MATHEMATICS ENCYCLOPEDIA

Squaring a 2-digit or more digits number ending in 5 Multiply the number before 5 with its successor and place 25 after the product.  Example square of 25 can be obtained by multiplying 2 with its successor 3 and placing 25 after the product, i.e. 625.  Square of 85 can be obtained by multiplying 8 with its successor 9 and placing 25 after the product, i.e. 7225.

Logarithm can be defined in many ways like: Cambridge dictionary definition: The number that shows how many times a number called the  base, has to be multiplied by itself to produce another number. Adding or taking away logarithm can replace multiplying or dividing large numbers. Oxford dictionary definition: A quantity representing the power to which a fixed number ( the base ) must be raised to produce a given number. Given a positive real number x, there is one and only one real number y such that e^y= x, we call this number y as logx. In general, the logarithm to the base a of the number x is the number y such that y=logx and you can write a^y=x. Logarithm is inverse operation of exponentiation. Logarithm to the base 10 is called common logarithm and the for the base e, it is called natural logarithm.

A set is well defined collection of objects. sets can be represented in two ways: Roaster or tabular form , e.g. set of vowels is represented as { a, e, i, o, u }  Set-builder form , e.g. V = { x : x is a vowel in English alphabet }, V denote the given set. It is read as " the set of all x such that x is a vowel in English alphabet ". The theory of sets was developed by German mathematician Georg Cantor ( 1845-1918 ). He first encountered sets while working on trigonometric series. A set does not change if one or more elements of the set are repeated , e.g. A = { 1, 2, 3 } and B = { 2, 2, 1, 3, 3 } are equal. Venn diagrams: Sets can be represented by means of diagrams known as Venn diagrams. These consist of rectangles and closed curves usually circles. Venn diagrams are named after English logician John Venn ( 1834-1883 ).

A series of numbers in which each number is the sum of the two preceding numbers. The first two numbers in the series are either 1 and 1 or 0 and 1. The series is written as 1,1,2,3,5,8,13,21,34,55,89, ........... The series was given by Italian mathematician Leonardo Fibonacci . When we make a squares with that numbers of the series we get a spiral. Golden ratio: When we take any successive number of the series, the ratio of the two numbers are very close to golden ratio φ. The value of ratio is approximately 1.618034....... November 23 is celebrated as Fibonacci day because the date format in mm/dd is 11/23 and it forms Fibonacci series.

Coordinate geometry is the study of geometry using a coordinate system. Let XOX' and YOY' be two mutually perpendicular lines intersecting at point O. The line XOX' is called the x-axis and the line YOY' is called y-axis, and the two lines together are called coordinate axes. The point O is the origin.  Quadrants: The two lines divide the plane in four parts XOY, X'OY, X'OY' and Y'OX called as first, second, third and fourth quadrants respectively. For a given point P, the point is expressed as P( x,y ). The x-coordinate is called abscissa and the y-coordinate is called ordinate. In I    quadrant: x>0, y>0 In II  quadrant: x<0, y>0 In III quadrant: x<0, y<0 In IV quadrant: x>0, y<0 The coordinate of origin is ( 0,0 ) Any point on x-axis is of the form ( x,0 ) Any point on y-axis is of the form ( 0,y ) The coordinate system was invented by French Mathematician Rene Descartes in 1637. He provided the link of the u...

Theodorus of Cyrene, who lived around 425 B.C., was a philosopher of ancient Greece. It is said that he discovered the construction below, which is therefore called " the wheel of Theodorus ". Also called as spiral of Theodorus, square root spiral, Einstein spiral or Pythagorean spiral. The spiral is started with an isosceles right triangle with base and height of unit ( equal to 1 ) length and the hypotenuse obtained is √2. The hypotenuse of second triangle is √3 and the third triangle hypotenuse is √4 and so on. Each triangle is formed using the hypotenuse of previous triangle. The 16th triangle has hypotenuse eqal to √17. Theodorus stopped at √17 because it the hypotenuse of last triangle that does not overlap the figure.

Euclid's division lemma: Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 <= r < b. Euclid's division algorithm is a technique to compute the highest common factor ( HCF ) of two given positive integers.  A lemma is a proven statement used for proving another statement.