Another example he gives is pi; the ever popular irrational number used in figuring out many things (circumference, area, etc…) about circles. It never repeats and therefore there is no true numerical representation of it. Only the symbol (π) does it justice.
Adams also presents a very famous paradox. It has to do with the simple idea that no matter how much a number is divided, it will never reach 0. Take 1 and divide it by 2, for example. 1, ½, ¼, 1/8, 1/16, ….. The only way for the fraction to equal 0 would be for the numerator to be 0, which it will never be. So using this paradox an applying it, we has humans should never be able to touch anything, since we need to cover half the distance, and then half of that… just like with the fractions above.
I enjoyed this question and answer because I love math. Being a math major will give you the ability to enjoy wrestling with the idea of infinity.