Math Is Fun Forum

I've seen a nice combinatorial 'proof' that 0^0 = 1.

"The cardinal number n^m is the size of the set of functions from a set of size m into a set of size n. If m is positive and n is zero, then there are no such functions, because there are no elements in the latter set to map those of the former set into. Thus 0^m = 0 when m is positive. However, if both sets are empty (have size 0), then there is exactly one such function — the empty function."

The only function between the empty set and itself is the identity function.

That's really nice

I've spotted it. Missed the 5 out when I replaced x's with t's.

Sometimes it helps just to see it written clearly

Thanks guys and girls

Oh, well if it is good luck with integrating it.

I think it's more likely to be x sec^x tanx. If so, use integration by parts (differentiate x, integrate the rest)

Look at the series expansions of e^x, and let x = iθ (simplify a bit)

Compare this to the series expansions of cosx and sinx. You should be able to see why e^iθ = cosθ + isinθ.

Easy mistake to make. You'll stop making it (as much) as you continue to do more maths (mainly because you work in radians pretty much the whole time).

We have that

I'm not sure, but can you not just say that because all the coefficients are real,

Which means

Consider the vertical/horizontal lines that separate the colums/rows. There are 9 vertical lines and 9 horizontal lines. In order to form a rectangle, you need to choose 2 vertical lines and 2 horizontal lines. How many ways are there of choosing 2 lines from 9? How can you therefore find the number of unique rectangles?