3.9 Complex numbers, algebraically
In section 3.6.2, I gave an example of extending the integers by considering elements of the form a + b√2. We can similarly extend R. complex number number$complex C`bold
The real numbers R do not contain the square roots of negative numbers. We define the value i as √(–1), which means i2 = –1. i`italic
For a and b in R, consider all elements of the form z = a + bi. This is the field of complex numbers C formed as R[i] = R[√(–1)]. We call a the real part of z and denote it by Re(z). b is the imaginary part Im(z). a and b are real numbers. Every real number is also a complex number with a zero imaginary part. complex number$arithmetic complex number$real part complex number$imaginary part Re() Im()
While we can always determine if x < y for two real numbers, there is no equivalent ordering for arbitrary complex ones that extends what works for the reals. ordered
