Intuitive Understanding of Euler’s Formula
The article is quite right that multiplying by <i>i</i> gives a rotation. But it doesn't quite explain the reason for this: it's because that's the whole point of defining imaginary numbers in the first place!<p>Remember you start off wanting to find a solution for the equation:<p><pre><code>    i^2 = -1
</code></pre>
This is actually easier to think about if you multiply it by a general real number r:<p><pre><code>    r i^2 = -r
</code></pre>
In other words you want <i>i</i> such that if you multiply <i>r</i> by <i>i</i> twice, it's the same as multiplying <i>r</i> by -1 once.<p>This is tough if you try to solve it by analogy with real positive numbers. If you picture the real number line (all the possible <i>r</i>) and multiply it by a positive number, let's say 4, then the whole thing stretches out quite a bit. It's pretty obvious that the way to break this operation into two equal parts is to stretch it a bit less (in this case, by a factor of 2).<p>The analogy of a stretch for -1 is a reflection: Imagine the whole number line collapsing in towards zero and bouncing back out again. But if you stop this half way then everything has just settled on zero, and doing that twice is obviously not going to get to the whole reflection. No other intermediate point seems any good either. (These are all the multiplications by <i>x</i> where -1 < <i>x</i> < 1.)<p>The key idea of imaginary numbers is to consider multiplication -1 to be a rotation by half a turn rather than a reflection. That is a lot easier to do half of! As soon as you have multiplication by -1 as a rotation by half a turn, it is obvious to identify <i>i</i> as rotation by a quarter turn.