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

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