3.9 Complex numbers, algebraically
In subsection 3.6.2a + b√2. We can similarly extend R.
The real numbers R do not contain square roots of negative numbers. We define the value i to be √−1, which means i2 = −1.
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].
3.9.1 Arithmetic
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.
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.
The equations for arithmetic are
| 0 | = 0 + 0i |
| 1 | = 1 + 0i |
