Given a number line with equally spaced tick marks one unit apart, we know how to measure rational lengths: the...
Continue reading...real analysis
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...Banach-Tarski Paradox
Did you know that it is possible to cut a solid ball into 5 pieces, and by re-assembling them, using...
Continue reading...Sierpinski-Mazurkiewicz Paradox
If you’ve seen the Banach-Tarski paradox, you know that it is possible to cut a solid 3-dimensional ball into 5 pieces...
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