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So that’s where our money goes…

14 Apr

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One of the most popular calculus textbooks used on college campuses across the nation is the one written by James Stewart. Just recently, I stumbled upon this website describing James Stewart’s home in Toronto.

From the Wall Street Journal: http://online.wsj.com/article/SB123872378357585295.html#project%3DSLIDESHOW08%26s%3DSB123869600484183257%26articleTabs%3Darticle

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Source for picture: http://www.ontarioarchitecture.com/twentyfirst.htm

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Descartes and the Cartesian Plane

10 Apr

The Cartesian Plane, also known as the Coordinate Plane, is named after the French mathematician René Descartes. He was born in La Haye, France on March 31, 1596. Descartes was taught at home up to the age of eight. At that point, he entered the Jesuit college of La Flèche, where he continued his schooling until graduating at the age of eighteen.

While at La Flèche, Descartes suffered health problems and because of this, his teachers allowed Descartes to stay in bed for most of the morning. Even though he missed almost all of his morning classes, he was still able to keep up with all of his studies. It has been rumored that Descartes’ inspiration for the coordinate plane came about because of all the time he spent in bed. The story goes that one day when Descartes was in bed, he noticed a fly crawling around on the ceiling. He tried to think of ways to describe where the fly was located and realized that he could do so by describing the fly’s position by its distance from each wall. Then he tried to relate the fly’s position to a point, and, well, one thing lead to another, and voilà! He came up with the coordinate plane and Cartesian coordinates!

Cartesian coordinates are used to locate a point in space by giving its relative distance from perpendicular intersecting lines. These perpendicular intersecting lines for the two coordinate axes of the Cartesian (coordinate) plane. Any point, line, or figure can be precisely located by referencing these axes. The horizontal axis is called the x-axis, and the vertical axis is called the y-axis. The coordinate plane is divided into four quadrants (as pictured below).

In the system that Descartes created, a coordinate pair (or an ordered pair) is what describes the location of a point in the coordinate plane. The coordinate pair, in general, is (x, y). The first value is the x-coordinate, which describes where on the x-axis the point is located. The second value is the y-coordinate, which describes where on the y-axis the point is located. By using the Cartesian coordinate system, any point in the plane can be described using a pair of coordinates.

The location of a city, country, or a ship at sea is given by a set of coordinates, and another application of ordered pairs is that computer graphic artists create figures and computer animations by referring to coordinates. What else are ordered pairs used for in the real world? Post responses below! 🙂

Source (MLA Format): “Descartes and His Coordinate System.” Book Rags. Web. 09 Apr. 2012. <http://www.bookrags.com/research/descartes-and-his-coordinate-system-mmat-02/&gt;.

Pythagoras and His Pythagorean Followers

27 Feb

The Pythagorean Theorem is known by many students as

a^2 + b^2 = c^2\!\,

But who came up with this? A Greek mathematician by the name of Pythagoras is credited for this discovery. He was born on the island of Samos, Greece in 569 B.C.. Around 518 B.C., Pythagoras settled down in a Greek colony in southern Italy called Crotona. It was here that he founded a religious and philosophical school where many of his followers, both men and women, lived and worked. These “Pythagoreans” were not allowed to have personal possessions and were vegetarians. The followers were also known as mathematikoi.

Pythagoras believed (and therefore his followers also believed) in the following:

  • All things are numbers. Mathematics is the basis for everything, and geometry is the highest form of mathematical studies. The physical world can understood through mathematics.
  • The soul resides in the brain, and is immortal. It moves from one being to another, sometimes from a human into an animal, through a series of reincarnations called transmigration until it becomes pure. Pythagoras believed that both mathematics and music could purify.
  • Numbers have personalities, characteristics, strengths and weaknesses.
  • The world depends upon the interaction of opposites, such as male and female, lightness and darkness, warm and cold, dry and moist, light and heavy, fast and slow.
  • Certain symbols have a mystical significance.
  • All members of the society should observe strict loyalty and secrecy.
[My favorite is that numbers have personalities, characteristics, strengths and weaknesses. Must be why 7 has a huge appetite and a problem with “numberalism” (get it? Because seven “ate” nine?? )]
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Anyway, because this society of Pythagoras’ was so secretive and the fact that they shared ideas and intellectual discoveries, no one knows for sure whether or not the theorems are actually ones Pythagoras came up with himself. Nonetheless, the Pythagoreans gave Pythagoras all the credit for the following:
  1. The sum of the angles of a triangle is equal to two right angles.
  2. The Theorem of Pythagoras (a.k.a. The Pythagorean Theorem) : For a right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides. The Babylonians are actually the ones who understood this 1,000 years earlier, but Pythagoras proved it and hence, he gets the credit.
  3. Constructing figures of a given area and geometrical algebra. (So, for example, they solved various equations by geometrical means)
  4. The discovery of irrational numbers is attributed to the Pythagoreans, but this seems unlikely to have been the idea of Pythagoras since it does not align with his philosophy the all things are numbers (he believed that number meant the ratio of two whole numbers).
  5. The five regular solids (tetrahedron, cube, octahedron, icosahedron, dodecahedron).
Here is a problem that requires you to use the Pythagorean Theorem to solve:
Scott wants to swim across a river that is 400m wide. He begins swimming perpendicular to the shore he started from, but he ends up 100m down river from where he started because of the current. How far did he actually swim from his starting point?
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Click here to find the answer.
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Source (MLA Format): Douglass, Charlene. “Biography of Pythagoras.” Math Open Reference. 2005. Web. 21 Feb. 2012. <http://www.mathopenref.com/pythagoras.html&gt;.

The “Father” of Algebra

21 Feb

Diophantus of Alexandria, a Greek mathematician, is considered by many sources to be the “father” of algebra. He is famous for two works, both of which are unfinished. One is about polygonal numbers, while the other more influential work was titled Arithmetica, which was the first known book to employ algebra in a modern way. It also inspired the rebirth of number theory.

Not much is known about his life, but we can approximate how old Diophantus lived to be from the following riddle:

Diophantus’s youth lasted one sixth of his life. He grew a beard after one twelfth more. After one seventh more of his life, he married. Five years later, he and his wife had a son. The son lived exactly one half as long as his father, and Diophantus died four years after his son.

How many years did Diophantus live? (Hint: Set up an algebraic equation equal to his age!)

For the answer to the riddle, click here.

Source (MLA Format): “Diophantus of Alexandria.” Encyclopædia Britannica. Encyclopædia Britannica Online. Encyclopædia Britannica Inc., 2012. Web. 20 Feb. 2012. <http://www.britannica.com/EBchecked/topic/164347/Diophantus-of-Alexandria>.