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Friday, June 23, 2017

#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







Thursday, June 22, 2017

Advanced Knowledge Problem of the Week: 6-22-17

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


Solution below.

Tuesday, June 20, 2017

Problem of the Week: 6-19-17 [Calculus]

Check out this week's problem of the week, finding the optimum way to craft a boxes net. Let us know how you did in the comments below or on social media!



Solution below the break.

Friday, June 16, 2017

New Publication: Some of Infinity by David Craft

Looking for a weekend read, or a gift for a mathematician in your life? 

Consider the latest publication from the Worldwide Center of Mathematics, Some of Infinity by David Craft. The book takes its time, meandering from topic to topic, numbers, infinity, fractals, calculus, topology, and takes the reader through these subjects from conception to completion. This way of going through each section makes for a good change of pace for anyone who reads a lot of math books, which can zoom through interesting points; and can be a thoughtful introduction to math material for anyone who doesn't.

Buy the digital or print version here.

Watch our review of the book:

#MathChops Episode 1: Proof of the Pythagorean Theorem

One of the cornerstones in Mathematics was proven by Pythagoras around 520 BC. Today we know this as the Pythagorean theorem, which states the sum of the squares of two sides of a triangle equal the square of its hypotenuse (a2 + b2 = c2). Pythagoras not only discovered this theorem, but he also started a philosophical and religious school where his followers worked and lived. They were known as the Pythagoreans and they lived by a specific set of rules, which dictated when they spoke, what they wore, and what they ate. Their lives were dedicated to universal discoveries and proving theorems. Pythagoras was the Master of these men and women, who were known as mathematikoi.
A graphic from Some of Infinity.

In our book, Some of Infinity, the author, David Craft, briefly talks about the Pythagoreans and goes on to prove the Pythagorean theorem. He touches on numerous sections of Mathematics such as Numbers, Infinity, Probability, Fractals, Calculus, and more. The idea for the book came about from trying to convince his friends that math is fun and cool. He does a very good job of portraying that math actually is fun and interesting, while keeping the reader engaged with cool puzzles and riddles.




Watch the proof here!