For example:Īdding (3 + 4i) to (-1 + i) gives 2 + 5i.Īgain, this is a visual interpretation of how “independent components” are combined: we track the real and imaginary parts separately. Addition with complex numbers is similar, but we can slide in two dimensions (real or imaginary). We’ve seen that regular addition can be thought of as “sliding” by a number. Magnitude measures a complex number’s “distance from zero”, just like absolute value measures a negative number’s “distance from zero”. See how it looks like the absolute value sign? Well, in a way, it is. The shorthand for “magnitude of z” is this: |z| So, a number z = 3 + 4i would have a magnitude of 5. Measuring Sizeīecause complex numbers use two independent axes, we find size (magnitude) using the Pythagorean Theorem:

I had an idea to write the imaginary part vertically, in fading ink, but it wasn’t very popular. Unfortunately, we don’t have nice notation like (3.4) to “merge” the parts into a single number. Complex numbers are similar: they have their real and imaginary parts “contained” in a single variable (shorthand is often Re and Im). We often write:Īnd it doesn’t bother us that a single number “y” has both an integer part (3) and a fractional part (.4 or 4/10). It is a bit strange how “one” number can have two parts, but we’ve been doing this for a while. We’re saying there’s a number “z” with two parts: 3 (the real part) and 4i (imaginary part). With complex numbers, there’s a gotcha: there’s two dimensions to talk about. In regular algebra, we often say “x = 3″ and all is dandy - there’s some number “x”, whose value is 3. This post will walk through the intuitive meanings. This insight makes arithmetic with complex numbers easier to understand, and is a great way to double-check your results. Imaginary numbers have an intuitive explanation: they “rotate” numbers, just like negatives make a “mirror image” of a number.

Dividing complex numbers