Mathematics

“There is a well defined way in which one can subtract one infinity from the other and get a finite answer.” Steven Hawking

     Infinity is not a number, and mathematics and infinity do not work well together.  In fact, no common mathematical operation works at all. You cannot add, subtract, multiply or divide using infinity as one of the operators.  One plus infinity is still infinity.  And infinity minus one is infinity.  If infinity is truly infinite, which it is by definition, then you can take away or add an infinite number of ones and the answer is still infinity.  And the result is the same no matter what real number you pick. Same for multiplication and division.  You can multiply infinity by however large a number you wish, even infinity itself, and the result won’t be any larger.  Infinity is already as big as it gets.  Well, you would think you could at least divide infinity into smaller pieces, but no. Infinity divided by two is half an infinity. So how big is half of something that is infinite, and the answer is always infinitely big.  Infinity is unlimited, and taking away half of it won’t affect it at all.  It has plenty enough stuff to give away half of itself but still have a enough left over to remain infinite.  In fact, the surrendered amount will itself be infinite and the two resulting infinities automatically converge into a single infinity again. Looking at it another way, if you tried to place an infinite number of widgets into two separate piles, one at time, alternating piles as you go, you would never run out of widgets.  You would be placing widgets into piles forever, without end.  There is no earthly way to measure how many of them would eventually end up in the two equal piles, because eventually would never come.

Odd or Even?  “Infinity can be mind-boggling.”

From Steven Strogatz’ New York Times Opinionator series —

In late February I received an e-mail message from a reader named Kim Forbes. Her six-year-old son Ben had asked her a math question she couldn’t answer, and she was hoping I could help:

Today is the 100th day of school. He was very excited and told me everything he knows about the number 100, including that 100 was an even number. He then told me that 101 was an odd number and 1 million was an even number, etc. He then paused and asked: “Is infinity even or odd?”

I explained that infinity is neither even nor odd. It’s not a number in the usual sense, and it doesn’t obey the rules of arithmetic. All sorts of contradictions would follow if it did. For instance, “if infinity were odd, 2 times infinity would be even. But both are infinity! So the whole idea of odd and even does not make sense for infinity.”

Kim replied:

Thank you. Ben was satisfied with that answer and kind of likes the idea that infinity is big enough to be both odd and even.

Although something got garbled in translation (infinity is neither odd nor even, not both), Ben’s rendering hints at a larger truth. Infinity can be mind-boggling.

http://opinionator.blogs.nytimes.com/2010/05/09/the-hilbert-hotel/