“The imaginary part of a complex number is real.”

This is what our maths teacher told us when we first learnt about complex numbers. You can imagine our class’s bafflement when asked to explain what it meant: what did it mean? We guessed pretty quickly that it was something philosophical – maybe it was telling us that imaginary numbers do exist, even though they’re intangible… or maybe it was even deeper than that…

But this is maths. Mathematics, the most logical, clear and abstract of all the sciences. The polar opposite of philosophy. We were wrong. Why would a maths teacher have a “deep statement” pinned above his whiteboard? Unless it wasn’t really deep…

It turned out, after a good deal of frantically whispered discussion between my classmate and I, to be very simple indeed. A complex number can be split into two parts: the real part, and the imaginary part. The imaginary part consists of a multiple of i , the square root of minus 1. However, when we show the complex number on, say, a graph, we can’t actually pinpoint the imaginary part, because it’s imaginary. Instead, we draw a graph with the real part (Re) on the x-axis and the (real) coefficient of i from the imaginary part (Im) on the y-axis. Everyone knows that the imaginary part on the graph needs to be multiplied by i, so it’s not a problem. And to be consistent, Im is defined as the coefficient of i whether you’re working with graphs or not.

Sometimes, just sometimes, maths is a lot simpler than it seems.

Have you ever found that something in maths has turned out to be a lot simpler than it looked? Tell us about your experience in the comments section below.


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