The traditional proof that the square root of 2 is irrational (attributed to Pythagoras) depends on understanding facts about the...
Continue reading...easy
These fun facts are at the easy level.
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...Van der Waerden Theorem
Can you color the integers red and blue such that there are no monochromatic arithmetic progressions (AP’s) extending infintely in...
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...Sum of Cubes and Beyond
We saw this wonderful identity in Sum of Cubes: 13 + 23 + … + n3 = (1 + 2 + … + n)2. Hence the set of numbers {1,2,…,n} has the...
Continue reading...Odd Numbers in Pascal’s Triangle
Pascal’s Triangle has many surprising patterns and properties. For instance, we can ask: “how many odd numbers are in row N...
Continue reading...Pretzel Unlinking
Imagine that each of the ropes in the two sets of links in Figure 1 are solid (with thickness) and...
Continue reading...Impossible Integral?
The following integral may be problematic for a freshman calculus student, even if armed with a list of antiderivatives: INTEGRAL0...
Continue reading...Pythagorean Triples
Which triples of whole numbers {a, b, c} satisfy a2 + b2 = c2 ? Such triples are called Pythagorean triples because they are integer...
Continue reading...High-Dimensional Spheres in Cubes
How good is your intuition in high dimensions? Take a square and divide it into its four quadrants. Inscribe a circle...
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