Introduction to Algorithm

Introduction to Algorithm:

Algorithm are defined in mathematics as one of the efficient method for solving many of the maths problems.Algorithm are having many number of sequence steps for solving the maths problem.Algorithm are mainly used for the purpose of calculation in mathematics and the data processing unit in the other fields.For learning an algorithm many of the instructions have been followed.

Example for Learning Algorithm:

Sample example for writing algorithm:

Input: Non-empty N.

Output: smallest number is N

Smallest ← N₀

For each item(Length(L)≥1), do

if item >smallest, then

Smallest← the item

Return smallest

here are many characteristics for learning an algorithm.They are mentioned below in the following,

1. By using algorithm we can get an accurate result.

2. By using algorithm, the efficiency of the product is good.

3. By using the algorithm, the calculation is very faster.

4. By using the algorithm, the time can be decreased.

Trigonometry Problems

Trigonometry Problems:

Question:

Answer:




problem 2:

Question:

Answer:

Identify Hexagon

Which of the figures is a Hexagon?

A ) Figure 1
B ) Figure 2
C ) Figure 3
D ) Figure 4

Steps to derive:

1 A six-sided polygon is called a hexagon.
[Hexa means six.]

2 Figure 1 has 5 sides. So, it is a pentagon.

3 Figure 2 is a closed figure formed by six sides. So, it is a hexagon.

4 Figure 3 is not at all a polygon, as it contains intersecting line segments.
[Polygons do not have intersecting line segments.]

5 Figure 4 has 4 sides. So, it is a quadrilateral.

6 So, Figure 2 is the only hexagon among the figures. This answer is given by choice B.

Hexagon and Pentagon:

Pentagon:

• A Pentagon is one of the polygon, The name of the five shaped image is known as pentagon. The pentagon has five sides. The pentagon can be divided as regular and irregular pentagon. A pentagon will have five sides and five interior angles.
• Always a pentagon will have five vertices.

Hexagon:

• A hexagon is one of the polygon, the hexagon has six sides,six interior angles.
• Always a pentagon will have six vertices.

Introduction to basic algebra

Basic algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures.
Together with geometry, analysis, topology, combination, and number theory, algebra is one of the main branches of pure mathematics. The part of algebra called elementary or basic algebra is often part of the curriculum in secondary education and introduces the concept of variables representing numbers. Statements based on these variables are manipulated using the rules of operations that apply to numbers, such as addition.In this article we shall discuss about basics of algebra

Sample Problem for Basic Algebra:

Example 1:

Solve the algebra equation.
|2x – 1| = 7
Solution:

Step 1: The absolute value of the given equation is already inaccessible.

Either 2 x-1 = +7 or 2 x - 1= -7

Step 2: Solve the equation 2 x – 1 =+7

2 x – 1 = 7

2 x = 8

X = 4

Step 3: Solve the equation 2x-1=-7

2 x – 1 = -7

2 x = -6

X = -3

The answers are 4 and -3. The answers may not be solutions to the equation.

Check the answer x=4 by replacement 4 in the given equation for x. If the left side of the equation sum value is equals the right side of the equation sum value after the replacement of x value, you have got the exact answer.

• Left side: |2 (4) – 1 | = 7

• Right Side: 7

Check the answer x = -3 by replacement -3 in the given equation for x. If the left side of the equation sum value is equals the right side of the equation sum value after the replacement of x value, you have got the exact answer.

• Left Side: | 2 ( -3) – 1 | = - 7
• Right Side: -7

The answer of the equation are x=4 and -3.

Writing Math

Normally, the writing math is classified into three types.

They are

• Word form
• Standard form
• Expanded form

Writing math in Word form:

Writing the given number in alphabetical form is called as word form.

Example for Word form:

Write 8795 in word form:

Solution:

The place value should be taken in account.

8000 = Eight thousand

700 = Seven hundred

90 = ninety

5 = five

8795 = Eight thousand seven hundred and ninety five.

Writing math in Standard Form:

Writing the given number in reduced form is said to be standard form.

Example for standard form:

Express 567000 in standard form:

Solution:

567000 = 5.67 × 10⁵

Here we have shifted the decimal to 5 places.

Writing math in expanded form:

Writing the given number in abbreviated form is said to be expanded form.

Example for Expanded form:

Expand the number: 9542

Solution:

9542 = 9000 + 500 + 40 + 2.

Classifying math diagrams - Square

Classifying math diagrams - Square:

• Square has four equal sides.
• The side length of the square is denoted as S.
• Total angle of the square is 360°
• All the four sides have the same angle 90°
• Square has two diagonals and both the diagonals are equal in length
• Area of the square is s²

Theorem of MatrIces

Theorem of Matrice:
From the definition of inverse of a matrix, we have (AB)(AB)^-1 = I

or A^-1 (AB)(AB)^-1 = A^-1 I (Pre-multiplying both sides by A^-1)

or (A^-1A) B (AB)^-1 = A^-1 (Since A^-1 I = A^-1)

or I B (AB)^-1 = A^-1

or B (AB)^-1 = A^-1

or (B^-1B)(AB)^-1 =B^-1A^-1

or I(AB)^-1= B^-1A^-1

or (AB)^-1 = B^-1A^-1

Properties of Arithmetic

Properties of Arithmetic:

The properties of arithmetic define when the order of operations can be changed in an initial calculation, so that we can get a second calculation which is equal to the first obtained result. The arithmetic operations that involving real numbers and/or real extents that uses the properties of arithmetic.

There are three types of arithmetic properties. They are,

• Associative property
• Commutative property
• Distributive property

The addition or multiplication of a group of values is similar to the how the numbers are grouped. There are three or more values involved in associative property. The parenthesis refers the terms that are considered one unit. The combination (Associative Property) are within the parenthesis. Therefore, the values are 'associated' together. For multiplication, the product is always same as the grouping. In Associative Property grouping the brackets are done first.

Associative property of addition:

a+(b+c)=(a+b)+c

Associative property for multiplication:

a×(b×c)=(a×b)×c

Example:

Find the associative property of 2+(3+9) and 2(85)

Solution:

2+(3+9)=(2+3)+9

2×(8×5)=(2×8)×5)

Commutative property is the property that altering the order of the values and does not alter the final result. It is a basic property of many binary operations. Many mathematical proofs based on it.

Distributive property of addition:

a+b=b+a

Distributive property for multiplication:

ab=ba

Example:

Find the Commutative property for 4+6 and 6×9

Solution:

4+6=6+4

6×9=9×6

Distributive properties used to distributing something or we can split something or broken up into parts.

This is defined as,

a(b+c)=ab+ac

Example:

Multiply 3 x 17 by using the distributive law.

Solution:

Given 3 x 17
=3(10 +7)
=3(10) + 3(7)
=30 + 21
Therefore 3 x 17=51.

Introduction to arithmetic terms

Introduction to arithmetic terms:
In mathematics the arithmetic terms are the basic one. It is used for calculating the additional, difference operations and etc. The arithmetic functions also used to computer science for writing coding. The basic math functions are including in this arithmetic terms. Here we will see about the basic four functions.

Functions:

The following 4 functions is the arithmetic terms.

• Addition
• Subtraction
• Multiplication
• Division

Trigonometry application

Trigonometry application:

Triangulation technique is used in astronomy to measure the distance of nearby stars.

• It is also used to measure the distance between the landmarks.
• It is used to calculate the oscillation (the waves that moves in a space. There are so many waves like sound waves, earthquake waves, matter waves and the light waves)
• The most important wave is the sin wave. It turns up all over the place, these wave forms can always be broken down into a series of superimposed sin waves of various frequencies. So this process is called as Fourier transform.

Table of Trigonometric identities

Table of Trigonometric identities:

reciprocal Identities: sin u = \frac{1}{csc u} cos u = \frac{1}{sec u} tan u = $\frac{1}{cot u}

csc u = \frac{1}{sin u} sec u = \frac{1}{cos u} cot u = \frac{1}{tan u}

sin u is the inverse function of cosec u

cos u is the inverse function of sec u

Tan u is the inverse function of Cot u

Cosec u is the inverse function of Sin u

Sec u is the inverse function of Cos u

Cot u is the inverse function of Tan u

Examples for Solve Length of side fire Triangle

Examples for Solve Length of side fire Triangle:

1) Solve the length of two side’s fire triangle is 29 cm and 30cm. If perimeter of the fire triangle is 81 cm, what is the length of third side of the fire triangle?

Solution:

Let third side length of the triangle is x cm then,

20 + 30 + x = 81

50+ x = 81

x = 81 -50

x = 31cm

Therefore length of third side of the fire triangle is 31cm.

How to find slope of a line?

Find the slope of the line where the points are ( 9,5) and (7,3) by using straight line tools.

Solution:

The formula for slope m = (y₁-y₂) / (x₁-x₂).

M = (5-3) / (9 -7).

M = 2 / 2.

= 1.

Thus the slope of line is 1.

General rules for multiplication

General rules for multiplication:

Positive x positive = positive

Negative x negative = positive

Positive x negative =negative

Negative x positive = negative

Positive x positive = positive:

Example:

x 4 =12

This two number are positive integers. So, answer is positive.

Negative x negative = positive

For example,

-4 x -3 = 12

This two number are negative integers .so, answer is positive.

Positive x negative =negative

For example,

4 x 3 =-12

One positive integer and one negative integer .so, answer is negative.

Negative x positive = negative

For example,

-4 x -3 =-12

One negative integer and one positive integer.so, answer is negative.

Solve the algebraic expression

Solve the algebraic expression:

4(x -1) + 2y - 5(x -y -4) + 5

Multiplying the integer terms

= 4x - 4 + 2y -5x + 5y + 20 + 5

Grouping the above terms

= -x + 7y + 21

Geometry Problem and Solution

Problem: Can two vertically opposite angles be supplement to each other? Draw a diagram to illustrate your answer


Let AB and CD bisect each other at right angles as shown in figure.







( Vertically opposite angles can be
supplements to each other)

Arithmetic Progressions

Arithmetic Progressions:
Consider the following lists of numbers :
(i) 1, 2, 3, 4, . . .
(ii) 100, 70, 40, 10, . . .
(iii) – 3, –2, –1, 0, . . .
(iv) 3, 3, 3, 3, . . .
(v) –1.0, –1.5, –2.0, –2.5, . . .
Each of the numbers in the list is called a term. In (i), each term is 1 more than the term preceding it. In (ii), each term is 30 less than the term preceding it. In (iii), each term is obtained by adding 1 to the term preceding it. In (iv), all the terms in the list are 3 , i.e., each term is obtained by adding (or subtracting) 0 to the term preceding it. In (v), each term is obtained by adding – 0.5 to (i.e., subtracting 0.5 from) the term preceding it. In all the lists above, we see that successive terms are obtained by adding a fixed number to the preceding terms. Such list of numbers is said to form an Arithmetic Progression ( AP ). So, an arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term.

Slope of a Line

Slope of a Line:
A line in a coordinate plane forms two angles with the x-axis, which are supplementary.
The angle (say) θ made by the line l with positive direction of x-axis and measured anti clockwise
is called the inclination of the line. Obviously 0° ≤ θ ≤ 180° (below Fig ). We observe that lines parallel to x-axis, or coinciding with x-axis, have inclination of 0°. The inclination of a vertical line (parallel to coinciding with y-axis) is 90°.


Definition 1 If θ is the inclination of a line l, then tan θ is called the slope or gradient of the line l. The slope of a line whose inclination is 90° is not defined. The slope of a line is denoted by m. Thus, m = tan θ, θ ≠ 90° It may be observed that the slope of x-axis is zero and slope of y-axis is not defined.

Perimeter and Area of a Circle

Perimeter and Area of a Circle:
circumference of a circle bears a constant ratio with its diameter. This constant ratio
is denoted by the Greek letter π (read as ‘pi’). In other words

Circumference/Diameter=π
circumference = π × diameter = π × 2r (where r is the radius of the circle) = 2πr


The great Indian mathematician Aryabhatta (A.D. 476 – 550) gave an approximate
value of π. He stated that π =62832/20000 which is nearly equal to 3.1416.
However, for practical purposes, we generally take the value of π as 22/7 or 3.14, approximately

area of the circle =1/2× 2πr × r = πr2