From your earliest days of math you learned that the order in which you add two numbers doesn’t matter: 3+5 and 5+3 give the same result. The same is true for the addition of any finite set of numbers. But what if you are adding...

Continue reading...# calculus

# 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...

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 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...

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 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...

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 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...

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 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?...

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 you know that it is possible to define a multi-dimensional version of Newton’s Method for functions from Rn to Rn? Here’s how it goes. The derivative...

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 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...

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 relates an expression involving all the positive integers to one involving just primes! And you can use it to prove there must be infinitely many primes....

Continue reading...# How many Rationals?

How many rational numbers are there? Yes, infinitely many, I hear you say. But how large is that infinity? Are there just as many rational numbers as integers? Well, this requires us to be precise about “just as many”. A mathematician would say that set...

Continue reading...