### Complex Numbers

The extension of the rational numbers, Q to, the "rational complex" numbers, or of the real numbers, R, to the set of complex numbers, C, is comparatively straight forward, and once again is done using ordered pairs. If <a,b> and <c,d> are two such pairs, (whether real or rational), we can define equality,addition and multiplication as follows:

• <a,b>+<c,d> = <a+c,b+d>
• <a,b>*<c,d>=<a*c-b*d,a*d+b*c>
• <a,b>=<c,d> iff a=c and b=d

Division is a little bit more complicated:

• <a,b>/<c,d>=<(a*c+b*d)/(c*c+d*d),(-a*d+b*c)/(c*c+d*d)>

The positive square square root of (c*c+d*d) of a complex number <c,d> is called its absolute value, and is a very important function defined for every number. Another key function is the complex conjugate <c,-d>

#### Gains and Losses

Complex numbers have some wonderful features. They are extremely useful when it comes to doing two dimensional geometry. Any polynomial equation can be solved using complex numbers: they form what is called an algebraically closed field.

However, they do have one very signifant disadvantage compared with all the other kinds of number: there is no natural way of ordering them. In other words there is no sensible way of saying whether one complex number is bigger or smaller than another.