MATHEMATICS ENCYCLOPEDIA

Topology ( Branch of mathematics ) Topology, branch of mathematics , sometime reffered to as "rubber sheet geometry" in which two ibjects are considered equivalent if they can be continuously deformed into one another through such motions in space as bending, twisting, stretching and shrinking while disallowing tearing apart or gluing together parts. The main topics of interest in topology are the properties that remain unchanged by such continuous deformations. Topology while similar to geometry differs from geometry in that geometrically equivalent objects often share numerically measured quantities, such as lengths, angles, while topologically equivalent objects resemble each other in a more qualitative sense. The area of topology dealing with abstract objects is referred to as general, or point set topology. General topology overlaps with another important area of topology called algebraic topology. These areas of specialization form the two major subdisciplines...

Klein bottle was first described by german mathematician Felix Klein in 1882. Klein bottle is obtained by identifying two ends of a cylindrical surface in the direction opposite that is necessary to obtain a torus. The surface is not constructible in three-dimensional Euclidean space  but has interesting properties, such as being one-sided like the mobius strip, being closed yet having no "inside" like a torus or a sphere and resulting in mobius strips if properly cut in two.

Mobius strip ( Amazing Geometry )  Mobius strip is a one sided surface that can be constructed by affixing the ends of a rectangular strip after having gived one of the ends a one half twist. The space exhibits interesting properties, such as having only one side and remaining in one piece when split down the middle.  The properties of the strip were discovered independently and almost simultaneously by two German mathematicians,  August Ferdinand Mobius  and  Johann Benedict Listing , in 1858. If you cut a mobius strip through the centre line, you will have one long strip with two full twist , rather than two seperate strips.

Konigdberg bridge problem ( Real life mathematical problem ) Konigsberg is a town on the Pregel River , which in the 18th century was a German town, but now is Russia.  The problem led to the development of the branches of mathematics known as Topology and Graph Theory . The problem of bridge is as follows: Within the town are two river islands that are connected to the banks with seven bridges as shown. People tried to walk around the town in a way that only crossed each bridge once,but it proved to be difficult problem.   Leonhard Euler, a Swiss mathematician in the service of the Russian empire Catherine the Great, heard about the problem. In 1736 Euler proved that the walk was not possible to do. He proved this by inventing a kind of diagram called a network, that is made up of vertices  ( dots where lines meet ) and arcs ( lines ) . He used four dots ( vertices ) for the two river banks and the two islands.These have been marked A, B and C, D. Th...

The Bermuda triangle ( Mysterious Geometry ) The Bermuda triangle is a part of the Atlantic Ocean bounded by Miami, Bermuda island and Puerto Rico where many ships and aircraft are said to have disappeared. The term "Bermuda triangle" was first coined by writer Vincent H. Gaddis in 1964. Also known as Devil's triangle . It is loosely defined region in the western part of the North Atlantic ocean where a number of aircraft and ships are said to have disappeared under mysterious circumstances. The triangle covers an area of about 500,000 square miles. The earliest disappearance in the Bermuda triangle is appeared in September 17, 1950, a article published in The Miami by Edward Van Winkle Jones. After two years Fate magazine published "Sea mystery at our Back door", by George X.

How much is a million? A million is one thousand thousand.  You can also read as ten hundred thousand.  Derived from the Italian word millione. In scientific notation, it is 10 6 .

Cartesian products of sets If you have two sets A and B. The cartesian product                                                      A Ⅹ B = { (a, b ) : a ∈ A, b ∈ B } Example:   Let A = { a, b } and B = { 5, 7, 9 }, then A Ⅹ B = { (a, 5 ), (a, 7 ), ( a, 9 ), ( b, 5 ), ( b, 7 ), ( b, 9 ) }. B Ⅹ A = { ( 5, a ), ( 5, b ), ( 7, a ), ( 7, b ), ( 9, a ), ( 9, b ) }. If either A or B is a null set, then A Ⅹ B will also be empty set, i.e. A Ⅹ B = Φ. Two ordered pairs are equal, if and only if the corresponding first elements are equal and the second elements are also equal. Let two ordered pairs ( a, b ) and ( c, d ) be equal, i.e.,               ( a, b )  = ( c, d ), if and only if a = c and b = d. If there are x elements in A and y element in B, then there are xy elements in A Ⅹ B, i.e., if n(A) = x and n(B)...