Linear Algebra and Group Theory

It seems as though everyone has different ways of doing simple calculations in their head, like adding multiplying, dividing. I was just wondering if any of you guys have a good method of doing this, I have been trying to decide which is the most efficient; maybe you guys can help me out. When I come across simple addition, lets say 37 + 17 I first break down the two numbers into multiples of 5, so I would figure 35+15 real quick then take the remaining numbers and add them, then add the two sums together so for me it's like 50+4=54 I can do this pretty fast but sometimes people can calculate things much faster. I also have a method for subtracting, that I will…

I need some help with some relatively easy number theory problems, I'm just not sure how to go about them...

When someone says "A is inversely proportional to B, it means A=k/B", where k is some proportionality constant. Or 2 inverse is 1/2, or 0.5, or 2^-1 = 1/2. Why is that sin inverse, although written as sin^-1 is not 1/sin? Likewise for cosine and tangent.

I have been reading about the public key cipher where the encrption is squaring your message mod n where n=pq (primes = 3 mod 4). The decryption scheme here relies on finding roots mod p,q - namely on the theorem that states: x^(p+1)/4 is a square root of either x or negative x mod p, if p is prime=3 mod 4. It was not mentioned in my text why the -x case is not relevant. Since the number that needs to be rooted is known to be a rsult of squaring, it is implied that the way they know the -x will not happen is that there is some theorem that states the following: let p be prime and p=3 mod 4, then there is no x such that x^2= -1 mod p. My question is: is there indeed …

I dont understand this at all, i dont seem to get the logic in this, ive seen my helpful little rules to memorize but i dont seem to recognize it when i face a factoring problem, is there any tricks or some logical concept on this topic?

Eh, I saw this problem on a quiz today and couldnt' remember for the life of me how to do it. I'm sure its something simple, but I'm way too lazy to get my notes out of my car Inverse Cosecant (2) = csc-1(2) =

Could someone explain string theory to me? I've heard of it, but never read up on it. I just got a book called "The Elegant Universe". Havn't started reading yet. I just want a general idea before I start reading.