advanced

These fun facts are the difficulty level advanced.

Liouville Numbers

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, not only are there infinitely many, there uncountably many transcendental numbers, as they are called. This may be seen by noting there are only countably many polynomials with integer...

Continue reading...

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 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...

Irrationality by Infinite Descent

The traditional proof that the square root of 2 is irrational (attributed to Pythagoras) depends on understanding facts about the divisibility of the integers. (It is often covered in calculus courses and begins by assuming Sqrt[2]=x/y where x/y is in smallest terms, then concludes that...

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 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...

Computability of Real Numbers

We can write a computer program that will successively print out the digits of the decimal expansion of Pi. We can also write one that will list the digits of the decimal expansion of the square root of 2. Similarly, there are computer programs that...

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 = zn, for integer powers n greater than 2? The French jurist and mathematician Pierre de Fermat claimed the answer was “no”, and in 1637 scribbled in the...

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...

Ordinal Numbers

One of the most useful properties of the whole numbers is that every non-empty subset has a least element; this allows us to begin a process of “counting” by successively choosing least elements: 0, 1, 2, 3, 4, … Any (totally) ordered set which has...

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 Super Bowl with probability 3/4. Assuming that Alice and Bob are both willing to accept any bet that gives them a positive expected value of...

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 is an arithmetic progression. Question: does every increasing sequence of integers have to contain an arithmetic progression in it? For instance, the sequence of primes...

Continue reading...