Here is a strange continuous function on the unit interval, whose derivative is 0 almost everywhere, but it somehow magically...
Continue reading...functions
Multidimensional Newton’s Method
You’ve probably heard of Newton’s Method from your calculus course. It can be used to locate zeros of real-valued functions. But did...
Continue reading...Sum of Cubes and Beyond
We saw this wonderful identity in Sum of Cubes: 13 + 23 + … + n3 = (1 + 2 + … + n)2. Hence the set of numbers {1,2,…,n} has the...
Continue reading...Continuous but Nowhere Differentiable
You’ve seen all sorts of functions in calculus. Most of them are very nice and smooth— they’re “differentiable”, i.e., have...
Continue reading...Euler’s Product Formula
Here is an amazing formula due to Euler:SUMn=1 to infinity n-s = PRODp prime (1 – p-s)-1 .What’s interesting about this formula is that it...
Continue reading...Connected Sums
A surface is any object which is locally like a piece of the plane. A sphere, a projective plane, a Klein bottle,...
Continue reading...Fourier Ears Only
Did you know that every sufficiently smooth function on an interval can be expressed as an infinite sum of sines...
Continue reading...Quick Square Roots
Here’s a very quick way to generate the square root of N. Let A0=N. Then generate a sequence of numbers A1, A2,...
Continue reading...Sine of (1/55555….)
Take your calculator, and enter a number consisting of several 5’s, such as 5555555. Now take the reciprocal of this number. Now take the...
Continue reading...Area of a Hyperbolic Wedge
The unit circle can be parametrized by (cos w, sin w). Given a point on it, the region cut out...
Continue reading...