Now, on to the subject of complex numbers. We have stayed away from this subject until now simply because we did not want to confuse things. Initially, it is easy to think of signals as “real valued.” ...

Real and imaginary: The time-dependent Schrödinger equation formulated using complex numbers. (Courtesy: iStock/piranka) Complex numbers are essential to achieve the most accurate quantum-mechanical ...

Mathematicians were disturbed, centuries ago, to find that calculating the properties of certain curves demanded the seemingly impossible: numbers that, when multiplied by themselves, turn negative.

Several years ago one of us, Toni, asked another of us, Marco, to come to his office at the Institute of Photonic Sciences, a large research center in Castelldefels near Barcelona. “There is a problem ...