Are there any real numbers that are NOT algebraic, i.e., expressible as the root of a non-zero polynomial with integer coefficients? In fact,...

Continue reading...# advanced

These fun facts are at the advanced level.

## Does Order of Addition Matter?

From your earliest days of math you learned that the order in which you add two numbers doesn’t matter: 3+5...

Continue reading...## Irrationality by Infinite Descent

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

Continue reading...## 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...## Fermat’s Last Theorem

There are lots of Pythagorean triples; triples of whole numbers which satisfy:x2 + y2 = z2. But are there any which satisfyxn + yn =...

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...## Ordinal Numbers

One of the most useful properties of the whole numbers is that every non-empty subset has a least element; this...

Continue reading...## Risk-Free Betting on Different Beliefs

Alice believes that the 49ers will win the Super Bowl with probability 5/8. Bob believes that Ravens will win the...

Continue reading...## Greedy to Avoid Progressions

An arithmetic progression is a sequence of 3 or more integers whose terms differ by a constant, e.g., 20, 23, 26, 29...

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