AKPotW: Showing that the group generated by an element of G is a subgroup of G [Algebra]

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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!

Next Episode

Problem of the Week: 6-6-17

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Problem of the Week: 5-30-17

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

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Problem of the Week: 5-9-17

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Problem of the Week: 5-2-1

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

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Problem of the Week: 4-25-17

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Problem of the Week: 3-28-17

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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!

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Problem of the Week: 3-14-17

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Advanced Knowledge Problem of the Week: 3-2-17

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Advanced Knowledge Problem of the Week: 2-24-17

Check out this week's AKPotW, and try to prove if a sequence converges! Let us know how you did in the comments!

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Advanced Knowledge Problem of the Week: 2-9-17

Check out our AKPotW on rings! let us know how you did in the comments below or on social media!

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Problem of the Week 2-14-17

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African Americans In Mathematics: Benjamin Banneker

                 Benjamin Banneker (1731-1806) of Baltimore County, Maryland was born a free man, but with plenty of familiarity to the brutality of slavery that was present at the time. Benjamin’s father, Robert, was a freed slave, and his mother, Mary, had parents who were both freed slaves. Mary’s mother, Benjamin’s grandmother, taught Benjamin to read at a young age and even pushed for Benjamin to be enrolled in a Quaker school. Benjamin’s school career did not last long, but his curiosity about mathematics was carried with him his whole life, a curiosity that would cause a great flow of scientific accomplishments.

Benjamin Banneker

          When Benjamin entered his twenties his passion for the sciences (ranging from mechanical engineering to astronomy) was bubbling. At this time, he had built a full sized grandfather clock modeled after a pocket watch, and was studying the cycles of eclipses. Benjamin continued to use his mathematical mind to create great things until his 40’s; by then, he built irrigation systems for his family farm, grain mills, and began to research bees and locusts. In 1772, the Ellicotts moved to a farm very close to the Banneker’s. The Ellicotts were Quakers; a faith that held that all races were equal and should be treated as such, and quickly noticed the brilliance of Benjamin Banneker.

            The Banneker family loaned many books to Benjamin, and encouraged him to begin calculating the exact times of eclipses to take place in the future. They also exchanged scientific research on surveying and much more. In 1791, Major Andrew Ellicott was asked to survey the land of Western New York by then Secretary of State Thomas Jefferson. Andrew suggested Benjamin as a more capable candidate for the position, and so began Benjamin’s rich correspondence with Thomas Jefferson.

            Benjamin became fairly close to Thomas, and wrote frequently about national issues and personal happenings. Benjamin quietly suggested that Thomas should do what he could to promote racial equality from his position in government. Some of these letters, along with scientific research, plans for cities, and personal commentaries were published in Benjamin’s Almanacs. The series of six annual almanacs were printed in the consecutive years leading up to the end of his life, and was the pinnacle of his scientific career.

The cover of Benjamin's 1795 Almanac

           

Sources:

https://en.wikipedia.org/wiki/Benjamin_Banneker#Mythology_and_legacy

http://www.biography.com/people/benjamin-banneker-9198038#early-years

Images

http://johnhopebryant.com/2012/02/black-facts-in-history-benjamin-banneker.html

http://www.pbs.org/wgbh/aia/part2/2h68b.html

Using Math to Create Something Beautiful

                   Think back to when you were first introduced to functions, thin lines depicting a single value output for each input in a domain.

A Function

For many people, the idea of what a function ‘looks’ like does not change much from this bland depiction of data. However, data can be crafted into something that carries much more information than just inputs and outputs, and in the right hands an enormous and messy set of data can be presented in a powerful way. Indeed, data representation is an important part of any scientific field.

            Compacting more data into inputs and outputs provides not only more information, but also a more stunning visualization of data. In a vector field, a single point can contain information about location, strength and direction of a force.  The more information a function tracks, the more stunning the display becomes, with 4 or even 5 dimensions represented on a graph of three-dimensional space and color.

Vectors depicting the strength and direction
of a magnetic field at discrete points.

A 3D graph, with a 4th color dimension

 Vector fields can even represent information that cannot be easily compiled into a simple function, which allows for out-of-the-ordinary occurrences in nature to be studied more carefully.

With developments in technologies that offer efficient data manipulation, the possibilities of what we can do with functions and data are more far-reaching than ever. Anne M. Burns of Long Island University Uses computers to create beautiful representations of functions.

Burns plots complex valued functions as a vector field, seen here.

The advantages of this technique transcend aesthetic purpose, and can be used to find roots of functions at a glance.

Attributing more dimensions to an occurrence is useful and can be beautiful, but what if the object or function in question is impossible to make sense of as it is? It is often handy to project or unfold an N-dimensional surface onto an (N-1)-dimensional surface. Most of the time, in calculus, a three-dimensional surface will be looked at as a two-dimensional projection on the xy, yz, or xz plane in order to set up an integral to find the volume of the object. In theoretical physics, this technique of reducing the dimension of mysterious happenings is used to speculate the nature of the universe. A common example, and perhaps the most accessible way to think of this process is the unfolding of a four-dimensional cube, the tesseract.

The nets of a 3D cube and a 4D hypercube above.

Dali's Corpus Hypercubus (1954)

This way of thinking about higher dimensions caught more than just the eyes of mathematicians and scientists. Salvador Dali, the great surrealist painter, was fascinated by the advances in science during the twentieth-century. In the 1950’s Dali was fascinated by nuclear physics and quantum mechanics, and found inspiration for many paintings in mathematics.

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At its core, mathematics does not only seek knowledge, but also pursues beauty in the natural world.

Works Cited

• The Function graphic was found on the page of Maret School's BC calculus page, and is spliced with Charlie Brown of Peanuts, created by Charles M. Schulz.
• The Magnetic field graphic was found on Vassar College's Wordpress blog, under a lecture by Prof. Magnes.
• 4D graph curtesy of user Blue7 on math.stackexchange. 
• Find all of Anne M. Burn's Work here.
• The Cube net image was found here. 

Any unwanted images in this article will be removed at the request of the owner.

Never Fear: ACT and SAT Help!

Standardized testing. 

Those are THE two words faced by countless high school students every year. The SATs, in particular, constantly hover above juniors and seniors once the school year starts. College applications are important, and as competition amongst students increases, it is becoming increasingly difficult for students to gain acceptance into their top choices. While extracurricular activities and GPA are important factors in adding substance to your college application, SAT scores are also a very significant part of the mix. No high school student wakes up on a Saturday morning yearning to take a 6 hour exam, but for those applying for college, the test is basically inevitable. 

One may expect that students who are successful in high school math classes would also perform well on the math portion of the SAT. Surprisingly however, this isn’t always the case. Students who excel in high level math courses are often discouraged and frustrated when their SAT scores fail to align with their school performance. Why does this disparity occur so rampantly? The focus of math classes is largely centered on method based thinking and displaying the correct work that led to the solution. This logical way of thinking is great for grasping complex concepts, but it does not match up with the format of the SAT. The multiple choice questions of the SAT do not allow for partial credit; only the correct answer gives the student points. Furthermore, SAT math questions are based on the fundamentals of mathematics. Even students in AP or high level math classes begin to shy away from the basics as they advance in courses or turn to their calculator. It is important for all students to review basic concepts and practice them throughout their SAT prep. 

Here at the Center of Math, we have a fair amount of combined experience in standardized testing. With these tips, we hope to increase your chances of a success as much as possible! 

Conway's Game of Life

This isn't about the board game with the spinner but rather something we consider to be much cooler! Devised in 1970, John Conway's Game of Life is an example of a cellular automaton, a discrete dynamical mathematical system. Despite its simple definition, it gives rise to patterns and objects that have very complex, even computer-like behavior. Keep reading to learn more about how the Game of Life works and what's been discovered about it! You can also dive right in with this free online resource here!

A large pattern in the Game of Life, known as a 'breeder,' the first discovered to exhibit quadratic growth. Colors for emphasis. Source.