Donald Knuth's letter about teaching O notation in calculus classes

Re:Donald Knuth's letter about teaching O notation in calcul... by yters on Wednesday April 16, 2008 @02:24AM (#164932)

The tricky thing is that infinity isn't a number as we normally think of numbers. There are different kinds of infinity, so infinity / infinity does not necessarily equal 1. It could be any number between 1 and infinity, inclusive.

To get this intuition, think of the integers. There are obviously an infinite number of them. Now think of the rationals. There are now an infinite number of numbers between 1 and 2, so there are more rational numbers than there are integers; in fact infinitely more.

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Re:Donald Knuth's letter about teaching O notation in calcul... by rms on Wednesday April 16, 2008 @02:34AM (#164947)

I believe set theory is the only area of mathematics where different orders of infinite actually mean anything.

So, in set theory, cardinality is a measure of the elements of a set. The cardinality of the infinite set of natural numbers is aleph-0. An infinite set has cardinality aleph-0 if it can be put in one to one correspondence with the naturals. http://en.wikipedia.org/wiki/Bijection This counterintuitively includes the rational numbers also. http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

The real numbers can't be put into one-to-one correspondence with the rational numbers. So the infinite of the reals truly is bigger than the infinite of the rational numbers.

Also see this ppt: http://www.cs.cmu.edu/afs/cs.cmu.edu/academic/class/15251/di...

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Re:Donald Knuth's letter about teaching O notation in calcul... by yters on Monday April 21, 2008 @12:34AM (#169076)

You don't think mathematics is unified, i.e. something true in one area may not be true in another? I don't think a lot of mathematicians believe this, and it isn't clear to me that "infinity" refers to two different things in the two different realms (the other possibility).

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