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