real 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 length m/n can be obtained by dividing a length m line segment into n equal parts (if you like, this can be done by straightedge...

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Rationals Dense but Sparse

Well we all know that between any two real numbers there is a rational. Mathematicians like to say that the rationals are dense in the real line… what this means is that any open set will contain some rational. So they are “everywhere” in the line, aren’t...

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Devil’s Staircase

Here is a strange continuous function on the unit interval, whose derivative is 0 almost everywhere, but it somehow magically rises from 0 to 1! Take any number X in the unit interval, and express it in base 3. Chop off the base 3 expansion...

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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 square? It seems that this should not be possible… after all, lines and planes are different dimensions. Surprisingly, such a curve is possible! This counter-intuitive object is...

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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 the power 2 and you’ll get 2. But what happens if you raise an irrational number to an irrational power? Can this ever be rational?...

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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 derivatives defined everywhere. Some, like the absolute value function, have “problem points” where the derivative is not defined. But is it possible to construct a continuous...

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Banach-Tarski Paradox

Did you know that it is possible to cut a solid ball into 5 pieces, and by re-assembling them, using rigid motions only, form TWO solid balls, EACH THE SAME SIZE AND SHAPE as the original? This theorem is known as the Banach-Tarski paradox. So why...

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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 and reassemble the pieces using only rigid motions to form two solid balls each the same size as the original. The construction depends on the Axiom...

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Tower of Powers

Consider an infinite “tower of powers” of x, defined by x^x^x^… = x^(x^(x^…)) Can we find a value of x so that this tower is equal to 2? Let’s write A for the value of the tower; then also A = xA from the definition, so...

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Koch Snowflake

Snowflakes are amazing creations of nature. They seem to have intricate detail no matter how closely you look at them. One way to model a snowflake is to use a fractal which is any mathematical object showing “self-similarity” at all levels. The Koch snowflake is constructed as...

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