Rational Numbers

A rational number is a number, the rational number in the form of a/b where a and b are two integers, with the bottom value (denominator) b not equal to zero. The b may be equal to one; every integer is a rational number. Rational number in the form of a/b. Here a numerator and b denominator. Numbers that are not rational, that is the number not in a/b form are called irrational numbers.

Students can also learn about other kinds of numbers like Prime Numbers, Composite Numbers, Whole Numbers etc.

Operation on Rational Numbers
Operations with the Rational Numbers are Addition, Subtraction, Division and Multiplication

There are three steps to add rational numbers:

• Step 1: To make the bottom numbers that is denominators are the same value.

• Step 2: Adding the top numbers that is numerators

• Step 3: Simplify the fraction when needed.

There are three steps to subtract rational numbers

• Step 1. To make the bottom number that is denominators are the same value.

• Step 2. Subtracting the top numbers that is numerators

• Step 3. Simplify the fraction.

There are three steps to multiply rational numbers

1. Multiplying the top numbers that is numerators.

2. Multiplying the bottom numbers that is denominators.

3. Simplify the fraction when needed.

There are three steps to divide rational numbers

1. Turn over the second fraction upside

2. Multiply the first fraction by that reciprocal

3. Simplify the fraction when needed.

Algebra is widely used in day to day activities watch out for my forthcoming posts on algebra 2 homework help and solving algebra word problems. I am sure they will be helpful.

Rational Numbers Examples

Add rational numbers:

1)      1/8+3/8

Step 1: The bottom numbers are already same numbers so go to step 2

Step 2: Add the top numbers 1/8+3/8=4/8

Step 3: simplify fraction 4/8=1/2

Answer is 1/2

2) 2/9+3/9

Step 1: The  deniminator numbers are already same  so go to step 2

Step 2: Add the Numerator numbers 2/9+3/9=5/9

Step 3: simplify fraction 5/9. (Not possible)

Hence the Answer is 5/9.

Subtract rational numbers:

1)      5/8 - 1/8

Step 1: The bottom numbers are already same numbers so go to step 2

Step 2: subtract the top numbers 5/8 - 1/8=4/8

Step 3: simplify fraction 4/8=1/2

Answer is 1/2

2) 6/7 – (-1/7)

Step 1: The Denominator numbers are already same so go to step 2

Step 2: subtract the Numerator numbers 6/7 – (-1/7)=7/7

Step 3: simplify fraction 7/7=1.

Ans: 1

Multiply rational numbers:

1)      5/5 X 1/2

Step 1: Multiply top numbers 5/5 X 1/2= 5/8

Step 2: multiply the bottom numbers 5/5 X 1/2 = 5/10

Step 3: simplify fraction 5/10 =1/2

Answer is 1/2

2) 4/5 X 5/2

Step 1: Multiply Numerator numbers 4/5 X 5/2= 20/5(2)

Step 2: multiply the Denominator numbers 4/5 X 5/2 = 20/10

Step 3: simplify fraction 20/10 = 2

Ans: 2

Divide rational numbers:

1/3 divides 1 / 6

Step 1: Turn over upset 1 / 6 => 6 / 1

Step 2: Multiply fraction by reciprocal 1 / 3 X 6/1

Step 3: simplify fraction 6 / 3 = 2

Answer is 2

2) 1/9 divides 1 / 36

Step 1: Turn over upset 1 / 36 => 36 / 1

Step 2: Multiply fraction by reciprocal 1 / 9 X 36/1

Step 3: simplify fraction 36 / 9 = 4

Ans: 4

Adding and Subtracting Rational Numbers

1. Add the following fraction to form rational numbers 4/8 and 20/8.

Ans: 3

2. Subtract the following fraction to form rational numbers 21/7 and 7/7.

Ans : 2

3. Multiply the given fraction to form rational numbers 2/3 and 21/14.

Ans : 1

4. Divide the following fraction to form rational numbers 1/10 divide by 2/60

Ans : 3

Introduction to greatest common factor of two number

Introduction to greatest common factor of two number:

In mathematics, the greatest common divisor (gcd), also known as the greatest common denominator, greatest common factor (gcf), or highest common factor (hcf), of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.

Through this article we are going to learn about how to find the greatest common factor of two numbers and some solved problems on greatest common factors of two numbers.

Greatest Common Factors of Two Numbers:

Steps to find greatest common factor:

Step 1: Write down the two numbers that we want to find greatest common factor.

Step 2: Find the factors of  given two numbers.

Step 3: Write down the common factor of both numbers.

Step 4: Choose the largest number from the common factor. That number is known as greatest common factor of  the two numbers.

Step 5: If there is no common factor , 1 is the GCF of the given two numbers.

Solved Problems on Greatest Common Factors of Two Numbers:

Problem 1:

Find the greatest common factor of the following two numbers 30, 60.

Solution:

Given, Numbers 30 , 60

We need to find the greatest common factor of 30, 60.

Step 1: Write down the two numbers that we want to find greatest common factor.

30 , 60

Step 2: Find the factors of  given two numbers.

Factors of 30 = 1, 2, 3, 5, 6, 10, 15, 30

Factors of 60 = 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30,60

Step 3: Write down the common factor of both numbers.

Common factors of 30 and 60 is 1, 2, 3, 5, 6, 10, 15, 30

Step 4: Choose the largest number from the common factor. That number is known as greatest common factor of  the two numbers.

Greatest common factor of 30 and 60 is 30.

Problem 2:

Find the greatest common factor of the following two numbers 180 , 256

Solution:

Given, Numbers 180, 256

We need to find the greatest common factor of 180, 256

Step 1: Write down the two numbers that we want to find greatest common factor.

180, 256

Step 2: Find the factors of  given two numbers.

Factors of 180 = 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180

Factors of 256 = 1, 2, 4, 8, 16, 32, 64, 128, 256

Step 3: Write down the common factor of both numbers.

Common factors of 180 and 256 is 1, 2, 4

Step 4: Choose the largest number from the common factor. That number is known as greatest common factor of  the two numbers.

Greatest Common factors of 180 and 256 is 4.

I am planning to write more post on how to multiply fractions with mixed numbers,and  simplifying fractions with exponents. Keep checking my blog.

Practice Problems on Greatest Common Factors of Two Numbers:

Problems:

1.Find the greatest common factor of the following two numbers 18 , 26

2.Find the greatest common factor of the following two numbers 56,90

Solution:

1. 2

2. 2

Introduction to Function rule definition

Introduction to Function rule definition:

Let us consider a set of items which are described in an ordered manner. The set expresses a relation between the items. This is general conclusion. If the relation is refined and could convey something better, then the relation becomes a function.

Let us discuss in detail with a simple example function rule definition.

Function Rule Definition – Basic Concept

Imagine a situation of a class teacher taking down the details of the ages of all the students. The teacher enters the record in an ordered manner, first the name of the student and his or her age next. The completed record now becomes a data set.

The items that come first in the set is called the domain set of the complete set. The items that come next is called the range set.

In this particular data set there is no condition exists between the items of the domain set and the items in the range set excepting that the information is ordered. Hence this data set expresses just a relation.                                                                                                                                         

Consider another situation. A data is collected for miles into kilometer conversion. The domain set contains items like 1mi, 2 mi, 3mi …….. and the range set consists of the equivalent measure in kilometers as 1.6 km, 3.2 km, 4.8 km….. In this set the relation between the items in both the sets is refined. That is, the items are related by a function.


Function Rule Definition – the Rule

Definition of function rule:Consider a set of domain elements and the corresponding range elements.

 If an item of the domain set corresponds to only and exactly one element in the range set, then the relation is a function.

Even if more than one element in the domain set correspond to one element in the range set, still the relation is a function. For example more than one student can be of the same age.

If an item in the domain set correspond to more than a single item in the range set, the relation is not a function. For example, a student corresponds different ages is meaningless.

Introduction to hexagonal prism

Introduction to hexagonal prism:

A hexagonal prism is defined as the prism which is composed of two hexagonal bases and six rectangular sides. Generally it is look like an octahedron.

The regular of the right hexagonal prism's edge length ‘a’ has surface area and volume

The surface area for the hexagonal prism is given by,

S = 3(2 + `sqrt(3)` )a² square units

The volume for the hexagonal prism is given by,

V = `(3)/(2)` `sqrt(3)`a³  cubic units

More about the Volume of Hexagonal Prism

Due to the structure of the hexagonal prism, the measures of the prism are varying by the methods. Another method is that it would be considering the side of the apothem as 'a', side as 's', height as 'h' in the hexagonal prism given below.



Surface area of the  regular hexagonal prism = 6 × apothem (a) × side length (s) + 6 × side length (s) × height (h).

It is represented in square units.

Volume of the hexagonal prism = 3 × apothem (a) × side length (s) × height (h).

It is represented in cubic units.

Examples for the Volume of Hexagonal Prism

Consider the hexagonal prism with the same length 5 cm .calculate the surface area and volume of the hexagonal prism.
Solution:

The surface area for the hexagonal prism is given by,

S = 3(2 + `sqrt(3)` )a²  square units

S=3 (2 + `sqrt(3)` ) 5²

S=279.90 cm2 is the required surface area of the hexagonal prism

The volume for the hexagonal prism is given by,

V = `(3)/(2)` `sqrt(3)`a³   cubic units

V = `(3)/(2)` `sqrt(3)`5³

V = 64.951 cm³  is the required volume of the hexagonal prism

Another method:

2. Consider the hexagonal prism for to calculate the surface area and volume of the hexagonal prism by considering the side of the apothem as '5', side as '4', height as '3' and all units are in centimetres.

Solution:

Surface area of the  regular hexagonal prism = 6 × apothem (a) × side length (s) + 6 × side length (s) × height (h) square units.

Surface area of the  regular hexagonal prism = 6 × 5 x 4 x 3

=360 cm² is the required surface area of the hexagonal prism

Volume of the hexagonal prism = 3 × apothem (a) × side length (s) × height (h) cubic units.

Volume of the hexagonal prism = 3 × 5 x 4 x 3

     =180 cm³ is the required volume of the hexagonal prism

Introduction to non invertible matrix

Introduction to non invertible matrix:

Many people talk about matrices but do you know what exactly a matrix is? Let's see. A matrix is a set of numbers arranged in a particular order in a rectangular array with  m rows and n columns in it.Below is an example of matrix in which a_ij are know as elements or entities.

Matrices are of different types. They are

• Row Matrix
• Column Matrix
• Square Matrix
• Rectangular Matrix
• Diagonal Matrix
• Scalar Matrix
• Identity Matrix
• Null Matrix

Define Non Invertible Matrix:

A matrix is said to be non-invertible matrix if and only if its determinant is zero.Non-Invertible matrices are also known as Singular matrices.They are also known as Degenerate Matrices.

Conditions for the matrix to be non-invertible are

1. It should be a square matrix,i.e.the number of rows and columns in a matrix should be equal.
2. The determinant of square matrix should be zero.

Que : How to calculate the determinant of a matrix?

Ans : Step1 - Let  A be a 2 x 2 matrix

Step 2 - Multiply the elements in the first and second diagonals and subtract the product of the second diagonal from the first.

Formula for finding the value of the determinant = `a_(11)` *`a_(22)`  -  `a_(12)` *`a_(21)`

det A = 4 * 5 - 2 * 10

          = 20 - 20

         = 0

Example Problems Based on Non Invertible Matrix:

Ex : 1 Calculate the determinants for the following matrices and check if they are non-invertible matrices.

 `[[5,6],[2,15]]`

Sol : Step 1 - Let  A = `[[5,6],[2,15]]`

Step 2 - Formula = `a_(11)` *`a_(22)`  -  `a_(12)` *`a_(21)` 

det A = (5x15) -(2x6)

detA = 75 -12 = 63 . Since det A is not equal to zero, the given matrix is not a non invertible matrix.

Ex: 2 : Calculate the determinants for the following matrices and check if they are non-invertible matrices.

 `[ [1,0,0],[-2,0,0],[4,6,1]]`    

Sol :  Step 1 : Let A be the given matrix.

Formula : `a_(11)` *`a_(22)`  -  `a_(12)` *`a_(21)` 

Step 2: Plugging in the values in formula

det A = 1[(0 * 1) - (6 * 0)] - 0[(-2 * 1) - (4 * 0)] + 0[(-2 * 6) - (4 * 0)]

det A = 0 - 0 + 0

det A = 0

Since det A is equal to zero ,the given matrix is a non-invertible matrix. 

Sample Variance

Definition of sample variance:

In 1893 the famous statistician Karl Pearson gave the definition of standard deviation as a measure of dispersion and gave a method for its calculation. Of all measures of dispersion the standard deviation is most commonly used. To understand what is sample variance, let us first see what is sample standard deviation.
Standard deviation: The positive square root of the number, obtained by dividing the sum of the squares of deviations from mean of observations of given data, by the number of observations is called the standard deviation of the data. It is denoted by s. Thus if x ¯ is the mean of n observations x_i (i = 1,2,3,…n), then,

With this background let us now define sample variance:
It is convenient to calculate the standard deviation in two steps. In fact, the number obtained by dividing the sum of the squares of deviations from the mean of observations of the given data by the number of observations is called the variance of the data. The variance is denoted by s^2 or s^2. After calculating the variance, its positive square root gives the standard deviation s. Thus, based on the above sample variance definition, we can write the sample variance formula as follows:

Calculation of variance for ungrouped data:

Direct method: Calculate the mean x ¯ of observations x1, x2, x3, … , x_n. Find the deviation x_i - x ¯ for each x_i. Calculate the square (x_i - x ¯)^2 for each x_i. Find the sum ?(x_i - x ¯)^2. Divide ?(x_i - x ¯)^2 by n to get the variance s^2.

If the values of x_i are relatively small and the mean is a fraction or a lengthy decimal, then we have an alternative formula for variance as follows:

Because of the presence of x_i^2 in the alternate formula for variance, finding variance by this formula could be tedious when x_i^2 are relatively large. The first method is also time consuming if the mean is a fraction (and not an integer). Fortunately, the numbers to be handled can be made small and the fractional mean can be avoided in calculations, by taking deviations of x_i^2  from an appropriately chosen number A, called the assumed mean.

Trigonometry is a branch of mathematics that deals with triangles, particularly those plane triangles in which one angle has 90 degrees (right triangles). Trigonometry deals with relationships between the sides and the angles of triangles and with the trigonometric functions, which describe those relationships.

Trigonometry has applications in both pure mathematics and in applied mathematics, where it is essential in many branches of science and technology. It is usually taught in secondary schools either as a separate course or as part of a precalculus course. Trigonometry is informally called "trig".

A branch of trigonometry, called spherical trigonometry, studies triangles on spheres, and is important in astronomy and navigation
Let's see one of the trigonometry examples
Question:-

solve for x

2 sin²x- sinx-1=0

Answer:-

We may use some trigonometry formulas to solve this.
Put sinx = z in the given equation ,you get

2z²-z-1 =0

2z²-2z+z-1=0

2z(z-1)+1(z-1)=0

(z-1)(2z+1)=0

either z-1=0 or 2z+1=0

so z=1 or z= -1/2

since z = sinx we get

sinx = 1 = sin π/2 or sinx = -1/2 = sin(π+π/6)

x= π/2 or x= π+π/6

so x= 90° or 210°