MATHEMATICS ENCYCLOPEDIA

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) = y...

Axiom Oxford dictionary definition: A statement or preposition which is regarded as being established, accepted, or self-evidently true. Cambridge dictionary definition: A statement or principle that is generally accepted to be true, but need not be so. Example: Peano axiom - For every natural number  n ,  S ( n ) is a natural number. Postulate Oxford dictionary definition: A thing suggested or assumed as true as the basis for reasoning, discussion, or belief. Cambridge dictionary definition : To suggest a theory, idea etc. as a basic principle from which a further idea is formed or developed. Example: Euclid's postulate- A straight line segment can be drawn joining any two points. Theorem Cambridge dictionary definition: A formal statement that can be shown to be true by logic. Oxford dictionary definition: A general preposition not self-evident but proved by a chain of reasoning; truth established by means of accepted truths. Examples: Pytha...

Perfect cuboid is an Euler brick having  integer side lengths integer face diagonal integer space diagonal The problem of finding such a cuboid is also called the brick problem, diagonals problem, perfect box problem, perfect cuboid problem, or rational cuboid problem. No perfect cuboid is known. Equivalent to solution of Diophantine equations: a 2 + b 2 = d 2 a 2 + c 2 = e 2 b 2 + c 2 = f 2 A lso an extra following equation has to be added: a 2 + b 2 + c 2 = g 2 ( space diagonal equation ) where a, b, c are the edges and d, e, f are the diagonals. The space diagonal is g.

Oxford dictionary definition: A four sided figure. Cambridge dictionary definition: A flat shape with four straight sides. A quadrilateral is a four sided figure bounded by four line segments and four angles in a plane. 'Quad' means four and 'lateral' means sides.   Consecutive or adjacent sides: Two sides of quadrilateral are consecutive, if they have a common point ( vertex ). Example: AB & BC, CD & DA etc. Opposite sides: Two sides of a quadrilateral are opposite, if they have no common end-point ( vertex ). Example: AB & CD, BC & AD etc. Consecutive angle: Two angles are consecutive, if they have a common arm. Example: ∠A & ∠B, ∠C & ∠D etc. Opposite angles: Two angles are opposite, if they don't have a common arm. Example: ∠A & ∠C, ∠B & ∠D etc. Angle sum property of quadrilateral The sum of all the four angles of a quadrilateral is 360 degree. ∠A + ∠B + ∠C + ∠D = 360°. Different types of quadr...

Definition of circle: A circle is a collection of those points in a plane that are at a given constant distance from a given fixed point in the plane. The fixed point is called the centre and the given constant distance is called the radius of the circle. A circle with centre O and radius r is usually denoted by C ( O, r ).  Thus in set notation we write C ( O, r ) = { X : OX = r } Position of point with respect to a circle: A point P is said to lie inside, outside or on the circle C ( O, r ) according as OP < r, OP > r, or OP = r respectively. Concentric circles: Circles having same centre with different radii are called concentric circles. Arc of a circle: A continuous piece of a circle is called an arc. Central angle: Let C ( O, r ) be any circle, then angle whose vertex is O is called the central angle. Chord: A line segment joining any two points on a circle is called a chord of the circle. Diameter: A chord passing through the centre of a circle i...

Once Professor G.H. Hardy visited Indian mathematician Srinivas Ramanujan in hospital in a taxi number 1729. Mr. Hardy described the conversation between them as: “I remember once going to see him when he was ill at Putney. I had ridden a taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. No, he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”” It can be written in following two forms: 1729 = 12 3 + 1 3 1729 = 10 3 + 9 3 There is also a film based on the above mathematicians, The film name is  "The man who kne w infinity" . See the poster belo w.