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

Solution below.

Check out this Problem of the Week.

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Solution below.

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

Solution below.

Check out this logic based Think Thursday Problem!

Solution below.

Check out this Problem of the Week.

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

Solution below.

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

Solution below.

Welcome to our first Think Thursday Problem!

This series aims to introduce logic based problems, puzzles, and other tricky brain teasers. The problems featured here are Math related, but do not require a extensive knowledge of Mathematics to solve. We hope you enjoy this new series!

Solution below.

Check out this Problem of the Week and enjoy this math joke.

*Because the constant was incapable of change.*

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Solution below.

Why did the variable break up with the constant?

Solution below.

Check out this Problem of the Week.

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Solution below.

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

Solution below.

The history of ‘e’ is a tangled one, one which would warrant an entire dedicated book to parse through mathematics to the original conception of the transcendental number. Even before e’s enigmatic beauty was fully unearthed, people using mathematics to solve real world problems encountered the number many times, and understood it enough to work it into their solutions. A good example of this is when e shows up in compound interest. Bankers found out that as the number of times one took annual compound interest grew to infinity, the rate of growth approached e! Watch the video to see two mathematical proofs of our statement, using two definitions of e.

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

Solution below.

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

Solution below.

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

Solution below.

Oh no, back to school is right around the corner!

We know it can be hard to jump straight back into Math classes during the first few days after a summer away. Lucky for you we have arranged some of of our Youtube channel videos into a helpful guide to make sure you are on your game in the first week of class. Check them out below!

We know it can be hard to jump straight back into Math classes during the first few days after a summer away. Lucky for you we have arranged some of of our Youtube channel videos into a helpful guide to make sure you are on your game in the first week of class. Check them out below!

Check out this Algebraic Problem of the Week.

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

Solution below.

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

Solution below.

Back in 300 BC, Euclid proved that there were an infinite number of primes. He used line segments to show that some line lengths could only be made up from single-unit line lengths and not lines with lengths of 2, 3, etc. These line lengths represented prime numbers. This proof has the same principle but is a little different than Euclid's and uses proof by contradiction. Take a look at this simple proof which shows that primes are infinite!

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Solution below.

Check out this Algebraic Problem of the Week.

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Solution below.

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

Solution below.

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

Solution below.

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

Solution below.

The mythology behind this fairly simple proof is what makes it one of the most popular proofs in math classes across the world. The story follows a young Carl Friedrich Gauss, whose first grade teacher asked the class to add up the numbers 1 to 100 in order to pass a good amount of time. Before the teacher had time to start grading papers, Gauss handed in his assignment. Watch the video to find out Gauss’ observation that is now one of the most famous math proofs out there.

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Solution below.

Check out this Mathematics inspired Alphametic Problem of the Week

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The two basic rules for solving alphametics are as follows:

Each letter must be represented by a different digit. If the letter is used more than once, it must be represented by the same digit.

Once you substitute digits for all your letters, you must end up with an accurate addition problem.

Solution below.

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

The two basic rules for solving alphametics are as follows:

Each letter must be represented by a different digit. If the letter is used more than once, it must be represented by the same digit.

Once you substitute digits for all your letters, you must end up with an accurate addition problem.

Solution below.

This problem of determining the area of a circle, or better defined as the area inside of a circle, was a huge dilemma in the field of mathematics. It was not until the mid 200's BC when Archimedes began to anticipate modern calculus and analysis though concepts of infinitesimals and exhaustion, which he used to solve this major challenge of finding the area of a circle.

Archimedes' method of finding the area is described as "squaring the circle", which is trying to find the square that has the same enclosed area as a circle of a given radius. Using this and also using a method where he approximated the area of a circle with other, known shapes such as squares and hexagons, Archimedes was able to determine the area inside of a circle. Take a look at the proof to see how Archimedes came up with the formula we know today:

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Solution below.

Check out this Problem of the week about Geometry and triangles within a circle. If you're interested in learning more about how you draw circles and what it says about your cultural background, read this article:

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Solution below.

## How do you draw a circle? We analyzed 100,000 drawings to show how culture shapes our instincts

Solution below.

The advent of graph theory, from the mind of Leonhard Euler, came from a long-standing problem for the people of Königsberg. The problem was that no couple had a long and happy marriage, if they were married in Königsberg. As tradition dictated, a newlywed couple had one chance to travel across Königsberg’s four land masses using each of the seven bridges once and only once. If the two lovers could complete this seemingly simple task, their marriage would be long and happy. Years went by and nobody could complete to task, until Euler constructed a mathematical object that broke the curse of Königsberg… a graph!

Watch the proof proposed by Euler below to learn how mathematical abstraction created a whole new field of math, which is now regarded as an important predecessor to topology. Euler’s invention itself is remarkable, but the implications to mathematical philosophy reveals something very deep in the heart of mathematics. Namely, the art of abstraction to gain a better understanding of certain truths inherent in life’s situations.

The Königsberg Bridge Problem, and its solving:

This week’s Top Pop Math Chop comes from Georg Cantor, who first solved this piece of set theory in 1891. He presented this as a mathematical proof which showed it was impossible to link infinite sets with an infinite set of the natural numbers. This is known today as Cantor’s diagonal argument, which he proved using binary numbers.

Cantor showed that if he has a list of binary numbers, takes one digit from each going diagonally, produces a new number, and swaps every single digit with a corresponding 1 or 0 (if is a 1 it becomes 0 and vice versa), that the number will be different than every other binary number listed before it. This is because in the first number the first digit is different, so it’s definitely different than the new number; in the second number the second digit is different than the second digit in the new number and so on.

You can do this same thing with real numbers, and produce infinite decimals between 0 and 1. This shows the real numbers are uncountable.

Check out the video below explaining Georg Cantor’s proof:

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This week's problem comes from our textbook, Worldwide Multivariable Calculus, so feel free to check it out or any other affordable texts we offer. Enjoy this problem and try to find its relation to July 4th!

Solution below.

This week's problem comes from our textbook, Worldwide Multivariable Calculus, so feel free to check it out or any other affordable texts we offer. Enjoy this problem and try to find its relation to July 4th!

Solution below.

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

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.

Solution below the break.

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

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

Solution below.

Solution below.

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.

Solution below the break.

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:

Buy the digital or print version here.

Watch our review of the book:

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!

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.

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!

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.

Solution below the break.

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.

Read more after the break.

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.

Solution below.

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.

Solution below the break.

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

Solution after the break.

Solution after the break.

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:

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.

Solution after the break.

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

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.

Works Cited:

http://www.history.com/topics/holidays/memorial-day-history

http://www.storyofmathematics.com/19th.html

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