# The Center of Math Blog

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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

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!

### New Series: Top Pop Math Chops

Top Pop Math Chops is the Worldwide Center of Mathematic's new series that will go into some popular, and often times important, proofs across many facets of mathematics; from simple geometry, to calculus and beyond. Some proofs you will recognize because you use the result in day-to-day mathematics, and we think it is important that the actual mathematics behind the proof is laid out clearly. The scope of this series is wide, ranging from ancient techniques to prove mathematical truths, to modern methods and intuitions.

We hope you enjoy our journey through the fun, important, and interesting proofs that every math enthusiast should know! keep in touch with @centerofmath on Facebook, Twitter, or G+ using #MathChops to let us know what you think, or if there are any proofs you think we should cover.

Watch the introduction episode now!

## Thursday, June 15, 2017

### Advanced knowledge Problem of the Week: 6-14-17 [Real Analysis]

Check out this exercise covering uniform and pointwise convergence of sequences of functions! Let us know how you did in the comments below or on social media!

Solution below the break.

## Wednesday, June 14, 2017

### Flag Day and Mathematics

Flag Day celebrates the adoption of the United States' flag on this day in 1777. The symbol for unity, spread over thirteen stripes and fifty stars, stands tall as a momentous proclamation of the United States' values. Over the years, with the growth of our country, the flag of the U.S. has also changed, from thirteen stars to fifty. With each revision of the flag's design, a great deal of thought goes into the arrangement of our star spangled banner; and while mathematics is not always considered in this process, we know math is capable of bringing to our attention beauty, so today we will consider how math could play into our flag.

## Tuesday, June 13, 2017

### Problem of the Week: 6-13-17

See how much you can say about the quotient rule for anti-differentiation in this Problem of the Week! Be sure to let us know how you did in the comments or on social media!

Solution below.

## Thursday, June 8, 2017

### Advanced Knowledge Problem of the Week: 6-8-17 [Abstract Algebra]

Check out this #AKPotW, and be sure to let us know what you think in the comments below or on social media!

Solution below the break.

## Tuesday, June 6, 2017

### Problem of the Week: 6-6-17

Let us know what you thought of this problem of the week in the comments below or on social media!

Solution after the break.

## Thursday, June 1, 2017

### Advanced Knowledge Problem of the Week: 6-1-17

Hello! And Welcome to the AKPotW, please let us know how you did in the comments below or on social media :-)

Solution below the break:

## Tuesday, May 30, 2017

### Problem of the Week: 5-30-17

Check out this problem of the week, and let us know how you did in the comments below or on social media!

Solution after the break.

### Mathematical Advancements During the First Memorial Day

The idea of honoring our fallen veterans began in the late 1800's after losing nearly 750,000 soldiers in the Civil War. In 1868, General John A. Logan, one of the leaders of a Northern Civil War veteran's organization, declared that there would be a nationally recognized day for the fallen soldiers. “The 30th of May, 1868, is designated for the purpose of strewing with flowers, or otherwise decorating the graves of comrades who died in defense of their country during the late rebellion, and whose bodies now lie in almost every city, village and hamlet churchyard in the land,” he proclaimed. He decided to call this special day Decoration Day and chose the date because there were no battle anniversaries on that day. Decoration Day would eventually be known as Memorial Day in 1971, which fully encompassed American military personnel who died in all wars. The memory of American troops stands for the protection of freedom within the world, a freedom to advance together as a collected human race.

Around the time of the Civil War and then the creation of Decoration day in the 19th century, there were numerous mathematicians who were able able to make vast advancements in the field.

George Boole: This British mathematician and philosopher was one of the few who focused on logic and reasoning as well as their usefulness in mathematics. A few years prior to the Civil War, Boole introduced a new form of algebra (now called Boolean Algebra or Boolean Logic) which only consisted of operators 'AND', 'OR', and 'NOT'. These algebraic operators could be used to solve logic problems as well as some mathematical functions. Boole also composed an approach to logical systems with a form of binary, where he would process two objects (yes-no, true-false, 1-0, etc.). Under Boolean Logic, 1 + 1 = 1 (True ∧ True = True). This was a major advancement in modern mathematical logic, but people at the time didn't recognize it as one. It wasn't until American logician Charles Sanders Pierce recognized Boole's work, revised some of it, and then elaborated on his ideas in 1864. Nearly seventy years later, Boolean logic would go on to be used for electrical switches to process logic and become the basis for computer science.

Bernhard Riemann: A very well-known mathematician, from Germany, who made great contributions to differential geometry, analysis, and number theory in the 1850s (a few years prior to the Civil War). In college, Riemann began to take stride under the wing of his professor, Carl Friedrich Gauss (ranked as one of History's most influential mathematicians). One of Riemann's contributions was that of elliptic geometry and also Riemannian Geometry, which essentially generalized the ideas of surfaces and curves. Riemann's new contributions changed how we view the higher dimensional world we live in. He would go on to break away from 2 and 3 dimensions, and look at n dimensional surfaces which would contribute to further conceptualize relativity on curved surfaces. Another big breakthrough for Riemann came from working with the zeta function, which Euler had first experimented with in the 18th century. Using the zeta function to build a 3-dimensional landscape, he noticed that "the zeroes" (where the graph dipped to zero) of his landscape had a connection to the way prime numbers are distributed. This relationship between his zeta function and prime numbers brought him instant fame in 1859, when his findings were published. Unfortunately, Riemann passed away at the age of 39 in 1866 and his incomplete findings on this relationship, known as the Riemann Hypothesis, remain unsolved 160 years later. A prize of \$1 million has been offered as a prize for a final solution by the Clay Mathematics Institute.

George Cantor: Also a German mathematician, became a full Professor at the University of Halle at the age of 34, which was essentially unheard of. One of his major contributions to mathematics is the first foundation of set theory, which helped explain the notion of infinity and became very common in all of mathematics. He also delved into the concept of the infinities of infinity, where he showed there may be infinitely many sets of infinite numbers. His work undoubtedly changed how mathematicians now view sets and the concept of infinity. This all started in the early 1870s when Cantor considered an infinite series of natural numbers (1, 2, 3, ...) and an infinite series of multiples of 10 (10, 20, 30, ...). He could clearly state that the the series of multiples of 10 was a subset of the series of natural numbers, but he could also recognize the sets could be matched one-to-one (1 with 10, 2 with 20, 3 with 30, etc). He used this process, which is known as bijection, to show that the sets were the same size. Cantor realized he could do the same sort of thing comparing rational numbers and natural numbers, concluding that they are of the same infinity even though fractions would seem to outnumber natural numbers. Cantor also looked at irrational numbers, and argued that there exists an infinite amount of irrational numbers between each and every rational number. In later eras, Cantor would go on and refine his set theory and introduce new ideas such as ordinality and cardinality.

Works Cited:

http://www.history.com/topics/holidays/memorial-day-history
http://www.storyofmathematics.com/19th.html

## Thursday, May 25, 2017

### Advanced Knowledge Problem of the Week: 5-25-17

Let us know how you did on this #AKPotW in the comments below, or on social media!

Solution below the break:

## Tuesday, May 23, 2017

### Problem of the Week: 5-23-17

Let us know how you did in the comments below, or on social media!

Solution After the Break:

## Thursday, May 18, 2017

### Advanced Knowledge Problem of the Week: 5-18-17

Let us know what you think of this AKPotW on social media, or in the comments below!

Solution below the break!

## Tuesday, May 16, 2017

### Problem of the Week: 5-16-17

Check out this week's problem, and let us know how you did in the comments below or on social media!

Solution below the break.

## Happy Mother's Day From the Center of Math!

No amount of mathematical talent in someone's nature can make them into a great mathematician, and in this article, we will acknowledge a mother behind a great mathematician who nurtured her son and sacrificed her own career to give him the best opportunity to succeed.

## Thursday, May 11, 2017

### Advanced Knowledge Problem of the Week: 5-11-17

Check out this week's problem, and let us know how you did in the comments below or on social media!

Solution below the break.

## Tuesday, May 9, 2017

### #NationalTeachersDay -- Rosenthal Prize and Edyth May Sliffe Award

On this national Teachers' Day we recognize the teachers we love.

Here's your chance, math masses! Be sure to share your appreciation. Know an amazing mathematics teacher? Check out the National Museum of Mathematics' 2017 Rosenthal Prize.

You might also like to check out the Mathematical Association of America's Edyth May Sliffe Award.

### Problem of the Week: 5-9-17

Check out this week's problem, and let us know how you did in the comments below or on social media!

Solution below the break.

## Tuesday, May 2, 2017

### Problem of the Week: 5-2-1

Check out this week's problem, and let us know how you did in the comments below or on social media!

Solution below the break.

## Friday, April 28, 2017

### WCoM to Begin Tradition of Dropping the Peano Axioms off Roof.

 Look at those natural numbers go!

## Thursday, April 27, 2017

### Advanced Knowledge Problem of the Week: 4-27-17

Check out this week's problem, and let us know how you did in the comments below or on social media!



Solution below the break.

## Wednesday, April 26, 2017

### Welcome to the Center: New Introduction videos on Facebook and YouTube

If you are familiar with the center of math, you probably know us by one or two avenues: you read our blog, which produces problems of the week and talks about math related news; or you have used one of our textbooks, and watched the videos that came along with it. Behind our mission, to bring accessible math materials into the world, is a community of mathematicians of all sorts. We would like to invite you to explore the Center of Math, and take some time to make that community grow. With new Facebook and YouTube introduction videos, finding your way around our little corner of the worldwide web has never been easier. The videos outline some features of the Worldwide Center of Mathematics' Facebook and Youtube pages, that you may want to know about.

Our Facebook page primarily serves as a place for our community to gather, and a hub to navigate around the Center of Math with ease. Visit the Facebook page and join the collective! Once on the page, you can click the learn more button to go to our main site, or use the menu on the left to explore more. Don't forget to like us on FB to stay in the loop.

The YouTube video helps serve as a starting point to our overwhelming library of math videos. Check it out yourself, and don't hesitate to dive right into our content! If you need some more direction, our 'playlists' tab is a great place to start, or just use the spotlight search to find a specific topic. Drop a subscribe if you like what you see, to never miss another math video.

These features are explained in greater detail in the welcome videos, so give them a watch, and we'll see you soon!

-Worldwide Center of Math.

## Tuesday, April 25, 2017

### Problem of the Week: 4-25-17

Check out this week's problem, and let us know how you did in the comments below or on social media!

Solution below the break.

## The Math Madness 2k17 Champion is Here!

The events leading up to this moment have been exciting. We've seen upsets, landslide victories, and to-the-teeth battles, but as each mathematician falls, the taste of the coming victory becomes stronger. On this day, the dream to be deemed champion is going to become a reality for either Cauchy or Ramanujan.

See who won after the break.

## Thursday, April 20, 2017

### Advanced Knowledge Problem of the Week: 4-20-17

Solution after the break.

## Tuesday, April 18, 2017

### Problem of the Week: 4-18-17

Check out this week's problem, and let us know how you did in the comments below or on social media!

Solution below the break.

## Thursday, April 13, 2017

### Advanced Knowledge Problem of the Week: 4-13-17

Check out this week's ADVANCED problem, and let us know how you did in the comments below or on social media!

Solution below the break.

## A Quick Overview of the Concepts Behind Differential Eqs

Details after the break.

## Tuesday, April 11, 2017

### Problem of the Week: 4-11-17

Check out this week's problem, and let us know how you did in the comments below or on social media!

Solution below the break.

## Thursday, April 6, 2017

### Advanced Knowledge Problem of the Week: 4-6-1

Check out this week's ADVANCED problem, and let us know how you did in the comments below or on social media!

Solution below the break.

## Tuesday, April 4, 2017

### WCoM Donates Statistics Textbooks

The Worldwide Center of Mathematics recently had some winter weather make its way into a stock room and slightly damage a number of Introduction to Statistics: Think & Do (Stevens).

The damaged books will not be sold and instead will be donated.

The recipient(s) include: Siem Reap at Life and Hope Association and Phnom Penh at People Improvement Organization. Both organizations are in Cambodia.

 Water-damaged books prepared for donation.

### Problem of the Week: 4-4-17

Check out this week's problem, and let us know how you did in the comments below or on social media!

Solution below the break.

## Thursday, March 30, 2017

### Advanced Knowledge Problem of the week: 3-39-1

Check out this week's ADVANCED KNOWLEDGE Problem of the Week! Let us know how you did!

Solution below the break.

## Tuesday, March 28, 2017

### Problem of the Week: 3-28-17

Check out this week's problem, and let us know how you did!

Solution below the break.

## Only Four Remain in This Week's Trial of Math!

Vote here for the final two!

## Friday, March 24, 2017

### Dr. Esole at The Center 3-31-17!

Mark your calendars for Friday, March 31st! Mboyo Esole is coming to the Worldwide Center of Mathematics to present his research on a new pushforward measure, and its applications. Find more information on the poster below.

## Thursday, March 23, 2017

### Advanced Knowledge Problem of the Week: 3-23-17

Check out this week's ADVANCED KNOWLEDGE Problem of the Week! Let us know how you did!

Solution below the break.

## Tuesday, March 21, 2017

### Problem of the Week: 3-21-17

Check out this week's problem, and let us know how you did!
Solution below the break.

## Monday, March 20, 2017

Math madness week one has drawn to a close, and so now where sixteen brilliant minds once stood, only eight remain. A few matches stood out with some nail-biting action, while most others were won in landslide fashion. Namely, the come-from-behind victory pulled off by Ada Lovelace to defeat Hausdorff really rocked the boat. I gotta say, the action in those few close matches may not have been enough to make for the most thrilling week of sports, but I'm looking at this week's bracket ant it is looking to be a thriller.

Our first match features Turing and Cauchy, two mathematicians who blew away their opponents last week. Fans are going to have to make a tough choice here, so I have a feeling it will come down to the bone on this limb of the bracket.

Coming up after that, it's the battle of physicists: Poincare and Maxwell. Will Poincare's work on chaotic systems beat out Maxwell's ordering and unification of electro-magnetic waves? I think Maxwell may be in for an upset defeat, as last week he struggled to edge out a win against Emile Borel, but only time can answer this question, who will move on to the final four?

This next match is a face-off between Lovelace and Ramanujan, and perhaps the most high profile game this week. These two mathematicians are world renowned, making this duel as close to a celebrity match this year's tournament will see.

And finally, Lebesgue and Abel will vie for a spot in the final four. While both competitors pulled off
a comfortable victory last week, neither one flat out blew away the competition, so I feel slightly lukewarm about this matchup. Either way, the victor will have to go up against Ramanujan or Lovelace, and that will be the real test.

Signing off from the Worldwide Center of Mathematics, this has been your introduction to week two of #MATHmadness2k17. Vote Here

## Friday, March 17, 2017

### Mathematical Things to do This St. Patty's Day Weekend

As spring slowly thaws the ice from the ground, we celebrate the life of the foremost patron saint of Ireland, Saint Patrick. The holiday is celebrated all over the world, with shamrocks and green clothing galore! Here is a list of a few (slightly) mathematical things you can do this weekend to celebrate.

## Thursday, March 16, 2017

### Advanced Knowledge Problem of the Week: 3-16-17

Check out this week's ADVANCED KNOWLEDGE Problem of the Week! Let us know how you did!

Solution below the break.

## Monday, March 13, 2017

### Problem of the Week: 3-14-17

Check out this week's Problem of the Week! Let us know how you did in the comments or on social media!

Solution below the break.