From your earliest days of math you learned that the order in which you add two numbers doesn’t matter: 3+5...

Continue reading...# analysis

## Dedekind Cuts of Rational Numbers

Given a number line with equally spaced tick marks one unit apart, we know how to measure rational lengths: the...

Continue reading...## Rationals Dense but Sparse

Well we all know that between any two real numbers there is a rational. Mathematicians like to say that the...

Continue reading...## Devil’s Staircase

Here is a strange continuous function on the unit interval, whose derivative is 0 almost everywhere, but it somehow magically...

Continue reading...## Space-filling Curves

Consider a square in the plane. Is it possible to draw a curve in the square that touches every point inside the...

Continue reading...## Rational Irrational Power

If you raise an irrational number to a rational power, it is possible to get something rational. For instance, raise Sqrt[2] to...

Continue reading...## 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...## Cantor Diagonalization

We have seen in the Fun Fact How many Rationals? that the rational numbers are countable, meaning they have the same cardinality as...

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...

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