Hi everybody,

How to show that exists a set $\displaystyle \mathbb{C}={a+ib, a,b \in \mathbb{R}}$ such as $\displaystyle i^2=-1$

Can you help me please??

Results 1 to 2 of 2

- Oct 31st 2010, 03:57 AM #1

- Joined
- Aug 2009
- Posts
- 62

- Oct 31st 2010, 04:27 AM #2

- Joined
- Apr 2005
- Posts
- 19,771
- Thanks
- 3028

Starting from what basis? If you are given a "symbol", i, such that $\displaystyle i^2= -1$, then the rest is easy. If not, then the usual construction of the complex numbers is:

Given the set of ordered pairs of real numbers, (x, y), define (x, y)+ (u, v)= (x+ u, y+ v) and (x, y)(u, v)= (xu- yv, xv+ yu) where x, y, u, v can be any real numbers.

Show that this set of pairs satisfies all of the properties of the complex numbers with (x, y)= (x, 0)+ (0, y)= x(1, 0)+ y(0, 1) and you identify (1, 0) with the real number 1 and (0, 1) with "i".