Functions:

Functions:

Introduction to Functions in Math:

The mathematical concept of a function expresses the intuitive idea that one quantity (the argument of the function, also known as the input) completely determines another quantity (the value, or the output). A function assigns a unique value to each input of a specified type.(source : Wikipedia)

In this article we are going to see about functions in math,types of functions in math and some sample solved problems on functions in math.

Types of Functions in Math:

There are several types of functions in math.the following are the some of the different types of functions,

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Composite functions
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Inverse functions
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Trigonometric functions
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Quadratic functions
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Linear functions
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Cubic functions
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Algebraic functions
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Periodic functions
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Even and odd functions

Next we are going to see some problems on functions in math.

Solved Problems in Functions in Math:

Problem 1:

Function f(x) =5x2 -3 and g(x) = 6x. Find f(g(x)).

Solution:

Given, f(x) =5x2 -3 and g(x) = 6x .

To find the f(g(x)) substitute x = g(x) in f(x),

f(x) =5x2 -3

Substitute x = g(x) = 6x

f(g(x)) = 5(6x)2 -3

= 5 ( 36x2 ) -3

= 180x2 - 3

Answer: f(g(x)) = 180x2 - 3

Problem 2:

Find the following function is odd or even f(x) = 10x2 + 5

Solution:

Given f(x) = 10x2 + 5

We need to find the function is odd or even.

If f(-x) = f(x) , the given function is even

If f(-x) = - f(x) , the given function is odd

So to find the function is odd or even ,Substitute x = -x in the given function,

f(-x) = 10(-x2 ) + 5

= 10x2 + 5

= f(x)

f(-x) = f(x)

So the given function is even function.

Answer: Given function f(x) =10x2 + 5 is even function.




Problem 3:

Find the inverse function of f(x) = 15 x - 8

Solution:

Given, f(x) = 15 x - 8

Let us take f(x) = y

That is y = 15 x - 8

To find the inverse function we need to solve for x,

y = 15 x - 8

Add 8 on both sides,

y + 8 = 15x - 8+ 8

y + 8 = 15x

15x = y + 8

Divide by 15 on both sides,

x = ( y +8) / 15

Now replace y = x and x =f--1 ( x)

f--1( x) = (x+8)/15

Answer: Inverse function of a given function is f--1( x) = (x+8)/15

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Sets

When we usually hear about the word Set,the first thing that would come to our mind would be a collection of something.In Mathematics, the basic meaning of set is a "well-defined collection of definite objects is called a set."

George Cantor is regarded as the "Father of Set theory".

The concept of "Sets" is basic in all branches of mathematics.

Set: Definition: well-defined collection of distinct objects is called a set.Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics.

Notation of Sets: Capital letters are usually used to denote or represent a set.

Representation of Sets: There are two methods of representing a set. (i) Roster Method (ii) Set builder form.

Finite and Infinite Sets: A set is finite if it contains a specific number of elements. Otherwise, a set is an infinite set.

Null Set or Empty Set or Void Set: A set with no elements is an empty set.

Singleton Set or Singlets: A set consisting of a single element is called a singleton set or singlet. The cardinality of the singleton set is 1.

Equivalent Sets: Two finite sets A and B are said to be equivalent sets if cardinality of both sets are equal i.e. n (A) = n (B).

Equal Sets: Two sets A and B are said to be equal if and only if they contain the same elements i.e. if every element of A is in B and every element of B is in A. We denote the equality by A = B.

Cardinality of a Set A: The number of elements in a finite set A, is the cardinality of A and is denoted by n(A).

Universal Set: In any application of the theory of sets, the members of all sets under consideration usually belong to some fixed large set called the universal set.

Subsets: If A and B are sets such that each element of A is an element of B, then we say that A is a subset of B and write A Í B.

Power Set: The family of all subsets of any set S is called the power set of S. We denote the power set of S by P (S).

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Algebraic Expression:

Introduction:In algebra an expression may be used to designate a value, which value might depend on values assigned to variables occurring in the expression; the determination of this value depends on the semantics attached to the symbols of the expression. These semantic rules may declare that certain expressions do not designate any value; such expressions are said to have an undefined value, but they are well-formed expressions nonetheless. In general the meaning of expressions is not limited to designating values; for instance an expression might designate a condition

Algebraic expression comes under basic algebra category in which algebra is one of the main branches in mathematics which deals with finding the unknown variable value with the references of known values. In algebraic expression the variables are represented with the help of alphabetic letters and the integers used in the algebraic expressions are considered as constants. The algebraic expressions can be easily solved by using arithmetic operations. The following are some of the example problems in algebraic expressions.

Algebraic Expressions Example Problems:

Example 1:

Derive the algebraic expression.

-2(m - 1) - 4m - 1 = 3(m + 2) - 2m

Solution:

Given
-2(m - 1) - 4m - 1 = 3(m + 2) - 2m

Multiply factors
-2m + 2 - 4m - 1 = 3m + 6 - 2m

Group like terms
-6m + 1 = m + 6

Subtract 1 to both sides
-6m + 1 - 1 = m + 6 -1

Group like terms
-6m = m + 5

Subtract m to both sides
-6m - m = m + 5 -m

Group like terms
-7m = 5

Multiply both sides by -1/7
m = - 5/7

m = - 5/7 is the solution to the given equation

Example 2:

Derive the algebraic expression.

-4(m + 2) = m + 9

Solution:

Given
-4(m + 2) = m + 9

Multiply factors in left term
-4m - 8 = m + 9

Add 8 to both sides
-4m - 8 + 8 = m + 9 + 8

Grouping the above terms
-4m = m + 17

Subtract m to both sides
-4m - m = m + 17 -m

Group like terms
-5m = 17

Multiply both sides by -1/5
m = -17/5

m = -17/5 is the solution to the given equation.

Hope you like the above example of
Algebraic Expression.Please leave your comments, if you have any doubts.

Simplification Example Problems

Simplification Example Problems:

Meaning of Simplification:


Simplification calculator is the online software in the Internet through which students in online give the question for simplification they get the answer to their questions. These simplification calculator is very much helpful for students to clear their doubts in their home work problems. Sample problems from algebra involves step by step simplification. The simplification calculator for algebra would be much useful for school students. let us see some sample problems from algebra for simplification calculator.

Example 1:


Find the sum of 2x4 – 3x2 + 5x + 3 and x + x3 – 6x2 – 1.

Solution:

Using the associative and distributive property, we obtain

(2x4 – 3x2 + 5x + 3) + (x3 – 6x2 + x – 1) = 2x4 + x3 – 3x2 – 6x2 + 5x + x + 3 – 1

= 2x4 + x3 – (3+6)x2 + (5+1)x + 2

= 2x4 + x3 – 9x2 + 6x + 2. (answer)

Example 2:

Factorize x2 – 8xy – x + 2y.

Solution:

x2 – 8xy – x + 2y = (x2 – 8xy) – (x – 2y)

= x(x – 8y) + (–1) (x – 2y)

= (x – 8y) [x + (–1)]

= (x – 8y) (x – 1). (answer).






Example 3:

Problem to find the variables x,y and z:

x + y + 2z = 8 ------> (1)

3x - y + 3z = 4 ------>(2)

2x + y + 4z = 6 ------>(3)

Solution:

Solve (1) and (2),

x + y + 2z = 8 ----> (1)

3x - y + 3z = 4 -----> (2)

add the above two equations.

we get 4x + 5z = 12 ------> (4)

solve (1) and (3)

(3) * 2 ----> 4x + 4y +8z = 16

(4) -----> 4x + 0y +5z = 12
(-) (-) (-)

3z = 4

[ z = 4/3]

substitute [z = 4/3] in (4) eqn

4x + 5 [(4/3)] = 12

4x = 12 - [20/3]

4x = [(36-20)/3] = [16 / 3]

x = [16/12] = [ 4/3]

substitute x = [4/3] , z = [4/3] in (1) eqn.

[4/3] + y + 2 [(4/3)] = 8

y + [12/3 ] = 8

y + 4= 8

y = 8 - 4 .

x = [4/3] , y = 4, z = [4/3] .

Example 4:

Find the sum of x3y + 2x2y2 – 3xy3 and 3x3 – 3x3y + y3 + 4xy3.

Solution:

(x3y + 2x2y2 – 3xy3) + (3x3 – 3x3y + y3 + 4xy3) = x3y + 2x2y2 – 3xy3 + 3x3 – 3x3y + y3 + 4xy3

= (x3y – 3x3y) + (2x2y2) + (–3xy3 + 4xy3) + (3x3) + (y3)

= –2x3y + 2x2y2 + xy3 + x3 + 3x3 + y3

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Linear transformations:

Linear transformations:

Introduction:

Linear transformation is the most important topic in the algebra chapters in mathematics. Linear transformation in mathematics is a rule for changing one figure shape (or matrix or vector) into another by using a formula with a specified format. This format must be a linear combination, in which the original components are modified through the formula of ax + by to form the coordinates of the transformed figure. Stretching it or compressing it, and rotating it. All the transformation has an inverse.

Algebra Linear Transformation: In this linear transformation section we are going to take look at a special kind of function that the study of Linear Algebra and has many applications in fields of mathematics, physics and as well as the engineering field. This basic definitions and facts are combined with this kind of function. We are looking for more number of examples.We can understand about the Linear Transformation with the help of few examples. Here we have using two examples for Algebra Linear transformation,

Example1: Zero transformation:

Given: the zero transformation is the transformation T: Sn -> Sm that maps every vector in this format as y in Sn to the zero vectors in Sm, which is T(y) = 0.

Solution:

This is the zero matrix problem so let us take m*n is zero matrix,

And then the matrix induced by this linear transformation is the m*n is zero matrix, 0 matrix is since,

T(y) = T0 (y) = 0y = 0

There fore, to make it clear form of this matrix is, using the zero transformation for matrix, and we are usually denoted by this in T0 (y).

Example 2: Identity transformation:

Given: the identity transformation is the transformation T: Sn -> Sn ( both are Sn because of identity transformations) that maps every vector in this form as y itself to the identity. Which is T(y) = y.

Solution:

Then the matrix induced by this transformation is the n*n matrix (this is identity matrix), so that I n since,

T(y) = Ti (y) = In y = y

There fore, to make it formation of identity transformation is Ti (y) to make clear for this, and we are usually denoted by this in Ti (y).

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Number Sense

Number Sense:

Introduction of Number Sense:Let me give an explanation about number sense in general,its very clear actually,as we can see the name of the concept itself is self explanatory,"Number Sense" number sense can refer to "an intuitive understanding of numbers, their magnitude, relationships, and how they are affected by operations.Many other definitions exist, but are similar to the one given. Some definitions emphasize an ability to work outside of the traditionally taught algorithms, e.g., "a well organised conceptual framework of number information that enables a person to understand numbers and number relationships and to solve mathematical problems that are not bound by traditional algorithms.

Let us now study about the Definition of Number Sense:

Number sense is defined as the proper understanding and usage of the numbers in the various places

It includes various things

1. Ability to find the relative values of a number.

2. How to use a number in different arithmetic operations like (addition, subtraction, multiplication and division).

3. Finding the problem solving strategies.

Number sense covers various topics

They are

Estimating and rounding of the numbers

Rounding and addition

Rounding and product

Rounding and division


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Law of Indices

Law of Indices:

Introduction:

Let us learn the meaning of Law of Indices in general,Indices is the concept of Algebra which is one of the major branch of mathematics.The Index of a number is the number of times multiplied by itself. If X and Y are positive integers and also X [!=] 0 then

X Y = X * X * X * X * X ........ Y times it is multiplied by same number

X is called the base and Y is the power. We read it as " X raised to the power Y " . The power is also called the " Index" or "Exponent".

A few examples of Powers : (i) 25 = 2 x 2 x 2 x 2 x 2 = 32 where index (or power or exponent) = 5 and base = 2 .

Its read as base 2 raised to the index 5 equals to 32

(ii) 23 = 2 x 2 x 2 = 8 where index = 3 and base = 2.

Its read as base 2 raised to the index 3 equals to 8

(iii) 33 = 3 x 3 x 3 = 27 where index = 3 and base = 3

Its read as base 3 raised to the power 5 equals to 27

(iv) 34 = 3x3x3x3 = 81 where index = 4 and base = 3

Its read as base 3 raised to the index 4 equals to 81

(v) 42 = 4x4 = 16 where index = 2 and base = 4

Its read as base 4 raised to the index 2 equals to 16

Indices or powers or Exponents are used to write statements involving repeated multiplication in shorthand.

Now let us learn about the specifics of the Law of Indices:For any expression like numbers , variables or functions having same base but different indices (or power or exponent) and also for the expressions consists of same index (or power or exponent) but different bases can be solved using Indices concept involving algebraic operations by following the below set of Laws of Indices.

Rule (1) Product Law for Indices : am x an = a m+n

consider an example, (a) 53 x 52 = 5 3+2 = 55 = 5 x 5 x 5 x 5 x 5 = 3125

(b) e 3t e -7t = e 3t x e -7t = e [3t + (-7t) ] = e [3t - 7t] = e -4t

(c) -4 4 -4 -1 = -4 [4 + (-1) ] = -4 [ 4 - 1] = -4 3

(d) (5/2) -3 (5/2) -9 = (5/2) -3 + (-9) = (5/2) -3 - 9 = (5/2) -12

(e) (q-2) (q 1/3) = q -2 + (1/3) = q (-6+1) / 3 = q -5/3

(f) (3) -1 (3) 5 = 3 -1+5 = 3 4

Rule (2) Quotient Law for Indices : am [-:] an =(a m / a n ) = a m-n

For instance, (a) 68 [-:] 65 = 6 8-5 = 6 3 = 6 x 6 x 6 = 216

(b) (4/3) -3 [-:] (4/3) 9 = (4/3) [-3 - 9] = (4/3) -12 = 1 / (4/3) 12 = (3/4) 12

(c) 5 -2 [-:] 5 -2 = 5 [-2 -(-2)] = 5 [-2 + 2] = 5 0 = 1

(d) g 6 [-:] g -7 = g [6 - (-7)] = g [6 + 7] = g 13

(e) Y (-7/2) [-:] Y (-1) = Y [ (-7/2) - (-1)] = Y [ (-7/2 + 1)] = Y [(-7+2)/2] = Y -5/2

Rule (3) Power Law for Indices : (a m) n = a mn

Example (a) (7 4) 2 = 7 4x2 = 7 8

(b) (2 -5 )4 = 2 (-5x4) = 2 -20

(c) (X -5) -3 = X (-5 x -3) = X -15

(d) [t (1/2)] 3 = t [(1/2)x3] = t (3/2)

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Congruent Triangles

Congruent Triangles:




Congruent Triangles:

The concept of Congruent Triangles is explained in detail here. Every geometric figure has a shape and a size. Two circles of different radii have the same shape but their sizes are different. But if we draw two circles of the same radius both the shape and size will be the same. Such figures with the same shape and size are called congruent figures. We can check whether two figures are congruent or not by the method of superposition. We say that two triangles ABC and DEF are congruent if we can make one triangle fit exactly on the other.

In the above figure, the two triangles are congruent and we write . The sides which are equal in length are called the corresponding sides and the angles which are equal in measure are called the corresponding angles.

Conditions for Congruent Triangles:

When two triangles have (i) the same shape, and (ii) the same size, then they are said to be congruent triangles.

Symbol for congruency is
Draw with AB = 4 cm, AC = 5 cm, and .

Also with DE = 4 cm, DF = 5 cm and

Draw these triangles on a tracing paper. Now place fig(ii) on fig.(i), so that D is placed on A, DE along AB and F on the same side as C.
It is noticed that coincides with exactly fits on ABC.


We Can put the entire explanation of Congruent Triangles in a Nut shell.


SAS Axiom:

If two triangles have two sides of one equal to two sides of the other, each to each, and angles included by these sides are equal, then the triangles are congruent.
AAS Axiom:

If two triangles have two angles of one equal to two angles of the other, each to each and also one side of one equal to the corresponding side of the other, then the triangles are congruent.
SSS Axiom:

If two triangles have three sides of one equal to three sides of the other each to each, then the triangles are congruent.
RHS Axiom:

If two right triangles have their hypotenuses equal and one side of one equal to one side of the other, then the triangles are congruent.

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Rhombus

Rhombus:

Introduction:

Rhombus is quadrilaterals with both pairs of opposite sides are parallel and all the sides are having same length. Rhombus is also called as equilateral parallelogram.
The rhombus is also called as diamond and rhomb. A square is also a rhombus, because the square is a quadrilateral with all sides equal.
All the opposite sides of the rhombus is parallel and all the opposite angles are equal.
One of the most important thing to keep in mind is the properties of a Rhombus.

• Opposite angles of the rhombus having equal measures.
• The diagonals of the rhombus intersect each other at right angles.
• Rhombus having two diagonals that are connecting opposite pairs of vertices.
• Rhombus is Symmetric along the diagonals
• Every Rhombus is parallelogram
• All the parallelograms are not a Rhombus.While studying about the Rhombus we cannot ignore to study about the area and the perimeter of the Rhombus.

The area of the Rhombus is same as the area of the parallelogram .That is the area is the multiplication of base and height.
The formula for calculate the area is,
A=b * h
The base of the rhombus is the length of one of its sides and the height is the perpendicular distance between the opposite sides. The perimeter of the rhombus is the total sum of its all the sides length. Generally the perimeter is sum of the length of the all sides. In the case of rhombus, all the sides are having equal length. So we can say that the perimeter is 4s, where s is length of the sides.
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Quadrilaterals:

Quadrilaterals:

A Quadrilaterals is a polygon with four sides (or 'edges') and four Vertices or Corners,its basically a figure formed by joining four points in an order is called a quadrilateral.Look around you and you will find so many objects which are of the shape of a
quadrilateral - the floor, walls, ceiling, windows of your classroom, the blackboard.....its very simple to identify a quadrilateral.The word Quadrilateral is made of the words quad (meaning "four") and lateral (meaning "of sides").
We can learn about Quadrilaterals by studying the objects around us as mentioned in the above example.The other way to understand quadrilaterals is by understanding the properties of a quadrilateral.

Let us now learn about the properties of a quadrilateral:

The sum of the angles of a quadrilateral is 360º.The other thing we must keep in mind while learning quadrilateral is that quadrilateral are of different shapes and sizes.
Rhombus:

A Rhombus or rhomb is a quadrilateral whose four sided all have the same length.The rhombus is often called a Diamond

Rectangle:

A rectangle is any quadrilateral with four right angles

Parallelogram:

A parralellogram is a quadrilateral with two pairs of parallel sides. In Euclidean Geometry, the opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure.

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Quadrilaterals:

Quadrilaterals:

A quadrilateral is a polygon having 4 sides. ABCD is a quadrilateral and AC and BD are its diagonals.The word quadrilateral is made of the words quad (meaning "four") and lateral (meaning "of sides").You may wonder why should we study about quadrilaterals (or parallelograms)

Look around you and you will find so many objects which are of the shape of a
quadrilateral - the floor, walls, ceiling, windows of your classroom, the blackboard,
each face of the duster, each page of your book, the top of your study table etc.


One of the easiest way to learn about Quadrilaterals is to learn about the Properties of a Quadrilaterals.


Two sides of a quadrilateral, which have no common point, are called opposite sides.
In the diagram, AB and DC is one pair of opposite sides.
AD and BC is the other pair of opposite sides.

Consecutive Sides of a Quadrilateral

Two sides of a quadrilateral, which have a common end point, are called consecutive sides. In the diagram,
AB and BC is one pair of consecutive sides.
BC, CD; CD, DA; and DA, AB are the other three pairs of consecutive sides.

Opposite Angles of a Quadrilateral

Two angles, which do not include a side in their intersection, are called the opposite angles of a quadrilateral.

Consecutive Angles of a Quadrilateral

Two angles of a quadrilateral, which include a side in their intersection, are called consecutive angles.

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Heron's Formula:

Let us learn about Hero's Formula and understand this concept better:

In geometry, Heron's (or Hero's) formula, named after Hero of Alexandria, states that the area A of a triangle whose sides have lengths a, b, and c is

where s is the semiperimeter of the triangle:

Heron's formula can also be written as:

Let us now look at one example of Heron's Formula:

Heron formula reduce the form on left to (ch)², or (cb)² − (cd)² from 4A ² = 4s(s − a)(s − b)(s − c). On the right side using b ² − d ² = h ² by the Pythagorean theorem.We get the form like,

(s(s − a) + (s − b)(s − c))² − (s(s − a) − (s − b)(s − c))²

Using this form (p + q) ² − (p − q) ² = 4pq.

cb = s(s-a) + (s-b)(s-c) and cd = s(s-a) - (s-b) (s-c).

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Lines and Angles

Lines and Angles:

Very early in school we were taught that a minimum of two points are required to draw a Line,there are different types of lines.Let us now look at the different types of lines,all lines have different properties,

Properties of line:

Line: A line can illustrate between two points only.

Parallel lines: These are straight lines in the similar plane and do not meet together. They may extend in any direction

Intersection: The intersection of two lines meet single point called as intersection point.

Similarly now let us learn about angles:

An angle is the amount of rotation point of intersection of two lines in order to make one line into correspondence with other. An angle is denoted by theta. Angles are usually measured in degree, radiations or gradations. The sign conversion the anticlockwise rotation is considered to be positive and clockwise rotation is considered to be negative.The other way to understand about angles is by studying the properties of angles.The size of an angle is calculated in degrees. When we say the angle ABC we denote the definite angle objects.. If we desire to talk regarding the size or compute of the angle in degrees, we must say 'the compute of the angle ABC- often written m∠ABC. However, many times we will see ∠ABC=34°. It should say m∠ABC=34°.

• Vertical angles: The vertical angles are opposite angles formed by two intersecting lines and congruent.
• Complementary angles: Angles are two angles whose measure, when added together, equal 90°.
• Supplementary angles: Angles are two angles whose measures, when added mutually, equal 180°.

• Adjacent angles: Angles share a general side and a general vertex and do not overlap. Two non-adjacent angles formed by transversal crossing parallel lines are alternate interior angles if they are between the parallel lines and on opposite sides of the transversal.
• Alternate exterior angles: Angles are a pair of angles located in outside a set of parallel lines and on opposite sides of the transversal.
• Corresponding angles: Angles are two angles in corresponding positions formed by a transversal crossing two lines.

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Areas of Parallelograms and Triangles

Areas of Parallelograms and Triangles:

The simplest way to understand what a Parallelogram is by using a figure.As we can see to the left is a perfect example of a parallelogram,In geometry, a parallelogram is a quadrilateral with two pairs of parallel sides. In Euclidean Geometry, the opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure.Let us now learn about some of the basic properties of a parallelogram:

• 
  
Opposite sides of a parallelogram are equal in length.Opposite angles of a parallelogram are equal in measure.The area, A, of a parallelogram is A = bh, where b is the base of the parallelogram and h is its height.Opposite sides of a parallelogram will never intersect.The area of a parallelogram is twice the area of a triangle created by one of its diagonals.The area of a parallelogram is also equal to the magnitude of the vector cross product of two adjacent sides.The diagonals of a parallelogram bisect each other.
• Let us now learn about Triangles,it is easy to understand about triangles and once we learn about them we can relate to objects of the similar shape around us:A plane figure bounded by three sides,or a polygon with three sides.sum of the interior angles of a triangle is equal to 180 degree exterior angles of a triangle are always equal to 360 degree,we can understand about a Triangle by studying about its Figure.

Areas of Parallelograms and Triangles:

The area of a parrellelogram can be calculated by using the formula given below:

Area = ½ bh

We are most commonly faced by questions as,how to find the area of a parrellellogram??? We can answer these questions by the following explanation:

• The area of a parallelogram can be determined by multiplying the bottom times the altitude.
• If a parallelogram has a base of length 5 inches and a height of 3 inches, its area is 5*3=15 square inches

The area of a parallelogram is given by the formula
Area = Base × Height
a = bh
Base: Any side can be measured as a base. If used to determine the area the equivalent height have to be used.
Altitude (height): The altitude (or height) of a parallelogram is the perpendicular distance from the bottom to the opposite side.
Example Problem:
Find the area of Parallelogram whose height is 5 cm and Base 20 cm.

Solution:
Area of Parallelogram = Height × base
= 5 × 20
Area of parallelogram = 100 cm²
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Real Numbers

Real Numbers

Introduction:

The union of the set of rational numbers and irrational numbers forms the set of real numbers.Let us now look at the properties of real numbers.The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real numbers may be thought of as points on an infinitely long number line.

I used to face the problem of finding or identifying the real numbers,but the easiest way to learn about real number is learning to identify them,usually a lot of problems are faced while studying real numbers.The other way to learn about real numbers is to solve more and more problems related to real numbers,basically we can do a lot of exercise related to real numbers.

Every nonnegative real number has a square root in R, and no negative number does. This shows that the order on R is determined by its algebraic structure. Also, every polynomial of odd degree admits at least one real root: these two properties make R the premier example of a Proving this is the first half of one proof of the fundamental theorem of algebra.

Hope you like the above example of Real Numbers.
Please leave your comments, if you have any doubts.

Constructions

Constructions:

The first thing that would come to our mind when we speak about construction vaguely would be the construction of buildings However, sometimes one needs an
accurate figure, for example - to draw a map of a building to be constructed, to design
tools, and various parts of a machine, to draw road maps etc.But in maths Constructions would purely refer to the constructions or drawing of figures,to draw such figures
some basic geometrical instruments are needed. You must be having a geometry box
which contains the following:

(i) A graduated scale, on one side of which centimetres and millimetres are
marked off and on the other side inches and their parts are marked off.

(ii) A pair of set - squares, one with angles 90°, 60° and 30° and other with angles
90°, 45° and 45°.

(iii) A pair of dividers (or a divider) with adjustments.

(iv) A pair of compasses (or a compass) with provision of fitting a pencil at one
end.

(v) A protractor.

Normally, all these instruments are needed in drawing a geometrical figure, such
as a triangle, a circle, a quadrilateral, a polygon, etc. with given measurements. But a
geometrical construction is the process of drawing a geometrical figure using only two
instruments – an ungraduated ruler, also called a straight edge and a compass. In
construction where measurements are also required, you may use a graduated scale
and protractor also. In this chapter, some basic constructions will be considered.These
will then be used to construct certain kinds of triangles.

Matrices

Matrices

Definition of Matrices:

A rectangular array of entries is called a Matrix. The entries may be real, complex or functions.
The entries are also called as the elements of the matrix.
The rectangular array of entries are enclosed in an ordinary bracket or in square bracket. Matrices are denoted by capital letters.

Example:

(i)


Matrices are one of the most powerful tools in mathematics.The
evolution of concept of matrices is the result of an attempt to obtain compact and
simple methods of solving system of linear equations. Matrices are not only used as a
representation of the coefficients in system of linear equations, but utility of matrices
far exceeds that use.

Now let us learn about the different types of Matrices.It is easy to understand Matrices if we learn its types.We might come across some very common questions in various math text book that ask about the types of matrices.

In this section, we shall discuss different types of matrices.

1) Column matrix:
A matrix is said to be a column matrix if it has only one column.

2)Row matrix
A matrix is said to be a row matrix if it has only one row.

3)Square matrix
A matrix in which the number of rows are equal to the number of columns, is
said to be a square matrix. Thus an m × n matrix is said to be a square matrix if
m = n and is known as a square matrix of order ‘n’.

4)Diagonal matrix
A square matrix B = [bij] m × m is said to be a diagonal matrix if all its non
diagonal elements are zero, that is a matrix B = [bij] m × m is said to be a diagonal
matrix if bij = 0, when i ≠ j.

5)Scalar matrix
A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal,
that is, a square matrix B = [bij] n × n is said to be a scalar matrix if
bij = 0, when i ≠ j
bij = k, when i = j, for some constant k.

6)Identity matrix
A square matrix in which elements in the diagonal are all 1 and rest are all zero
is called an identity matrix.

7)Zero matrix
A matrix is said to be zero matrix or null matrix if all its elements are zero.

Hope you like the above example of Matrices.
Please leave your comments, if you have any doubts.

Inverse Trigonometric Functions

Inverse Trigonometric Functions:

The meaning of word "Trignometry" comes from two greek words trigonon and metron where trigonon means triangle and metron means measure.Trignometry deals with the relation between the angles and sides in a triangle.It is a branch of Mathematics mostly useful for the measurements of areas, heights and distances.

The inverse trigonometric functions or cyclometric functions are the so-called inverse functions of the trigonometric functions, though they do not meet the official definition for inverse functions as their ranges are subsets of the domains of the original functions.

The most important thing that we need to learn about the Trigonometric Functions is its Properties.

Properties of Inverse Trigonometric Functions:

The inverse functions of trigonometry are other than specified as cyclometric functions the both functions are also specified as inverse functions of the trigonometric. The properties of trigonometric should encompass the functions and angles. The majority trigonometric functions are sin, cos, tan. Let us see about the properties in this Blog.
1. Sine functions- The trignometric sine function is written as sin,and function is, f(x) = a*sin(bx+c)+d
2. Cosine funtcions-The trignometric cosine function is written as cos,and function is
f(x) = a*cos(bx + c) + d
3. Tangent functions - The trignometric tangent function is written as tan, and function is f(x) = a × tan(bx+c) + d
We can understand the meaning of Trigonometric Functions better with the help of the figure given above:
Example of Trigonometric Functions:
Signs of 6 Trignometric Functions:

* The entire coordinate plane is divided into four quadrants and are named in counter clock-wise direction.

* First quadrant ranges from 0 º to 90 º and the second quadrant ranges from 90 º to 180 º.

* Third quadrant ranges form 180 º to 270 º and the fourth quadrant ranges from 270 º to 360 º.

* Let P(x , y) be a point in the coordinate plane.

Coordinate Geometry

Coordinate Geometry:
Introduction:

Rene' Descartes' (1596-1665), a French philosopher and mathematician, introduced a method by which the position of a point can be corresponded with an ordered pair of real numbers. These pair of real numbers are called the Coordinates. This method is the new idea of combining two branches of mathematics, Algebra and Geometry. The combination of these two branches of mathematics was called Algebraic Geometry, Coordinate Geometry or Analytical Geometry.

Coordinates:

The coordinates of a point are quantities, which determine the position of the point. If for instance, a point P lies somewhere on a straight line XX', then its position may be defined by a single number.

Rectangular Coordinate System:

The position of a point in a plane is determined by two coordinates.

Rectangular Coordinate Method

Introduction: Rectangular coordinate is also called Cartesian coordinate plane or the xyplane. The system of representation of points in the plane by ordered pairs of numbers is called the Cartesian or rectangular or x y-..

Cartesion System Summary

Summary - In coordinate geometry, the tools of algebra are used in studying geometry by establishing 1-1 correspondence between the points in a plane and the ordered pairs of real numbers. If P(x 1 ,y 1 ) and Q(x 2 ,y 2 ) be any two points in the plane, then If A(x 1 ,y 1 ), B(x 2 ,y 2 ) ..

Arithmetic Progression

Arithmetic Progression

Solving online progression is very interesting since we can find the nth term of the particular sequence in much easier way. In this article we shall learn about steps involved in progressions solving. Moreover we will see in detail about different types involved in progression.

There are three types of Progression in math,

• Arithmetic progression
• Geometric progression
• Harmonic progression

Definition:

It is a sequence of numbers in which each term except the first term can be calculated by adding constant number (common difference) to the immediately preceding number.

The General form of the arithmetic sequence is,
a, a+d, a+2d, a+3d………..
Here a is the first number and d is the common difference.
To find the nth term of an arithmetic progression we can use the following formula,
a_n=a+ (n-1) d


Definition:
It is a sequence of numbers in which each term of the sequence except the first term can be calculated by multiplying the preceding term by means of a constant factor (common ratio).
The General form of the Geometric progress is,
a, ar,ar²,ar³,………
n^th term of the geometric progression is,
a_n=ar ^(n-1)


Definition:
In mathematics, a harmonic progression is a progression formed by taking the reciprocals of an arithmetic progression.Equivalently, a sequence is a harmonic progression when each term is the harmonic mean of the neighboring terms.
The general form of the harmonic progression is ,
a , a , a , a .............
1+d 1+2d 1+3d