Problem of the Week: 7-4-17 [Geometry]

Be sure to let us know how you did in the comments below or on social media!

Solution below.

Episode 3: All Horses Are the Same Color -- Equine Monochromaticity [#MathChops]

Obviously, this theorem is false, but it is a good way to show off your math chops and confuse a friend who may be taking an introductory course in math reasoning. This ‘proof’ is purely for fun, but does point out an important part of inductive proofs, which is that the assumption for the ‘n’th case must imply our statement is true in the ‘n+1’th case for any arbitrary n. Take what you will from this proof, but it reminds me of a joke I heard once.

A mathematician, physicist, and engineer are on a train in spain and see a white horse. The engineer remarks, “all horses are white!” to which the physicist and mathematician shake their heads. “No no no,” says the physicist, “what this means is that some horses in spain are white.” to which the mathematician shakes his head. The mathematician thinks for a little, and says “In passing we saw a white horse grazing in the plains of spain; therefore, there exists at least one horse in spain, of which at least one side is white.” and the three go about their day.

Proof

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Advanced Knowledge Problem of the Week: 6-29-17

Be sure to let us know how you did in the comments below or on social media!

Solution below.

Problem of the Week: 6-27-17 [Linear Algebra]

Check out this #PotW about properties of orthogonal matrices! as always, let us know what you think about it in the comments below or on social media!

Solution below the break.

#MathChops Episode 2: Proof That the Irrationals Are a Dense Set Within the Reals

The first conception of this episode was to prove that the rationals are a dense within the reals, which is an algebraic proof showing that between any two real numbers, there is a rational number. This proof does not define the real numbers, and treats them as some empirical fact that you know; yet, once the real numbers are constructed, the proof is really trivial. The proof used in this episode utilizes an analytic definition of dense sets: if a set `A’ along with its limit points equals the `B’, then `A’ is a dense set within `B’. You will see that we construct the reals in such a way that the rationals are dense within the reals. But first, a little background.

First, we construct the natural numbers using Peano’s Axioms, and the integers can be constructed many different ways from the natural numbers (think including additive inverses). From the integers, the rational numbers are all ratios of two integers. These ratios can be thought of as finite decimal expansions, and we will construct the real numbers using Dedekind cuts. To define a real number, we chop the number line at the end of an infinite decimal expansion, and call the set of all rational numbers less than that cut the real number. Now of course, this is defining any real number as the limit point of a rational sequence, making the closure of the rationals (the rationals along with their limit points) the reals. The proof that the irrationals are a dense set within the reals is less obvious.

Construction of the rationals, from AMS.

We need to find an irrational sequence that converges to a rational number (let’s choose 1,  and get any rational number by multiplying our sequence). After some thought, the sequence

a_n = 1 + \frac{\sqrt{2}}{n} is a sequence of irrational terms whose limit point is a rational number. Thus, the irrationals are a dense set within the reals.

Proof

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Advanced Knowledge Problem of the Week: 6-22-17 [dynamics]

Check out this problem on dynamical systems! Let us know how you did in the comments below or on social media!

Solution below.