Why are constants ignored in asymptotic analysis?

Question

Why are constants ignored in asymptotic analysis?

Answer 1

It's because of the linear speedup theorem for Turing machines.

If you show me a Turing machine that solves a problem of size n in f(n) steps, and specify a constant c > 0, I can make a Turing machine that solves the same problem in c f(n) steps (or one step, if c f(n) < 1). For example, by taking c = ½ my machine can solve the same problem in half as many steps. Or, by taking c = ¹/_1000000, my machine can solve the same problem in only a millionth as many steps!

This result makes constant factors uninteresting (theoretically speaking: obviously in practice they still have some interest).