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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