This post is a special shout-out to my former students who are now taking calculus (if you don’t know any calculus, just hang tight… there will be more calculus-less math goodness coming your way soon). This post is 100% money-back guaranteed to not really help you at all on the AP exam! But don’t worry, it’s still awesome.
First, consider this equation:
Perhaps you’ve seen it before… among other things, it gives us a very neat connection between complex numbers expressed in polar coordinates ( is the complex number with polar coordinates ) and expressed in rectangular coordinates. But to see where this equation comes from, you need some calculus!
Let’s set . The key observation is that differentiating z is the same as multiplying it by i (remember, ):
So we write this observation as a simple differential equation; after swapping and z we can integrate:
And since is equal to 1 when is zero, C must be equal to 1. Therefore we have , as promised! Note that this makes sense with our key observation from before: differentiating gives , the same as multiplying by i.
Now for a really good time, set . Noting that , we end up with the Most Beautiful Equation in the World:
Almost makes you cry, doesn’t it? It relates five of the Most Important Numbers in the World (0, 1, e, i, and ) using three of the Most Important Operations in the World (addition, multiplication, and exponentiation) and nothing else. Beauty, simplicity, elegance–it’s all right here.
And for a short time only, this equation can be yours for the low, low price of the number of grains of rice on the last square of a chessboard if you put just one grain on the first square, two grains on the second square, four grains on the third, and merely double the number of grains each time. Order now, supplies are limited!