# Maths

Mathematics is the base of human civilization. From cutting vegetables to arranging books on the shelves, from tailoring clothes to motion of Planets - Mathematics applies everywhere.

### 1. Addition of Rational Numbers

(a) Consider the addition of rational numbers with same denominators: p/q and r/q

#### Rational Numbers

When an integer a is divided by another non-zero integer b. The following cases arise:

#### Integers

While dealing with natural numbers and whole numbers, it is not always possible to subtract a number from another. For example, (2 – 3), (3 – 7), and (9 – 20) are all not possible in the system of natural numbers and whole numbers. Thus, it needed another extension of numbers which allow such subtractions.

#### Natural Numbers

The counting numbers 1, 2, 3, ... constitute the system of natural numbers. These are the numbers which are used in day-to-day life.

#### Numbers

Counting things is easy now. You can count objects in large numbers and represent them through numerals. It is not as if we always knew how to convey large quantities in conversation or through symbols. Many thousands years ago, people knew only small numbers.

#### Basic Geometry

The term ‘Geometry’ is the English equivalent of the Greek word ‘Geometron’. ‘Geo’ means Earth and ‘metron’ means Measurement. According to historians, the geometrical ideas shaped up in ancient times, probably due to the need in art, architecture and measurement. These include occasions when the boundaries of cultivated lands had to be marked without giving room for complaints.

#### Prime Factorisation

When a number is expressed as a product of its factors, the number is said to be factorised.

#### Tests for Divisibility of Numbers

You can find a pattern that can tell whether a number is divisible by 2, 3, 4, 5, 6, 8, 9, 10 or 11.

#### Even and Odd Numbers

In the numbers 2, 4, 6, 8, 10, 12, 14, ... each of them is a multiple of 2. These are called even numbers.

#### Prime and Composite Numbers

There are numbers, having exactly two factors 1 and the number itself. Such number are 2, 3, 5, 7, 11, and so on. These numbers are prime numbers.

#### Highest Common Factor (HCF)

You can find the common factors of any two numbers. For example, the common factors of 12 and 16 are 1, 2 and 4. The highest of these common factors is 4.

#### Lowest Common Multiple (LCM)

The Lowest Common Multiple (LCM) of two or more given numbers is the lowest (or smallest or least) of their common multiples. For example, the common multiples of 4 and 6 are 12, 24, 36, ... . The lowest of these is 12. So, the lowest common multiple of 4 and 6 is 12.

#### Factors and Multiples

A factor of a number is an exact divisor of that number. For example, 1, 2, 3 and 6 are exact divisors of 6. They are called the factors of 6.

#### Whole Numbers

Given any natural number, you can add 1 to that number and get the next number i.e. you get its successor. The successor of 16 is 16 + 1 = 17, that of 19 is 19 +1 = 20 and so on. The number 16 comes before 17, we say that the predecessor of 17 is 17 – 1 = 16, the predecessor of 20 is 20 – 1 = 19, and so on. Does 1 have both a successor and a predecessor?

### Arithmetic Mean

A is called the arithmetic mean of the numbers a and b if and only if a, A, b are in AP.

#### Harmonic Progression

A sequence of non-zero numbers is said to be in harmonic progression (HP) if their reciprocals are in AP.

#### Geometric Progression

A geometric progression (GP) is a sequence of numbers in which the first term is non-zero and each term, except the first, is obtained by multiplying the term immediately preceding it by a fixed non-zero number. This fixed number is called the common ratio and it is denoted by r.

#### Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which each term, except the first, is obtained by adding a fixed number to the immediately preceding term. This fixed number is called the common difference, which is generally denoted by d.

#### Binomial Theorem

A Binomial is an algebraic expression of two terms which are connected by the operation: + or −.

#### Principle of Mathematical Induction

Induction is the process of inferring a general statement from the truth of particular cases.

#### Combinations

The word combination means selection. Suppose you are asked to make a selection of any two things from three things: a, b and c.

#### Circular Permutations

A circular permutation is one in which the things are arranged along a circle. It is also called closed permutation.

#### Permutations When Objects Can Repeat

The number of permutations of n different things, taken r at a time, when each may be repeated any number of times in each arrangement, is nr.

#### Permutations of Objects Not All Distinct

The number of mutually distinguishable permutations of n things, taken all at a time, of which p are alike of one kind, q alike of second such that p + q = n, is n!/p! q!.

#### Scalar or Dot Product

The scalar product of two vectors is a scalar quantity. Therefore, the product is called scalar product.

#### Direction Ratios

Any three numbers proportional to direction cosines of a vector are called its direction ratios.

#### Direction Cosines

Let P(x, y, z) be any point in space with reference to a rectangular coordinate system O (XYZ). Let α, β and γ be the angles made by OP with the positive direction of coordinate axes OX, OY, OZ respectively. Then cos α, cos β, cos γ are called the direction cosines.

#### Resolution of Vector

If P is a point in a two dimensional plane which has coordinates (x, y).

### Triangle Law of Addition of Vectors

Triangle Law of Addition of vectors states that, if two vectors are represented in magnitude and direction by the two sides of a triangle taken in the same order, then their sum is represented by the third side taken in the reverse order.

### Zero or Null Vector

A vector whose initial and terminal points are coincident is called a zero or null or a void vector. The zero vector is denoted by .

#### Introduction To Vectors

Physical quantities are divided into two categories: Scalar quantities and Vector quantities.

#### Inverse Trigonometrical Functions

The quantities sin−1x, cos−1x, tan−1x are called inverse circular functions. sin−1x is an angle θ, whose sine is x.

#### Solution of Triangles

A triangle has six parts or elements - the sides a, b, c and the angles A, B, C.

#### Triangle Area Formula

In any triangle ABC, area of the triangle is

#### Sub-multiple Half Angle Formula

In any triangle ABC,

#### Projection Formula

In any triangle ABC,

#### Cosine Formula

In any triangle ABC,

#### Napier’s Formula

In any triangle ABC,

#### Sine Formula

In any triangle ABC,

a/sinA = b/sin B = c/sinC = 2R

### sin θ = sin α

θ = nπ + (−1)nα, where n ∈ Z

#### Equation of the Form: a cosθ + b sinθ = c

a cosθ + b sinθ = c where c2 ≤ a2 + b2

Divide each term by √(a2 + b2)

### sin θ = 0

The general solution of sin θ = 0 is given by

θ = nπ, n ∈ Z where Z is the set of all integers.

#### Trigonometrical Equations

An equation involving trigonometrical function is called a trigonometrical equation. To solve these equations, you are required to find all replacements for the variable θ that make the equations true.

#### Trigonometric Conditional Identities

If A + B + C = π, then

#### Trigonometric Multiple Angle Identities

Identities involving sin 2A, cos 2A, tan 3A, etc. are called multiple angle identities. To derive these identities you can use sum identities.

#### Trigonometric Compound Angles: A + B and A − B

The relation f(x+ y) = f(x) + f(y) is not true for all functions of a real variable. All the six trigonometric ratios do not satisfy this relation.