Math Is Fun Forum

But my book says

Thanks.

The techniques used to find the nth roots of unity can be extended to find the nth roots of any complex number.

Suppose that

is an nth root of

then

The book then says that

and

, but I don't see why this would necessarily be the case?

Wouldn't you have to say that

and

?

Thanks.

I've just been doing some investigtion and I've found something quite nice.

The number of distinct factors a number has is given by the product of one more than the maximum power of each of the prime factors.

i.e. If

then the number of distinct factors n has is given by

.

This is relatively easy to prove. We have

. There is a total number of α_1 + 1 numbers that can be made from taking powers of p_1 from 0 to α_1, and similarly a total number of α_2 + 1 numbers that can be made from taking powers of p_2 from 0 to α_2. Each number from the first α_1 + 1 numbers can be paired with each of the α_2 + 1 numbers, giving a total of (α_1 + 1)(α_2 + 1) numbers. These can then be paired with the α_3 + 1 numbers that can be made from powers of p_3, giving a total of (α_1 + 1)(α_2 + 1)(α_3 + 1) numbers. This process can then be continued to give the desired result. 

For example, if n = 40 = 2^3 x 5^1, the number of distinct factors is (3+1)(1+1) = 8. This can be verified using the principals involved in the proof.

1       1
2       5
2^2
2^3

The possible factors of 40 are therefore: (1*1, 1*5), (2*1, 2*5), (2^2*1, 2^2*5), (2^3*1, 2^3*5), a total of 8.

Rather than me setting exercises for others to do, I'd like people to give me some challenging questions It's a thread where people can share nice problems, but I'd like the benefit of seeing if I can get to the result myself too.

By challenging questions I mean ones which require insight and clever tricks, or ones with nice results, rather than something that requires a lot of knowledge and can be done by appyling heavy theorems. 

Content-wise they can contain anything that doesn't assume too much university-level knowledge, although I would like to learn some things along the way so don't hold back.

I'm mainly looking at people like Jane, mathsy and Ricky, but anyone else is of course welcome

Thank you.

For the curve with equation y = arsinh(x+1), find

a) the coordinates of its point of inflexion P
b) an equation of the normal to the curve at P.

But this derivative is never 0???

Thanks.