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.

Pythagoras theorem: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides, i.e. if a, b and c are the base, height and hypotenuse respectively, of a triangle then a^2 + b^2 = c^2. Pythagoras was ancient Greek philosopher who has credit of this theorem. Pythagorean triplet: If three natural numbers a, b and c are related as a^2 + b^2 = c^2, then a, b and c are called pythagorean triplet. 

Linear equation in one variable:  An equation of the form ax+b=0 or, ax=b, where a and b are real numbers such that a is not equal to zero, is called linear equation in one variable, e.g. 2x+3=0, 7x=14 etc. Linear equation in two variable: An equation of the form ax+by+c=0 or ax+by=c, where a, b, c are real numbers, a and b are not equal to zero and x,y are variables, is called Linear equation in two variable, e.g. 3x+5y+10=0, 6x+2y=5 etc. The graph of linear equation is a straight line. A linear equation has infinitely many solutions. The equation of x-axis and y-axis are y=0 and x=0 respectively.

The word percent, symbolically written as % mean in every hundred or per hundred, e.g. 25% means 25 per hundred. Percent, fractions and decimals are linked to each other, e.g. 25% can be written as 25/100 or as 0.25.

For any real number a and any positive integer n, we define a^n as                    a^n = a*a*a*............*a ( n factors ) a^n is called the nth power of a. The real number a is called the base and n is called the exponent. Example: 2^3 = 2*2*2, 5^2 = 5*5, 10^4 = 10*10*10*10 etc.

A combination of constants and variables connected by the signs +, -, /, * is called an algebraic expression. The different parts seperated by the above signs are called terms of the expression. Examples of algebraic expressions are 5x - 7y, 3x + 2, 7x + 4y + 9xy - 5, 2x^2 + 6x + 4 etc.

A symbol in algebra having a fixed value is called a constant, whereas a symbol which can be assigned different values is called a variable. 1/2, √3,  -5, π etc. are all constant whereas x, y, z, p, r etc. are all variables.

Cubes The cubes of a number is the number raised to the power of 3, e.g. the cube of 4 is 4^3= 4*4*4= 64. Perfect cube: A natural number n is a perfect cube if there exist a natural number m such that m*m*m = n, i.e. m^3 = n. Cubes of all odd natural numbers are odd, e.g. 5^3 = 5*5*5 = 125. Cubes of all even natural numbers are even, e.g. 2^3 = 2*2*2 = 8. Cubes of all negative numbers are negative. Cube roots The cube root of a number a is that number which when multiplied by itself three times gives a. The cube root of 8 is 2 because 8 = 2*2*2. The cube root of 27 is 3 because 27 = 3*3*3.

Oxford dictionary definition: The highest number that can be divided exactly into each of two numbers. Cambridge dictionary definition:   The highest number that a set of two numbers can be divided by exactly. 4 is the highest common factor of 8 and 12 because 4 can divide 8 as well as 12 and it the largest number that can divide both 8 and 12. 2 and 1 can also divide both 8 and 12 but 4 is bigger than 2 and 1, hence 4 is the hcf of 8 and 12. There is no other number which could divide both 8 and 12.

Definition of LCM: We can define least common multiple of two integers x and y is the smallest positive integer which is divisible by both numbers x and y. It is  denoted as         LCM ( x,y ). It is also known as lowest common multiple or smallest common multiple.

Squares If a number is multiplied by itself, the product so obtained is called the square of that number, e.g. 2 square is 2*2= 4, 3 square is 3*3= 9, 12 square is 12*12=144 etc. The square of a number is a number raised to the power of 2. The square of an even number is always even, e.g. 16*16=256. The square of an odd number is always odd, e.g. 19*19=361. Perfect squares: A natural number is called perfect square or square number if it the square of some natural number, e.g. 25 is perfect square because 25 is the square of 5. Perfect squares never ends in 2,3,7 or 8. The number of zeroes at the end of a perfect square is always even, e.g. 400*400=160000, 50*50=2500 etc. For any natural number n, n^2=n*n= sum of first n odd natural numbers, e.g. 5^2=5*5= 1+3+5+7+9=25. Square roots The square root of a number n is that number which when multiplied by itself gives n as the product, e.g. the square root of 25 is 5. Symbol of square root is √  . Hence √16=4, √25=5, √36

Ratio Ratio compares quantity of the same kind in the same unit. Ratio between two quantities a and b, is written as a : b and expressed as a fraction a/b. You can read the ratio between a and b as 'a is to b'. a and b are called the terms of the ratio. a is called antecedent and b is called the consequent. Ratio has no unit. Ratios are compared in the same ways as the fractions are compared. Proportion Two equal ratio forms a proportion, e.g. 1/3 = 4/12 and can be written as 1 :3 : : 4 : 12. In general, if four numbers a, b, c, d are in proportion, i.e., a : b : : c : d then a/b = c/d. a and d are called extremes and b and c are called the means.  Product of extremes = product of means i.e. ad = bc. Three quantities a, b and c are in proportion, if a : b : : b : c. Mean propotional between a and c is square root of ac.

Number line is also called as real number line. A line on which numbers are marked at intervals, used to illustrate simple numerical operations. A drawing thaty represents all the numbers that exist, including those greater than and less than zero and irrational number such as π. Every point on the number line represents a real number.

A fraction is simply a number representing part of a whole. A fraction is a numerical quantity that is not a whole number e.g. 1/2. A fraction is a number results from dividing one whole number by another, e.g. 1/4, 3/7, 8/3 etc.  3/8 is a fraction. It is read as three-eighth. Here 8 represent the number of equal parts into which the whole has been divided. 3 here is the number of equal parts which have been taken out. Here 3 is called the numerator and 8 is called the denominator. Fractions with same denominators are called like fractions, e.g. 1/3, 2/3, 5/3 etc. Fractions having different denominators are called unlike fractions, e.g. 4/7, 2/3, 8/17 etc. Fractions having numerator bigger than the denominator are called improper fractions, e.g. 3/2, 4/3, 8/3 etc. Fractions having numerator smaller than denominator are called proper fractions, e.g. 1/2, 2/3, 5/9 etc.

A decimal numbers is a number having two parts seperated by decimal point.The part to the left of decimal point is called the whole number part and to the right of the decimal point is called the decimal part. Examples of decimal numbers are 3.7, 1.2, 14.6 etc. The part to the right of the decimal point is less than 1. Place value: Places to the left of decimal point are ones, tens, hundreds, thousands etc. and places to the right of decimal point are tenths, hundredths, thousandths etc. Adding extra zeroes to the right of decimal point does not change its value, e.g. 3.9= 3.90= 3.900 and so on.

A factor of a number is an exact divisor of that number, e.g. factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24 itself because these numbers can divide 24. 1 is factor of every number. A multiple of a number is obtained by multiplying the number by natural numbers, e.g. the multiples of 4 are 4, 8, 12, 16, 20, 24,.... etc.