Complex Numbers

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Chapter 7 Complex Numbers

We have motivated the passage from the naturals to the integers to the rationals to the reals by acknowledging at each stage that there are certain kinds of equations that we'd like to solve and processes we'd like to follow to specify numbers: we passed from \(\NN_0\) to \(\ZZ\) in order to be able to subtract; we passed from \(\ZZ\) to \(\QQ\) in order to be able to divide; we passed from \(\QQ\) to \(\RR\) in order to be able to form limits.

But we aren't done: there are still equations we cannot solve. The equation \(x^2+1=0\) “should” have \(2\) solutions \(z\) and \(-z\text{,}\) but it has no real solutions. Introducing two new solutions \(i, -i\) while keeping the field structure in place, produces the field \(\CC\) of complex numbers.