The four color theorem is a mathematical theorem that states that, given a map, no more than four colors are required to color the regions of the map, so that no 2 regions that are touching (share a common boundary) have the same color. This theorem was proven by Kenneth Appel and Wolfgang Haken in 1976, and is unique because it was the first major theorem to be proven using a computer. This proof was first proposed in 1852 by Francis Guthrie when he was coloring the counties of England and realized he did not need more than four colors to color the map. Either he or his brother published this theorem (you only need four colors to color a map) in The Athenaeum in 1854. Many people had tried to solve this and had failed, two notables who had tried were, Alfred Kempe (1879) and Peter Guthrie Tait (1880). Many mathematicians kept failing until around the 1960s – 1970s when German mathematician Heinrich Heesch developed a way to use computers to solve proofs. And by 1976 Kenneth Appel and Wolfgang Haken, at the University of Illinois had stated that they had proven the theorem. They had used two technical concepts to prove that there was no map that had the smallest possible regions that required five colors. The two concepts were: 1. An unavoidable set contains regions such that every map must have at least one region from this collection. 2. A reducible configuration is an arrangement of countries that cannot occur in a minimal counterexample. If a map contains a reducible configuration, then the map can be reduced to a smaller map. This smaller map has the condition that if it can be colored with four colors, then the original map can also. This implies that if the original map cannot be colored with four colors the smaller map can't either and so the original map is not minimal. What they had done was use mathematical rules and procedures to prove that a minimal counterexample to the four color conjecture could not exist. They had to check around 1900...

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Pythagorean Theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle (right-angled triangle). In terms of areas, it states:
In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).
The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:[1]
where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.
These two formulations show two fundamental aspects of this theorem: it is both a statement about areas and about lengths. Tobias Dantzig refers to these as areal and metric interpretations.[2][3] Some proofs of the theorem are based on one interpretation, some upon the other. Thus, Pythagoras' theorem stands with one foot in geometry and the other in algebra, a connection made clear originally byDescartes in his work La Géométrie, and extending today into other branches of mathematics.[4]
The Pythagorean theorem has been modified to apply outside its original domain. A number of these generalizations are described...

...Pythagorean Theorem was one of the earliest theorems known to ancient civilizations. This famous theorem is named for the Greek mathematician and philosopher, Pythagoras. Pythagoras founded the Pythagorean School of Mathematics in Cortona, a Greek seaport in Southern Italy. He is credited with many contributions to mathematics although some of them may have actually been the work of his students.
The Pythagorean Theorem is Pythagoras' most famous mathematical contribution. According to legend, Pythagoras was so happy when he discovered the theorem that he offered a sacrifice of oxen. The later discovery that the square root of 2 is irrational and therefore, cannot be expressed as a ratio of two integers, greatly troubled Pythagoras and his followers. They were devout in their belief that any two lengths were integral multiples of some unit length. Many attempts were made to suppress the knowledge that the square root of 2 is irrational. It is even said that the man who divulged the secret was drowned at sea.
The Pythagorean Theorem is a statement about triangles containing a right angle. The Pythagorean Theorem states that:
"The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides."
Figure 1
According to the Pythagorean Theorem, the sum of the areas of the two red...

...Thales’ Theorem
Thales’ Theorem simply states that if three points exist within a circle, and one of those points is the diameter of the circle, then the resulting triangle will always be a right triangle. This simple idea can become very useful for certain applications such as, identifying the center of a circle with its converse. On the triangle the vertex of the right angle always terminates at the ends of the diameter line. By locating the two points of the diameter line and drawing a line connecting them, the center can be found directly between them. This is the case if and only if it is a right triangle. Thales’ theorem is a special case of the inscribed angle theorem. It was attributed to Thales of Miletus and proved in the third book of Euclid’s Elements, 33rd proposition. According to history the Indian and Babylonian mathematicians knew this for special cases prior to Thales. It is even said that he learned of inscribed triangles during his journey to Babylon. It is mostly attributed to him due to his initial proving, utilizing his results of Isosceles base angles being equal and the total sum of angles within a triangle equally 180 degrees. Thales’ is the subject of two stories regarding astronomy and geometry. In the first, he predicted an eclipse in the year 585 BCE. In the next, Thales took observations from two land points and used his knowledge of geometry to determine the distance of a ship....

...only colour in real life is white? White light is indeed the bearer of all colours. The colours we see are reflections bouncing off an object or are the light source itself. The light sources change and the colours with them. Colour is not a stable affair. Colour is light alone, but our experience is so direct that we trust our eye and believe that a colour is inherent to an object. Imagine living in a world where everything is white in colour!
However, the world we live in is, thankfully, full of colours, and they trigger emotional responses as they influence our feelings and us. Interestingly, colours can sway thinking, change actions, and cause reactions. It can irritate or soothe your eyes, raise your blood pressure or suppress your appetite.
Colours are the basic things in life, yet it plays an important role in our life. However, most of us take it for granted. For example, each time we wake up to a brand new day, we would take it for granted that we can see the colours of our bedrooms. At night when we go to sleep, we do not count our blessings for the beautiful colours that we have been looking at all day.
We are fortunate that we are able to see the many wondrous colours around us. Some cannot see them. They are colourblind. All that they can see is black and...

...Colours for living and learning
The Universe is a magnetic field of positive and negative charges, constantly vibrating and producing electro-magnetic waves. Each of these has a different wavelength and speed of vibration; together they form the electro-magnetic sphere. White light when seen through a prism or water vapour splits into the colours of the spectrum. Of all the electro-magnetic sphere it alone can be seen. Radio waves, infra-red waves, ultra-violet waves, x-rays and gamma waves can not be seen.
Colour through age development
There is a cycle of changing colours that affects our mind and body through the different stages of life. These are reflected in our changing colour preferences. Children have colour likes and dislikes according to individual character and stage of development.
Over a hundred years ago the pioneering educationalist Rudolph Steiner believed that people were surrounded by particular colours that had a spiritual influence and objective effect on their emotional life as well as benefiting physical health and mental well-being. Followers of the Steiner philosophy introduced his principles into their networks of schools and communities. It was believed that surrounding children with soft pastels and rounded architectural forms at the kindergarten level (2-7 years), progressing to central shared learning areas with stronger more vibrant...

...In mathematics, the Pythagorean theorem — or Pythagoras' theorem — is a relation in Euclidean geometry among the three sides of a right triangle (right-angled triangle). In terms of areas, it states:
In any right-angled triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).
The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:[1]
where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.
The Pythagorean theorem is named after the Greek mathematician Pythagoras (ca. 570 BC—ca. 495 BC), who by tradition is credited with its discovery and proof,[2][3] although it is often argued that knowledge of the theorem predates him. There is evidence that Babylonian mathematicians understood the formula, although there is little surviving evidence that they used it in a mathematical framework.[4][5]
The theorem has numerous proofs, possibly the most of any mathematical theorem. These are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The theorem can be generalized in various ways, including higher-dimensional spaces, to...

...Fermat's Last Theorem
Fermat's Last Theorem states that no three positive integers, for example (x,y,z), can satisfy the equation x^n+y^n=z^n if the integer value of n is greater than 2. Fermat's Last Theorem is an example a Diophantine equation(Weisstein). A Diophantine equation is a polynomial equation in which the solution must be an integers. These equations came from the works of Diophantus who was a mathematician who worked methods on solving these equations. Fermat's Last Theorem was based on Diophantus's work. A more common Diophantine equation would be Pythagorean Theorem, where the solution would be the the Pythagorean triples(Weisstein). However, unlike Pythagorean Theorem, Fermat's Last Theorem has no practical real world applications.
Fermat had scribbled on the margin of Arithmetica, the book that inspired his theorem, that he had a proof that would not fit on the margin of a book. From the 1600's-mid 1900's this proof remained unsolved. It was eventually solved by Andrew Wiles. Andrew Wiles as a child always loved math, he would always make up problems and challenge himself....

...Pythagorean Theorem: Proof and Applications
Kamel Al-Khaled & Ameen Alawneh
Department of Mathematics and Statistics, Jordan University of Science and Technology
IRBID 22110, JORDAN
E-mail: kamel@just.edu.jo,
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Idea
Investigate the history of Pythagoras and the Pythagorean Theorem. Also, have the opportunity to
practice applying the Pythagorean Theorem to several problems. Students should analyze information on
the Pythagorean Theorem including not only the meaning and application of the theorem, but also the
proofs.
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1
Motivation
You’re locked out of your house and the only open window is on the second ﬂoor, 25 feet above the ground.
You need to borrow a ladder from one of your neighbors. There’s a bush along the edge of the house, so
you’ll have to place the ladder 10 feet from the house. What length of ladder do you need to reach the
window?
Figure 1: Ladder to reach the window
1
The Tasks:
1. Find out facts about Pythagoras.
2. Demonstrate a proof of the Pythagorean Theorem
3. Use the Pythagorean Theorem to solve problems
4. Create your own real world problem and challenge the class
2
2.1
Presentation:
General
Brief history: Pythagoras lived in the 500’s BC, and was one of the ﬁrst Greek mathematical thinkers.
Pythagoreans were interested in...