The traditional proof that the square root of 2 is irrational (attributed to Pythagoras) depends on understanding facts about the...

Continue reading...# irrational

## 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...## Computability of Real Numbers

We can write a computer program that will successively print out the digits of the decimal expansion of Pi. We...

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...## 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...## Square Root of Two is Irrational

An irrational number is a number that cannot be expressed as a fraction. But are there any irrational numbers? It was known to...

Continue reading...## 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...## Pi Approximations

Pi is the ratio of the circumference of a circle to its diameter. It is known to be irrational and its decimal...

Continue reading...## Pick’s Theorem

A lattice point in the plane is any point that has integer coordinates. Let P be a polygon in the plane whose vertices have...

Continue reading...## Memorizing Pi

The digits of Pi are fascinating. As the ratio of the circumference of a circle to its diameter, Pi has...

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