Fix some number N. What fraction of the integers less than or equal to N are prime? Thinking about it,...
Continue reading...calculus
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...Volume of a Cone in N Dimensions
One of the first geometric formulas we learn in plane geometry is that the area of a triangle is: Area of a...
Continue reading...Gamma Function
The Gamma function is an amazing integral: Gamma(x) = INTEGRALt=0 to infinity tx-1e-t dt . Using integration by parts, you can show...
Continue reading...Arclength Surprise
Consider a unit circle, and any arc S on the unit circle in the first quadrant. No matter where S is...
Continue reading...e is irrational
If e were rational, then e = n/m for some integers m, n. So then 1/e = m/n. But the series expansion for 1/e is 1/e =...
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...Chords of a Unit Circle
Take N equidistant points on the unit circle. Pick one of those points, then draw chords from it to all...
Continue reading...Euler’s Formula
The five most important numbers in mathematics all appear in a single equation!ei*Pi + 1 = 0.In fact, this is a...
Continue reading...Derivative Paradox
Here’s a fun (but untrue) fact. You know from calculus that the derivative of x2 is 2x. But what’s wrong with the following...
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