Complex Numbers in Quantum Theory

The representation of real measurable quantities sometimes requires the use of imaginary numbers.

In physics pure imaginary numbers are rarely used on their own. They are always used alongside real numbers as a complex pair, with one part real and the other part imaginary.

But a complex pair is more than just a pair of numbers.  That one is real and one is imaginary means they can cross multiply across the pair, the two parts multiplying with both parts of another complex number.

Every complex number has another complex number than when they are multiplied together, make a real number. If you want to produce a real number from a complex number, this is how you do it---you multiply the number by this other special number. 

These real numbers obtained this way from complex numbers give rise to many measured quantities in the physical sciences, especially in quantum theory.