The subject of finding algorithms that estimate the magnitude of a complex number, without having to perform one of those pesky square root operations, has been discussed many times in the past on the comp.dsp newsgroup. That is, given the complex number R + jI in rectangular notation, we want to estimate the magnitude of that number represented by M as:

On August 25th, 2009, Jerry (Mr. Wizard) Avins posted an interesting message on this subject to the comp.dsp newsgroup (Subject: "Re: Complex versus real numbers"). In his message Jerry said that in the ol' days before hand calculators, experienced slide rule users performed rectangular-to-polar conversion using:

I'm not rightly sure why the algorithm in Eq. (2) seemed so startling, and appealing, to me. Maybe because it (surprisingly) seemed to translate a square root operation into forward/inverse trigonometric, and ratio, operations. (All of which could be performed on a slide rule.) In any case, this "neat" algorithm, as they say in the U.S. Military, "works fine and lasts a long time."

Can you figure out why Eq. (2) is true? Once you do, you'll see a good example of how a simple idea can appear to be complex, ... oops, I mean complicated.