Method to Find the Degree of a Given Polynomial

There are equations and expressions in mathematics. The degree of a polynomial will be nothing but highest exponent that is present in a particular term of it. There can be variables as well as constants in it. Variables are terms whose value change but constants are terms which have the same value throughout. The exponents or power will appear only for the variables. So, only the variables’ exponents have to be checked and the constants left alone. These variables and constants will be separated by the basic arithmetic operations like addition, subtraction. There can multiplication symbol also. So, degree of polynomial is basically concept which tells about the highest power or exponent of the variable present in it. The degrees of polynomials will be there if there are more than present. The procedure applied or the concept is same. The variables have to be looked into as constants do not have a role to play in finding the degree.

Sometimes there can be more than variable in the same term. In that case all the powers or the exponents of the variables must be added and then the degree has to be found out. This can done by applying the laws of exponents. When two exponents are on terms which are multiplied with each other the exponents are added up. Before finding the degree of the given expression one must see to it is converted it into its standard form. Only then the degree must be found out. Otherwise the answer obtained will be wrong. So, one has to be very careful while finding the degree if the standard form is not given. Standard form is nothing but converting the expression into a form when the terms in it is just separated the arithmetic symbols.  Once this is done the basic requirement to find the degree is fulfilled and the degree can easily found out.

The process of finding the degree is quite a simple process and does not take much time, if the expression is already given in its standard form. If it is not given in the standard form then it must be first converted in to standard form and then the degree has to be found out. This make a little more time than the usual procedure. But to convert the expression into the standard form and finding the degree is more interesting .

How to write a letter to a friend?

Letter writing is an important part of primary school education. Though letter writing in real life has overshadowed by emails and messages, in schools letter writing is still valued in the stepping stone towards learning. A letter to a friend is an informal letter where the person approaches his or her friend directly about any topic with considering any formal etiquette. Let’s have a look at a sample of letter written to a friend.
7th November
Dear x,
I hope you are doing fine. I am also doing well. Since we met last year during Christmas holidays, this is my first letter to you. I am waiting for Christmas holidays. Christmas is approaching and I wish to meet you again this year. I have already prepared my wish list and I hope each of them is fulfilled. This year during Christmas I also have to buy a lot of gifts, as my nephew and nieces from my elder sister are coming to stay with us during Christmas holidays. I am so confused with what I should buy for these kids. I think I should buy some Barbie, Disney or Chota Bheem toys online. A few good sites have encouraged me to buy Barbie, Disney and Chota Bheem toys online. Also I am planning to gift a Little Mommy talking doll to my niece. As girls love dolls, I hope she will enjoy playing with the Little Mommy talking doll . I have also heard MeeMee products are hugely popular. I can even buy Mee Mee products online. Let me know if you are aware of some other good ones. We are planning to go to a resort after Christmas for New Year celebration and you are heartily invited. Please try to come; we will have a lot of games and fun.
Give my love to uncle and aunty and ask them to visit us some day. Looking forward to receive your suggestions and plans for Christmas and wish to see you during the holidays. Take care.
With love,
Your friend y

This is a sample of letter written to a friend.

Adding and Subtracting Positive and Negative Numbers

Positive and Negative numbers are important concepts in arithmetic. A number that is greater than 0 (zero) is called positive number. On the other hand, any number that is less than 0 are called negative numbers. A positive number can be written with the plus sign in front of the number or just as a number. For example: 1, 2, 3, 4 …….n. A negative number is written with the minus sign in front of the number. For example: -1, -2, -3, -4…..-n. There are different rules of operations for positive and negative numbers. The basic sets of rules are the addition and subtraction rules of arithmetic for positive and negative numbers. Let’s have a look at the same in this post.

Adding Positive and Negative Numbers
Positive + Positive = Positive: The sum of two positive numbers is equal to a positive number. For example: (2 girls’ vest ) + (3 girls’ vest) = 5 girls’ vest.
• Positive + Negative = Negative: The sum of one positive and one negative number results a negative number. For example: (2 toddler shoes) + (- 5 toddler shoes) = - 3 toddler shoes.
• Negative + Negative = Positive: The sum of two negative numbers results a positive number. For example: (- 2 online baby stores ) + (- 2 online baby stores) = 4 online baby stores.
• Negative + Positive = Negative: The sum of one negative and one positive number results a negative number. For example: (-3 apples) + (5 apples) = - 2 apples
Subtracting Positive and Negative Numbers
• Negative – Positive = Negative: When a negative number is subtracted by a positive number, the result is negative. For example: (-8baby shoes) – (3 baby shoes) = -11 baby shoes
• Positive – Negative = Positive: When a positive number is subtracted by a negative number, the result is positive. For example: (8 baby slings) – (- 5 baby slings) = 13 baby slings.
• Negative – Negative = Negative: When two negative numbers are subtracted, the result is negative. For example: (- 5 oranges) – (- 5 oranges) = - 10 oranges.
These are the rules of addition and subtraction for positive and negative numbers.

Difference of Sets

Difference of Sets is the operation on set(s). if there is a st A and a st B then st difference A – B  gives the elements of st A that are not in st B. similarly by Difference Set B – A gives the elements of st B that are not in st A.

As shown in above Venn diagram A – B is the closed curve with blue outline. This shows clearly that A– B is composed of those elements which do not belong to st B. set diffrnce A – B can also be stated as st A – (A∩B).  A∩B is the part which is common in A and B. so A – B is said to be that part of A which has exclusion of common part of A and B or which excludes common part with st B.

A – B = {x: x ∈A,and  B}
For example if there exists a st S1 {1, 2, 3, 4, 8, 10} and st S2 {2, 6, 7, 4, 8, 5} then S1 – S2 gives the elements of S1 that are not in S2 {1, 3, 10}. As you see this difference does not contain any element of S2 and also it contains only those elements of S1 that are not in st S2 or that are uniquely in st S1. Similarly S2 – S1 or above example gives a st = {6, 7, 5}.

Symmetric Difference of Sets is another operation done on sts as explained below:
Given two sts A and B then by symmtric Difference Sets A and B gives those elements of A and B which are either in A st or in B st but not in both st. We can also say that it gives the union of two sts but excluding the common element of them or intersection of them. This operation is represented as: A ∆ B = (A ∪B)-(A ∩ B ) as shown in diagram given below:

   
Example: if st A = {2, 3, 4, 5, 7} and st B = {3, 6, 7, 8, 9} then A ∆ B = {2, 4, 5, 8, 9}. Elements 3, 6, 7 are not there as they are common in both the sts. Another way to do it is find union and intersection of both sts and subtract intersection from union as:  A ∪B  = {2, 3, 4, 5, 6, 7, 8, 9} and A ∩ B = {3, 6, 7}. So (A∪B)-(A∩B ) ={2,4,5,8,9}
R Set Difference of B and A gives elements of B but not of A.

Tenses – Past, Present and Future

Tense is one of the most important concepts in English Grammar learning. Tense in English grammar is a category that refers to a situation in time. This concept of time is classified into three types of tenses, namely: Present Tense, Past Tense and Future Tense. Let’s have a look at each of the type of tenses.
Present Tense:
Present tense is a type of tense that refers to the current time. In simple terms, the tense that is a verb referring to the action or expression of present is called present tense. For example:
• She is listing down return gift ideas for the party. (Here, it is present tense because ‘she’ is listing down the return gift ideas in the present time.)
• Sana is exploring the online infant shopping India collection. (Here, it is present tense because ‘Sana’ is exploring the online infant shopping India collection in the present time.
• I am eating. (Here, ‘I’ am eating at the present time.)
Past Tense:
Past tense is a type of tense that is a verb referring to an action or expression in the past. For example:
• Mary’s cousin bought nappy pads online sometimes back. (Here, the sentence is referring that Mary’s cousin has bought nappy pads online in the past.)
• I had a heavy lunch this afternoon. (Here, it is past tense as the sentence is referring to the lunch happened in the past)
• I was eating. (Here, I was eating at the past time.)
Future Tense:
Future tense is a type of tense that expresses actions or state something in future. For example:
• India will be having many new online stores in the near future. (Here, the verb is referring to a situation in future.)
• My niece will start going to school from next year. (Here, niece will go to school in future.)
• I will eat. (Here, I will eat in future)
These are the three types of tenses in English Grammar.

Introduction to rectangle images:

Rectangle is one of the 2D object and also quadrilateral regular polygon. It contain four sides. Opposite of the sides are equal in rectangle Image. Rectangle image having four sides. Each side are intersected with 90 degree Two sides are lengths of the rectangle. Two sides are width of the rectangle. In rectangle images Opposite sides of lengths are equal and also opposite sides of width are equal. Rectangular changes should be based on length of the rectangle and width of the rectangle.

Basic Concepts of Rectangle Images:

 From the above image AB,BC,CD,DA These are Sides of the rectangle Image.

AB and CD are Parallel sides of the rectangle Images

AB and CD These are Length of the Rectangles

BC and AD These are Width of the rectangle Images

AC and BD Are Diagonals of the Rectangle Images (Rectangle Image Contains Two Diagonal AC and AD .Two Diagonals Are Equal length o f Rectangle Image)

 Area and Permeter of Rectangle:

Area of the rectangle:Area of the rectangle is Prodouct of Length and  width

Area of rectangle= Length*Width

Perimeter of the rectangle=2(Length + Width)

Example Problems in Rectangle Images:

Ex 1: Find the area of a rectangle whose length is 10m and width is 25 m .

Solution: Area of the rectangle is Product of Length and  width

Area of rectangle= Length*Width    =10*25,  Area = 2500 m^2  

Ex 2: Find the Perimeter of the rectangle image?

Solution: Perimeter of the rectangle=2(Length + Width)

From the given image Length =20,width= 5

Perimeter = 2(5+20)    =2(25)    =50cm

Ex 3: Find the area of a rectangle whose length is 1/2m and width is 8/15 m .

Solution: The area of a rectangle is multiplication of Their length and Width

Length= `1/2` in

Width=`8/15`

Area = Length * width = `1/2 xx 8/15 = 8/30 = 4/15`

Area= `4/15` in^2

Ex 4: Find the Perimeter of the rectangle image,Their length is 12 and width is 2

Solution: Perimeter of the rectangle=2(Length + Width)

From the given image Length =12,width= 2

Perimeter = 2(12+2)      =2(14)  =28cm

What to understand by diameter of a circle

What is a circle?
In a third grader’s language, a circle is a closed figure with curved side or no sides. A circle is defined by its centre and its radius. The centre of a circle is the point exactly in the middle of the circle such that every point on the circle is at the same distance from the centre of the circle. The radius of the circle is the distance between the centre of the circle and any point on the circle. Since all points on a circle are at the same distance from the centre of the circle, there can be infinitely many radii (plural of radius) of a circle.

What is the diameter of a circle?
When we draw two radii from the centre of a circle to any two different points on a circle, an angle is formed such that the two radii are the rays of the angle. This angle can be an acute angle, an obtuse angle, a right angle, a straight angle or also a reflex angle. When this angle is a straight angle, that is, when the angle at the centre subtended by the two radii is 180 degrees, the two radii form a straight line that passes through the centre of the circle and touches two points on the circle. Such a line is called the diameter of the circle. In other words, a diameter of a circle is a line that passes through the centre of the circle and touches any two points on the circle as well.

Diameter of a circle can also be defined in terms of the chord of a circle. A chord of a circle is a line segment that joins any two points on the circle. If a chord is such that it passes through the centre of the circle then it is called the diameter of the circle.
Just as in radii, a circle can have infinitely many diameters as well. By the definition of a diameter stated above, we also see that all the diameters would pass through the centre of the circle. Therefore we can say that all diameters of a circle are concentric.

How to find the diameter of a circle?
From the definition of a diameter above we saw that the length of the diameter would be two times that of the radius. Therefore the diameter of a circle formula can be written like this:
d = 2r
Where, d  is the diameter of the circle and r is the radius of the circle.

Simplifying Algebraic Expressions

Algebraic expressions are the representations in order to neglect writing in terms of words of relation between various terms. There are two variations in algebraic expressions which can be said as the rational and irrational algebraic expressions. A rational expression is nothing but an algebraic expression which can be written in a quotient of polynomials, for example x2 + 2x + 4. Meanwhile, an irrational expression is the one that in which is not rational, for example square root of (x+4).

Algebra Simplifying Expressions
In general, the algebraic expressions contain numbers and alphabetic symbols. Simplifications of some lengthy algebraic expressions are needed in order to shorten and solve the problem. While making that simplification, an equivalent expression would be finally arrived which is simpler than the previous. It obviously means that the simplified expression is smaller than the normal.

How to simplify Algebraic Expressions?

• There are no standard procedures for the simplification of algebraic expressions since there can be so many methods can be made which differ from person to person while doing. But, those can be grouped in some form as three types as follows.
• Expressions which can be simplified without any kind of preparation immediately.
• Expressions which requires preparation before doing the simplification.
• Expressions which cannot be simplified to any form.
• Also there are some rules are methods that can be used while doing the simplification. Before going into the example for simplifying, we shall see some of the rules while doing them.
• While doing the addition of fractions:  When the denominator is common for all the fractions, then add all the numerators and sum it and divide it by the common denominator.
• Simplify the expression as much as possible to the extent.
• Order of Expression: When there are confusingly operations were given, for example 8+4(2+3)2-7, then the order of doing simplification as follows. 1) Simplify the terms inside the parenthesis 2) Evaluate the powers and exponents if it is there, 3) Multiply or divide 4) Add or subtract. 

Examples of simplifying Algebraic Equations

• The below some examples would give an idea of how to simplify expressions. 2 x+3y+6-3x-2y-2+4yx. This expression can be easily simplified by adding all the like terms and combining finally. Hence it can be rewritten as (2-3+4) x + (3-2) y + (6-2) and finally it gives 3x+y+4.
• To simplify the expression of previously said example of 8+4(2+3)2-7, after first process we will get, 8+4(5)2-7, then 8+4(25)-7, then 8+100-7 and finally we will get answer as 101.

Properties of numbers

While counting any quantity we use numbers 1,2,3….and so on. In various calculations using numbers the four basic operations used are the addition, subtraction, multiplication and division. Based on these operations there are properties of numbers which make the calculations simpler. These properties lay the foundation to work with different equations and hence it is important to get familiar with them. To the question What are the Properties of Numbers, we can say that the basic property number are the commutative property, associative property, distributive property and Identity.

Commutative Property: A given operation is said to be commutative if when the numbers are interchanged the value of the result remains the same. When numbers are added in whatever order the result remains unchanged and hence addition operation is commutative.

For Example:  3+4=7 also 4+3=7.
When numbers are multiplied in any order the result remains unchanged and hence the multiplication operation is commutative.

For Example: 3x7=21 also 7x3=21. Subtraction operation and division operation are not commutative as the result changes when numbers are interchanged; 7-4=3 but 4-7 = -3; 12/3=4 but 3/12=1/4, here the results are different when numbers are interchanged.

Associative Property: A given operation is said to be associative if when the change in the grouping does not change the result.
When numbers are added the grouping of numbers does not change the result and hence addition operation is associative. For example: 3+(4+5) is same as (3+4)+5. When numbers are multiplied the grouping of numbers does not change the result and hence multiplication operation is associative. For example: 2x(4x5) is same as (2x4)x5.
Subtraction operation and Division operation are not associative.

Distributive Property: This property helps to multiply the number outside with each of the terms inside the parenthesis thus helps to remove the parenthesis. For example: 3(a+b)= 3.a + 3.b= 3a+3b; (x-2)(y+3) each of the terms in the first parenthesis is multiplied with each of the terms in the second parenthesis giving the required product, xy+3x-2y-6.

Identity Property: When a zero is added to any number the result is the same number and hence zero is called the additive identity. For example: 3+0=3. When a number is multiplied by one the result is always the same number and hence one is multiplicative identity. For example: 4x1=4

The other property of numbers is multiplicative inverse, the product of any number and its inverse is always one, a x 1/a=1; 1/a is the multiplicative inverse of a. Zero property, when any number is multiplied with zero the product is zero, ax0=0

Numbers in Hexadecimal form

It was the time of representation of things using something else, in short a metaphor which is used to represent things. Numbers has so many forms and Hexadecimal Number is one in that line of representation of numbers. Hexadecimal Number System is usually the numbers from 0 – 9 and A, B, C, D, E, F which represents 10, 11, 12, 13, 14, 15 respectively. The hexadecimal numbers are the numbers with a base 16. The base or the subscript to avoid confusion, similarly we will be using the subscript 10 to denote the decimal number. An example for hexadecimal number is 125AB16. The numbers or values corresponding to each hexadecimal number in terms of decimal, binary or octal in a chart like representation are the Hexadecimal Number Chart. This chart will help us in easily identifying the value of the number in decimal form which every single can understand, it can be named as a compiled chart.

Now let us try to answer the question of How to convert Hexadecimal to Decimal. To do the same we have to follow the steps below which will give the solution: First of all the place value of decimal numbers should be mastered, unit place value is considered as 0, tens place value is considered as 1 and so on. Then hexadecimal should be written with equal space. Then number starting from the left should be multiplied by 16 raised to the power of the place value it is assigned. Finally all the numbers should be added according to their place value.

Let us consider a problem which will give a better grab, Convert the hexadecimal number B432 into decimal number. B = 11*16^3 = +45056, 4 = 4*16^2 = +1024, 3 = 3*16^1 = +48, 2 = 2*16^0 = + 2. The answer is  4613010

Thus the above problem has helped us in understanding the way to convert hexadecimal to decimal. The Hexadecimal Numbering System is not easy for a layman to read or understand the way it, the conversion is to enable every single. The next concept will be Adding Hexadecimal Numbers, to make this concept clear let us consider the following example: ABC + 12A = BE6

The above addition cannot be done when the number which results to more than 16, in that case the number should be subtracted from 16 and a one should be carried over to the next adding digits to be added and goes on.

Vision Statement Examples

Introduction to stem and leaf plot

We are having lot of methods to show the data’s.  We can use the graphs, charts and tables for showing the data. Stem and leaf plot is one of the method which is used for showing large amount of data. It is like the histogram. The main difference between stem and leaf plot and histogram is we can understand the mean, mode and median from this. We will see some examples for back to back stem and leaf plot.

Terms for back to back Stem and Leaf Plot:

If we are having large number of data we will use the back to back stem and leaf plot. Just like the series of data’s. Normally a series mean it will contain more number of data’s. We can analyze the data’s using the back to back stem and leaf plot. In this we have to identify the data’s using its place value. Here the largest place value is known as stem and the other values are leaf. We already know about end to end connection method. Back to back stem and leaf method is like that only. In this we will write the leaves in both side of the stem. In the left side of the stem we are having the rounded leaves and right side we are having the truncated leaves.

Sample Problem for back to back Stem and Leaf Plot:

Back to Back Stem and Leaf Plot problem 1:

First we will see how to make a stem plot for a value consider the value 89

Solution:

Stem     is 8 and the leaf is 9.

Likewise we have to construct the back to back stem and leaf

Here we will see how to make a back to back stem and leaf plot

Let us consider a value 149.

So for this value the rounded leaf is 15 and the truncated leaf 14

Here the stem is 1

So the Back to Back Stem and leaf plot is

Rounded leaf (left)                       Stem                    Truncated leaf (right)

15                                               1                                         14

Back to Back Stem and Leaf Plot problem 2:

Draw the back to back stem and leaf plot for the following data's: 159, 260, 398, 401, 931

Solution:

Let us take the above set in ascending order. 159, 260, 398, 401, 931.

The back to back stem and leaf plot will be like the following


Rounded leaf stem Truncated leaf
16 1 15
26 2 26
40 3 39
40 4 40
93 9
93

Parts of Speech

Parts of Speech are a very important lesson in English grammar. It is a linguistic category of words that is classified based on the behavior of the lexical term. There are eight parts of speech in English grammar – Noun, Pronoun, Verb, Adjective, Adverb, Preposition, Conjunction and Interjection.

Noun: Noun is a part of speech that refers to a concrete entity. Nouns are names of people, places, and things and so on. For example: Johnson & Johnson is the most popular brand for kid’s clothes. Here ‘Johnson & Johnson’ is the name of the brand and a noun in the sentence.

Pronoun: Pronoun is a part of speech that is used as a substitute of noun. As for example: Today is Tina’s birthday and her father gave her a Barbie doll. In this sentence ‘her’ is a pronoun that is used instead of the noun Tina.

Verb: Verb is a part of speech that signifies an activity or a process being performed or done by the noun. As for example: Tina is playing with her Barbie doll. Here, the activity Tina is performing is ‘playing’ and therefore playing is the verb.

Adjective: An adjective is a part of speech that qualifies or modifies a noun or pronoun by stating something about it. An adjective generally precedes the noun or pronoun that it modifies. As for example: Tina is a beautiful girl. Here ‘beautiful’ is an adjective as it is describing the noun Tina.

Adverb: An adverb is a part of speech that modifies an adjective, verb or other adverb. As for example: Mother quickly bought two tops for girls from the market. Here quickly is modifying the verb bought and therefore quickly is the adverb.

Preposition: A preposition is a part of speech that describes a relation between other word and phrases in a sentence. As for example: Mother bought tops for girls from the nearest kid’s store. Here, from is the preposition.

Conjunction: A conjunction is a part of speech that joins two words or phrases. As for example: Ram and Hari are good friends. Here, and is the conjunction.

Interjection: An interjection is an exclamation used to denote emotions. As for example: Alas, the old man is dead. Here, Alas is the interjection that denotes the emotion of sadness.

Countably infinite

Countably infinite on the face of it appears to be a contradiction, which it is not.  In everyday language we may be using the term countable to signify a countable (and hence finite) number. But in mathematical terms countably infinite has a very specific and well defined meaning. In order to understand what countably infinite means,  we first need to understand certain concepts and a little about the history of counting.

Countably Infinite :history of Counting

Primitive man had no use for counting large numbers. While hunting or fighting other tribes he found it enough to indicate one, two or more. This sense of one, two or many is also found among the animals. You will almost never find a lonely animal picking a fight with a large group, while it will willingly fight a one to one battle with another animal. If you observe carefully, you will find that at times a lonely animal does fight two after an initial hesitation. This tendency shows that animals too do have a primitive sense of counting and numbers.

With the development of civilization various systems for counting developed, which have ultimately evolved as numbers the way we know them. The concept of infinity also came into existence with this evolution and the meaning of " infinite " was taken as something that cannot be counted.

Finite and Infinite

Concepts of finite and infinite became known and initially none imagined that there could be different degrees of  " infinite"  too. Researches on number theory, set theory and analysis have proved otherwise. Thus we know today that there are finite and infinite numbers but among the infinite numbers there too are various degrees .  Thus, in present day mathematics infinite cannot be just treated as a number that cannot be counted.

Definition of Countably Infinite

Countably infinite in mathematics means a set of elements which can be mapped one to one on to the set of natural numbers. In other words for which a one to one correspondence can be found between all its elements and natural numbers without skipping any natural number and without assigning either two elements to the same natural number, or assigning two natural numbers to the same element..

Not all infinite sets are countably infinite. It requires a very high degree of skills in mathematics and logic to understand how. But it has been established logically and mathematically that not only a higher degree of infinite exists beyond countably infinite, but for any degree of infinite, there exists one which is beyond it and hence larger.

Dividing Rational Numbers

To review, rational number is any number that can be written as a fraction of integers. We shall understand by using example, 3 and 4 are the integers so 3 divided by 4 would be considered as a rational number. Fractions as we know are rational numbers and so are whole numbers. Let us take 3 divided by 1 is 3 and is the same thing so dividing rational numbers using the multiplicative inverse.

Dividing Rational Numbers or How to Divide Rational Numbers – Rational numbers are divided using the multiplicative inverse. When we divide fractions, we find the multiplicative inverse of the divisor; often it is also known as reciprocal. Reciprocal is when we replace the numerator by denominator and denominator by numerator then the resultant fraction. For example: - For a fraction 7/8, its reciprocal will be 8/7.

To Divide Rational Numbers, we find the multiplicative inverse or reciprocal of divisor. For example, if we have to divide 2/5 by ¾ then ¾ is considered as dividend and 2/5 as the divisor. So in this case reciprocal of divisor will be 5/2. Now instead of dividing we will directly multiply 5/2 by 3/4. Therefore in simple words 3/4 is multiplied by the multiplicative inverse of 2/5. When we multiply fractions we multiply straight across the top, 3 times 5 is 15 and 4 times 2 is 8. Thus 15/8 is the solution.

We also understand that in Division of Rational Numbers, we have to turn the fraction upside down and then multiply the first fraction by the resulted reciprocal. If we have two rational numbers 2 /3 and 3/4 and we need to divide ¾ by 2/3 so ¾ will be called as divisor, its reciprocal will be 4/3, which is nothing but done upside down. Then next step will be multiplying the first fraction with the reciprocated fraction. That is 2/3 multiplied by 4/3. Here we shall multiply 2 by 4 and divided by 3 multiplied by 3 and get 8/9 as the solution. We also will understand how we simplify the fractions, 48/108 we simplify by 2 the whole fraction as it is divisible by 2 we get 24/54. Another simplification by 2 gives us 12/27. Here we understand that 12 and 27 will not be simplified by 2 anymore as it’s not divisible by 2, thus which could be another number, 3 is the other number which is divisible so we get 4/9. Thus we simplify the fractions.

Binary Numbers

Binary Numbers Tutorial – Binary numbers are used in computer programming and are used in all modern computer based devices. In Binary numeral system, there are two numeric values which are used to represent the numbers and those two values are 0 and 1. The numerical value represented in every case depends upon the value assigned to each symbol. For example: - 0 is denoted as 0, 1 is denoted as 1 and 2 is denoted as 10. Binary numeral system is also termed as base – 2 system because in this the base 2 is used to give a numeric value to a number.

Binary Representation of Numbers – Binary numbers are represented using two numerical values only that are 0 and 1. We know that in decimal system we use base 10 to represent numbers, the binary works on the same principle but the only difference is that it uses base as 2. For Understanding Binary Numbers, let me show you an example.
For example: - if we have a three digit number 432 then we know that number 4 holds the ones place which means 4 multiplied by 10^0, 3 holds the tens place which means 3 multiplied by 10^1 and 4 holds the hundreds place which means 4 multiplied by 10^2. In Binary system, we apply the same method but the only difference is we use base 2 instead of base 10. For this number we use 2^0, 2^1, 2^2 and so on.

To convert a Binary number to a decimal number, we follow the same pattern. For example: - if we have 100111 then it can be expanded as 1 X 2^5 + 0 X 2^4 + 0 X 2^3 + 1 X 2^2 + 1 X 2^1 + 1 X 2^0 which can be simplified as 32 + 0 + 0 + 4 + 2 + 1 = 39. Hence 39 can be written as 100111.

Multiplying Binary Numbers – Binary numbers are multiplied the same way as we multiply numbers in the decimal system. For example: - 1 X 1 = 1, 1 X 0 = 0 and 0 X 1 = 0. Therefore if we have to multiply 101 and 11 then it will be 1111

Dividing Binary Numbers – Binary numbers are divided the same way as we divide the numbers in decimal system. For example: - 11011 divided by 101 gives 101 as a quotient and 10 as a remainder.

All about Box and Whisker Plot

Definition of Box and Whisker Plot
Statistics tells us that any group of data given for statistical analysis is clustered around a middle or central value.  We can define Box and Whisker Plot as a box drawn graphically by marking the given data points on the graph in such a way that the box so obtained shows the middle part of the data values. The box and whisker plot is represented in box and whisker graph. Definition of Box and Whisker plot also states that it is the representation of data distribution along a graphical number line using quartiles and median. The Box and Whisker plot can be drawn either horizontally or vertically as needed.

How to Create a Box and Whisker Plot?
Once the given data is arranged in ascending order, we can create a Box and Whisker plot by finding the median of the data (Q2) which divides the given data into two halves. The median of the two halves, Q1 and Q3 is found out and it divides the data into quarters or quartiles. Now a box is drawn from Q1 to Q3. The whiskers are drawn at the first and last data value and then they are attached to the box using a line. As the Box and whisker plot visually represents median, the 25th quartile, the 75th quartile, the smallest data value and the highest data value of the given data values, it is also called as five-number summary. These values represent the middle or the centre of the data distribution, the spread of the data and the overall distribution range.

Example of Box and Whisker Plot 
We can see a wide range of applications of box and whisker plot in real life.  Some of the real life examples of the box and whisker plot are comparison of the marks, weight or height of the students in a class, comparison of the rate at which the different commodities are sold, comparison of the weight and the operating time of the phones, comparison of the height climbed by skyscrapers, comparison of cow’s daily milk yield etc.  The box and whisker plot can also be applied on data showing the rainy days, weight of the dogs, weekly earning etc.

Box and Whisker Plot Online
Box and whisker plot can be drawn using free online tools available in the internet. Box and Whisker Plot Online software helps us to draw the box and whisker plot easily for the given data. It is very useful for students to learn for the data required. The data can be user specified or the students can even use the inbuilt data available.

Line Equation

A line equation is nothing but an algebraic equation where every term in that equation will either be a product of a constant and a single variable or simply a constant. These equations will have one or more than one variables. They play an important role in applied mathematics. Equation of a line may simply be said as the linear equation which will not include exponents. In modeling many phenomenons, these equations are used in reducing non linear equations to linear equations from the assumption that the interested quantities will vary from background state only to a small extent.

The most common form used in order to find an equation of a line is given by, Y= mx + b.

The above given equation is used to describe the straight line on a particular coordinate plane. There are many equations representing the equation of a straight line, but this is the common form. This representation is known as Slope and Intercept form. In above said equation, ‘x’ and ‘y’ represents the coordinates of any points lying on the line, ‘m’ is known as the slope of the line(steepness) and ‘b’ is known as the Intercept which is a point formed when the line is crossing the y axis.

Standard Form Line Equation
There are many ways in order to express the equation of a line which are having their own pros and cons. There is a standard form in order to represent the equation of the line, which is given as Ax + By = C, where both A and B are not equal to zero. The equation finally is represented in this form giving out the standard form to represent. This standard form is essential when one needs to graph out the line or in order to find the line y intercept or x intercept. Also, some of the other forms for finding equation of a line are given as follows:

• Point-slope form: y – y1 = m(x-x1), where x1 and y1 are any points present on the line and ‘m’ is the slope, which is nothing but the proportionality constant. 
• Two point form: When there are two points lying on the line, then the formula to find the equation of the line is (x2–x1) (y-y1) = (y2-y1) (x-x1). Here x1, y1 and x2, y2 are two points present on the line, where x1 and x2 are not equal. Here the value y2-y1 divided by x2-x1 is nothing but the slope formula, which when simplified will give the point slope form. 
• Intercept form: This equation is a modification of standard form, where A is 1 divided by ‘a’ and B is 1 divided by ‘b’. This equation states that the sum of x divided by ‘a’ and y divided by ‘b’ will be equal to one.

Algebra Problem

Algebra problems present a challenge to many students who continue to be mystified by the subject throughout high school. The key to understanding algebra is a combination of understanding the concepts, practice, memorizing the equations, and practice.
Since algebra is a new branch of math with concepts and methodology students are completely unfamiliar with, it’s quite important to start students off on a positive note. It’s highly likely that they have already formed a negative impression based on reports from older siblings and friends. The most effective thing that tutors can do is to present algebra as a simple subject which any student can learn with ease.

Of course, presenting the course material in a way that makes it comprehendible to every student is another story altogether. Some students take to algebra like a duck to water, after only a few classes while others tend to take longer. Many students go through high school math with a hazy overview of algebra concepts and many unanswered doubts.  Algebra problems in particular, take some time and guided instruction before students can start solving them on their own, which students may or may not receive in class.

With the number of students in each class, longer and more complicated curriculum to finish, and teachers pressed for time, individual attention for each student is largely an unmet criterion. On the bright side however, there are plenty of learning aids available in the market today which are designed keeping students’ learning issues and hurdles in mind. Many of the learning aids, products and services are focused on providing help with math, particularly algebra theory and problems.

Students who are keen on getting extra help with the subject should seriously consider using one of these services to learn better, understand concepts, solve algebra problems easily, and score better grades. Most of them are available on the internet so all you really need is a computer with an internet connection and you’re all set to access math help, anytime you need it. Students who need personalized instruction can make use of written and video tutorials or live tutoring services. If you’re looking at sharpening your problem solving skills, peruse hundreds of worksheets and practice questions which have different types of questions of varying difficulty. Algebra geniuses can make use of these services to keep challenging themselves, test their knowledge and learn advanced concepts, which may not be covered in class.

Introduction to Statistics Examples

The study of data is called Statistics.  Collections of observation of an individual or a number of individuals is called data.

Collection of data:  There are two types of data namely Primary data and Secondary data.

Primary Data:  The data which is collected by the investigator with a definite object for his own purpose is called Primary Data.

Secondary Data:  The data which is collected by someone other than the investigator is called Secondary Data.

Statistics Examples: Measures of Central Tendency

Measures of Central Tendency:

A numerical value which represents approximately the entire statistical data is called Measures of Central Tendency of the given data.

The different ways of measuring central tendency of a statistical data are

Mean,  Median  and  Mode.

Statistics Examples: Mean

Mean :

The mean of a set of data is the same as finding average.

Mean = `(Sum of all observations )/(Total Number of Observations)`

`Mean of ungrouped data:`

` Mean = ``sum_(i = 1)^n` `f_(i)` `x_(i)`
                ------------------------------
                   `sum_(i = 1)^n` `f_(i)`

Ex :

Find the mean of the following data:

x f
25 25
35 20
45 15
55 15
75 10

Solution:

Construct another tabe:

x f fx
25 25 625
35 20 700
45 15 675
55 15 825
75 10 750
85 3575

` Mean = ``sum_(i = 1)^n` `f_(i)` `x_(i)`
                 ---------------------------------
                    `sum_(i = 1)^n` `f_(i)`

Mean =  `sum`fx /  `sum`f
=3575 / 85
=42.06

Statistics Examples: Median

Median for Raw data:

Arrange the set of datas in ascending or descending order.  The middle most value is the Median.

Rule 1:  If n is odd, the median = `(n + 1)/(2)` th term

Rule 2 :  If n is even, there are two middle terms ie `(n)/(2)`  th term and  `(n)/(2)` + 1 th term.

In this case , the arithmetic mean of these two terms is the median.

Median =     `(n)/(2)`  th term  +  `(n)/(2)` + 1 th term
                        ----------------------------------------
                                                  2

Ex 1:
Find the median of 6, 7, 2, 5 and 10

Sol:
Arrange the given datas in ascending or descending order:
2, 5, 6, 7, 10
Here n= 5 ( odd number)
Median =  `(n + 1)/(2)` th term =  `(5 + 1)/(2)` th term
=   `(6)/(2)` th term
=  3 rd term
=   6

Ex 2:
Find the median of : 6, 11, 15, 7, 19, 8, 4, 10

Sol :
Arrange the given datas in ascending or descending order:
4, 6, 7, 8, 10, 11, 15, 19
Here n = 8 ( even)
Median =    `(n)/(2)`  th term  +  `(n)/(2)` + 1 th term
                     ----------------------------------------
                                              2

Median  =   `(8)/(2)`  th term  +  `(8)/(2)` + 1 th term
                     ----------------------------------------
                                             2

Median  =   `(4th term + 5th term)/(2)`
 = ``(8 + 10)/(2)`
= 18 / 2
= 9

Statistics Examples : Mode

Mode:  Mode is the repeated value of the given data

Ex: Find the mode for the given data:  34, 56, 21, 56, 71, 98, 22, 56

Sol: In the given data 56 is repeated thrice.  So the mode is 56.
Mode for tabulated data:

Number  7 8 9 10 11 12 13 14 15
Frequency 3 7 11 14 13 17 12 8 6

Sol:  Since the frequency of number 12 is maximum
Mode = 12

Continuity of a function

Limit of a function can be found from the graph of that function, besides other methods. Some of the graphs are continuous. So what is continuity?  That means they can be drawn without lifting pencil from the paper. See some examples below:

The functions that all the above graphs represent are continuous. Now look at the following graphs:

 All the above graphs are not continuous or discontinuous as they cannot be drawn without lifting the pencil from the paper. With this understanding now let us try to define continuity.

Definition of continuity:
If the domain of a real function f contains an interval containing a and if lim (x->a) f(x) exists and lim (x->a) f(x) = f(a), then we say that f is continuous at x = a.

Thus, if lim (x->a+) f(x), lim (x->a-) f(x) and f(a) all exist and are equal, the f is said to be continuous at x = a.

If f is not continuous at x = a, we say that it is discontinuous at x = a.

(1) In the following picture, x is not defined at x = 2. Therefore f is discontinuous at x = 2.

(2) In the following picture, f(-1) is defined, but the left hand limit and the right hand limit at x = -1 are not equal. So the function is discontinuous at x = -1

(3) The above picture, at x = 1, both left and right hand limits exist and are equal but the limit of the function is not equal to f(1) so the function is again discontinuous at the point x = 1.
In simple words we can state continuity as follows:
A function is said to be continuous at any point x = a if the following three conditions are met:
(a) f(a) exists
(b) lim (x->a-) f(x) = lim (x->a+) f(x) = lim(x->a) f(x)
(c) quantities in (a) and (b) are equal.
If any of the above conditions is not met, we say that the function is discontinuous at the point x = a.

Mean median and mode

What is mean, median and mode?
In many statistical situations, like the distribution of weight, height, marks, profit, wages and so on, it has been noted that starting with rather low frequency, the class frequency gradually increases till it reaches its maximum somewhere near the central part of the distribution and after which the class frequency steadily falls to its minimum value towards the end. Thus, the central tendency may be defined as the tendency of a given set of observations to cluster around a single central or middle value and the single value that best represents the given set of observations is called the measure of central tendency.  Mean, median and mode are all measures of central tendency.

Define mean, median and mode:
Mean: The average value of a set of data is called the mean. If x1,x2, x3, …. Xn are n values of a given variable then the mean value, represented by μ, would be sum of these x values divided by n.
Median: The middle value of a data set is called the median. It is represented by ‘Me’.
Mode: In a data set of various values of a variable, the number that occurs maximum number of times is called the mode. In other words the value with maximum frequency is called the mode.

How do you do mean median and mode?
Mean median and mode problems usually involve calculating mean median and mode. That can be done using the following formulas:
Mean = μ = [∑xi]/n
Median = Me = middle value obtained after arranging the values in ascending order.
Mode = Mo = the value that occurs most number of times or the value with maximum frequency.

Solved example: Find the mean, median and mode of the following numbers: 5, 4, 5, 5, 6, 7, 8, 9, 6, 8
Solution:
Mean = μ = [∑xi]/n = [5+4+5+5+6+7+8+9+6+8]/10 = 63/10 = 6.3
Median: Firs arrange the data in ascending order. So we have:
4, 5, 5, 5, 6, 6, 7, 8, 8, 9. The two middle numbers are 6 and 6. The average of these numbers is (6+6)/2 = 6. Therefore,
Me = 6
Mode: The number that occurs most number of times is 5. Therefore,
Mo = 5

Identities of Pythagorean Theorem

Pythagorean Theorem is derived from the right angle triangle. This theorem is very important and useful for all calculus part and geometrical part. In this article we discuss about Pythagorean identities. It is also known as Pythagorean trigonometric identities mean expressing the Pythagorean Theorem in terms of trigonometric functions. It also includes formula of sum of angles and it shows the basic relations between sine and cosine functions and from this basis other identities are derived. For deriving the identities we also have to know the theorem and the theorem states that in a right angle triangle the square of hypotenuse is equal to sum of square of base and square of height.

There are 3 Pythagorean identities. We discuss about all three identities in theory and mathematical forms. Suppose we have given a unit circle, we mark a point on the circle the point is vertex of the right angle and it is represented by the coordinates. Suppose the coordinates are (sinθ, cosθ).the given circle is unit circle so hypotenuse is 1. And the two legs of right angle triangle in a unit circle are sinθ and cosθ. By using Pythagorean Theorem we can write as (sin^2θ+cos^2θ=1). This basic equation is known as first Pythagorean identity.

For the second Pythagorean identity we start from the first Pythagorean identity. First write the identity (sin^2θ+cos^2θ=1). Now we divide each term by (cos^2θ). While dividing we have to remember that (sinθ/ cosθ= tanθ) and (1/ cosθ= secθ). After dividing we get (tan^2θ+1=sec^2θ). This equation is known as second Pythagorean identity.

Now we derive the third Pythagorean identity. For this again we have to use the first Pythagorean identity. First we write the equation (sin^2θ+cos^2θ=1). Now we divide each term by (sin^2θ). While dividing the equation we have to remember some trigonometric formula such as (cosθ/ sinθ= cotθ) and (1/ sinθ= cosecθ). After dividing each term we get (1+ cot^2θ=cosec^2θ). This equation is the third Pythagorean identity.
Pythagorean identities list are...
1. (sin^2θ+cos^2θ=1)
2. (tan^2θ+1=sec^2θ)
3. (1+cot^2θ=cosec^2θ)
Some Pythagorean identities problems. First problem is, suppose we have given secx= (-2/3) and tanx>0 then we have to find values of other trigonometric functions.  Solution of this problem is first find (cosx=1/secx=-3/2), then by using first identity find (sinx). After this we can easily determined the remaining functions.
Second problem is suppose we have to solve (sinθcos^2θ-sinθ). A Solution of this problem is that first take out the common term and then we use the first Pythagorean identity. Finally we get the result as (-sin^3θ).

What is an ogive?

What is an ogive?
Ogive in mathematics it comes under the part of statistics. It is a way to show the data in a single line. Representation of all the data with help of single line diagram. We can show an olive by graph.  An ogive can be used to show the result at any time. What changes are occurring in curve point by point we can understand by an ogive? Either any particular value curve increases or decreases we clearly understand by ogive curves. It also helps to describe the slopes of curve.

We know the frequency distribution in statistics. An ogive is the graph of all frequencies of a particular frequency distribution. All these frequency make a continue series. In any graph we have X-axis and y-axis. X-axis denotes boundaries and Y-axis denotes frequencies.

There are two types of ogives. First type is less than ogive; it means that we plot the graph between less than cumulative frequencies and upper limits of boundaries. It is an increasing curve. Second type is more than ogive, it means that we plot graph between higher values from cumulative frequencies and lower limits of boundaries. It is a decreasing curve. Ogives are very useful in several areas such as median, quartiles, and deciles, percentiles etc. ogives also to differentiate from a given set that which value is above and which is below from a particular value. We also compare between frequency distributions.

Definition of ogive
Ogive is a curve in frequency distribution. It shows the relationship between cumulative frequencies and boundaries.
In architecture ogive means a diagonal rib of a vault. Similar to conic shape, in staring it is in pointed and then it gradually become wider. For example we can say, in astronautics conical head of any missile or any rocket.

Ogive in statistics
Statistics ogive shows a single line curve. All data are mentioned in that single line graph. Graph may be increasing or decreasing depends upon the data values. If we want that all the values individually categorized then an ogive gives ideal curve. We can understand statistics ogive by some examples.
Suppose we have give data in two columns. First shows range of class(15-20, 20-25, 25-30, 30-35, 35-40, 40-45, 45-50) and second column shows frequency(4,6,10,12,15,3,5). We have to find median quartile and decile. First we calculate cumulative frequency then separate higher and lower values. After this make a table and finally we get result.

More about Quartiles

Quartiles are the values that divide the given data arranged in ascending order into subdivisions of twenty five percent, fifty percent and seventy five percent. First quartile is the twenty fifth percentile also known as the lower quartile. Second quartile is the 50th percentile also known as the median and the third quartile is the seventy fifth percentile also known as the upper quartile. The lower quartile or the first quartile is the middle value or the median of the first half of the data values arranged in the numerical order. It is denoted as Q1. Q1 =( ¼). (n+1)th value of the data set, here n is the total number of data values. The second quartile denoted by Q2 is the median of the data set arranged in the numerical order. Q2 = median = (1/2)(n+1)th value of the data set. The third quartile or the upper quartile is denoted by Q3. Q3 is the (3/4)(n+1)th value which is the middle value or median of the upper half of the data set.

We know that range of a given data set is the value got by calculating the difference between the highest and lowest values in the data set. So, range = highest score – lowest score.  Now Quartile Range is a bit similar to range, it is the difference between the upper quartile (Q3) and lower quartile (Q1). Quartile Range is given as (Q3 – Q1) for a particular data set which is also called the Inter Quartile Range denotes as IQR. So, IQR = (Q3 – Q1). Let us learn about Quartile Deviation which is the absolute measure of dispersion. It is also called the semi Inter Quartile Range and is half of the Inter Quartile Range. It is written as Q.D in short, Q.D. = (1/2) (Q3 – Q1)

For example let us find the lower quartile, median, upper quartile, inter quartile range and quartile deviation of the data set 15, 18, 14, 20, 26, 16, 18.
First arrange the data values in the numerical order:
14, 15, 16, 18, 18, 20, 26
The number of values, n = 7
Lower Quartile = (1/4)(n+1) = (1/4)(7+1) = 8/4 = 2nd value which is 15(Q1) in the data set
Second Quartile = Median = (1/2)(n+1)= (1/2)(7+1)=8/2= 4th value which is 18(Q2)in the data set
Third Quartile= (3/4)(7+1)=3. 8/4 = 3.2 = 6th value which is 20(Q3) in the data set
Inter Quartile Range = IQR = (Q3 – Q1)= (20 – 15) = 5
Semi Inter Quartile Range = Quartile Deviation= Q.D. = (1/2)(IQR)= (1/2) (Q3 – Q1) = 5/2 = 2.5

Step by step math solution

Line Plot Graph made simple
Line Plot Definition
A data represented on a number line with marks like ‘x’ or any other mark which shows the frequency of a value in the data is defined as the line plot. For example:
The line plot below shows the marks of 20 pupils in a class.

The ‘x’ marks show the frequency of the marks obtained by the pupils
Let us now take a quick look at how to make a line plot
• First we need to gather the information. Once the information is ready we look for the data sets which occur often that is the data which is frequently shown. Something like the favorite flavor of certain people or the number of pets a group of people have.
• The data is to be sorted and then a chart is created so as to organize the list. We then name the chart for convenience.
• This is an important step which involves determining the scale. The scale might not have the labels that are not the data values as per the given information and hence we need  to decide the scale depending upon the frequency of the data items for which a numerical scale is used which begins with the least number and ends in the highest number in the data set.
• Now we draw a horizontal line which is similar to  a number line according to the chosen scale.
• Finally we start marking ‘x’ above the line corresponding to the number on the scale as per the data we have. Once the markings are done, the line plot for the given data is ready for further analysis.

Let us now make line plot graphs using a line plot example
Given are the costs of 15 books sold in a book store, represent a data as a line plot graph.
$20 $15 $9 $15 $9 $20 $9 $20 $20 $35 $25 $20 $9 $30 $15
We need to decide on the scale, the lowest value is 9 and the highest value is 35. So, the scale should start from 5 and end in 35

Solve by the addition method

Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign (+). For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples. Therefore, 3 + 2 = 5. (Source: Wikipedia)

Example Problems for Solve by the Addition Method:-

Problem 1:-

Solve 453 + 213 by the Addition method.

Solution:-

In the following step by step process of addition method

Step 1:-

               453
            + 213
           -----------
           -----------
The above equation 453 is adding to 213 in between (+) plus operation. In basic addition process start with right side value to left side value

Step 2:-

               453
            + 213
           -----------
                   6
           -----------
Adding the right side values 3 and 3. 3 is equal with 3 in 3+3 =6. Then move to next value.

Step 3:-

               453
            + 213
           -----------
                 66
           -----------
Adding the next two values 5 and 1. 5 adding with value 1 in 5+1 =6. Then move to next value.

Step 4:-

               453
            + 213
           -----------
               666
           -----------

Adding the last two values 4 and 2. 4 adding with value 2 in 4+2=6. We get the final answer is 666.

Problem 2:-

Solve 654 + 323 by the Addition method.

Solution:-

In the following step by step process of addition method

Step 1:-

               654
            + 323
           -----------
           -----------

The above equation 654 is adding to 323 in between (+) plus operation. In basic addition process start with right side value to left side value

Step 2:-

               654
            + 323
           -----------
                   7
           -----------

Adding the right side values 4 and 3. 4 adding with value 3 in 4+3 =7. Then move to next value.

Step 3:-

               654
            + 323
           -----------
                  77
           -----------

Adding the next two values 5 and 2. 5 adding with value 2 in 5+2 =7. Then move to next value.

Step 4:-

               654
            + 323
           -----------
               977
           -----------

Adding the last two values 6 and 3. 6 adding with value 3 in 6+3=9. We get the final answer is 977.


Practice Problems for Solve by the Addition Method:-

Problem 1:-

solve 421 + 167 by the addition method.

Answer:- 588

Problem 2:-

solve 217 +171 by the addition method.

Answer:- 388

Problem 3:-

solve 383 + 71 by the addition method.

Answer:- 454

Problem 4:-

solve 152 + 38 by the addition method

Answer:- 190

Problem 5:-

solve 43 + 24 by the addition method.

Answer:- 67

Frequency Distribution in Statistics

In mathematics frequency distribution is used in statistics. Mean of a frequency distribution is that the arrangement in which sets of value occurs and in the values one or more variable takes place. Frequency distribution is in the form of either graphical or tabular. Each value in the table contains frequency or count of values, how many times they occur. The values of frequency in group or interval forms.  After summarizing the entire values frequency distribution table is formed. Mean of frequency distribution is also that it shows the total number of observations within a given interval. The interval is either exclusive or exhaustive. The size of intervals generally depends on the data which we have to analyze and calculate. One thing we have to remind that the intervals must not be overlapped to each other.

Now we discuss that how to construct frequency distribution tables. We use some steps to make a frequency distribution table. In step one; we determine the range of given data. Range of given data means the difference between the higher value and the lower value. In step two, we decide that which data can be grouped means formulation of approximate number of groups. There are no particular rules for step two. It can be 5 groups to 15 groups. But there is one formula for this (K=1+3.322logN), where K is the no of groups, logN is the total number of observations.

In step third, we decide the size of intervals.  The size of interval is denoted by (h). To determine the size we can use a formula (h= range/number of groups). If result is in fraction then we choose next higher value. In step fourth, we decide start point means starting from the lowest value and in the ascending order. In step fifth, we determine the remaining groups. It is determined by adding the interval size corresponding to all values. In step sixth, we distribute all the data into their groups. For this we use tally marks method because it is suitable for tabulating the observations into their respective groups. By using these six steps we can construct a frequency distribution table.

Now we come to standard deviation for frequency distribution. It is a measure of variation or measure of dispersion amongst the data. In place of taking absolute deviation we may square each deviation and obtained the variance. The square root value of variance is known as standard deviation for given values of frequency.

Derivatives of inverse trigonometric functions

Like many functions, the trigonometric functions also have inverse.  Just like how we can find the derivative of trigonometric functions, we can also find the derivatives of inverse trigonometric functions. In this article we shall take a quick look at inverse trig functions derivatives. To be able to find the standard formulae for derivatives of inverse trig functions, we would need the formula for derivative of any general inverse function.

That would be like this:
f’(x) = 1/g’(f(x)), where f and g are the inverse functions of each other.
Let us first try finding the inverse trig function derivatives of the sine function. The sine function inverse is written as arc sin (x). Therefore the function would look like this: y = arcsin(x),
Therefore, x= sin (y) for –pi/2 ≤ y ≤ pi/2. We fix this domain for the sine function to ensure that our inverse exists. If we don’t restrict the domain, then y could have multiple values for same value of x. For example, sin (pi/4) = 1/sqrt(2) and sin (3pi/4) is also 1/sqrt(2). With that in mind, we can write the relationship between sin and arc sin as follows:
Sin(arc sin (x)) = x and arc sin(sin(x))  = x
Thus here our f(x) = arc sin(x) and g(x) = sin (x), then using the derivative of inverse formula that we stated above we have:
f’(x) = 1/f’(g(x)) = 1/cos (arc sin (x))
This formula may to be applicable in practice. So let us make a few changes in there. We had earlier x = sin y so y = arc sin (x). Using that, the denominator of our derivative would become
Cos(arc sin (x)) = cos y
Next we use our primary trigonometric identity which was:
Cos^2 (x) + sin^2 (x) = 1
Thus, Cos^2 (y) + sin^2 (y) = 1 from this we have
Cos (y) =√( 1- sin^2 (y)) = √(1 – (sin y)^2). But we have already established that sin y = x. So replacing that we get,
Cos (y) = √(1 – x^2). So now plugging all that back to our equation of derivative of inverse trig functions for sine f’(x), we have:
f’(x) = 1/cos(arc sin(x)) = 1/cos y = 1/√(1-x^2)
Therefore we see that derivative of arc sin (x) = 1/√(1-x^2).
The derivatives of other trigonometric inverse functions of arc cos, arc tan etc can be derived in a similar way. They are:
d/dx arc cos (x) = -1/√(1-x^2) and d/dx arc tan (x) = 1/(1+x^2)

All about the hypotenuse of a right triangle

Like every triangle, a right angled triangle would also have three sides. However, in a right triangle one of the angles is a right angle. That means one of the angle measures 90 degrees (or 𝛑/2 radians). Since the sum of angles in any triangle has to be 180 degrees, in a right triangle as one angle is already 90 degrees, the sum of the other two angles have to be 90 degrees. That means that the other two angles are compliments of each other. It also means that the other two angles have to be acute angles. A typical right triangle would look as follows:

The longest side is called the hypotenuse. The side that is adjacent to the know angle is called the adjacent side and the side opposite to the known angle is called the opposite side. By default the hypotenuse will always be the side opposite the right angle.

The adjacent and opposite sides together are also called the legs of the right triangle. The length of the hypotenuse of a right angled triangle can be found using different methods, depending on what part of the triangle is given to us.

To find the hypotenuse of a right triangle given the length of the legs:
If the hypotenuse = c and the legs are ‘a’ and b. If a’ and b’ are known, then we can calculate the length of the hypotenuse using the Pythagorean rule as follows:
C^2 = a^2 + b^2
Finding hypotenuse of a right triangle given one of the angles and the adjacent side:
In the picture below,

Suppose the angle marked in red is x and the adjacent side = a, then the length of the hypotenuse H can be given by the formula:
H = a/Cos (x)
Formula for the hypotenuse of a right triangle given one of the angles and its opposite side:
Again from the picture above, if we are given the opposite = b instead of the adjacent side. Then the formula for the hypotenuse can be written as follows:
H = b/sin (x), where x is again the angle marked in red.
Thus as we saw above there are more than one ways to find the length of the hypotenuse of a right triangle.

Trapezoidal Numerical Integration

Introduction to numerical integration:  As we have seen, the ideal way to evaluate a definite integral a to b f(x) dx is to find a formula F(x) for one of the antiderivatives of f(x) and calculate the number F(b) – F(a). But some anti derivatives are hard to find, and still others, like the antiderivatives of (sin x)/x and sqrt(1 + x^4), have no elementary formulas. We do not mean merely that no one has yet succeeded in finding elementary formulas for the anti-derivatives of (sin x)/x and sqrt(1 + x^4). We mean it has been proved that no such formulas exist. Whatever, the reason, when we cannot evaluate a definite integral with an anti-derivative, we turn to numerical methods such as the trapezoidal rule and Simpson’s rule.
Numerical Integration Methods are Trapezoidal rule and Simpson’s rule. Let us now describe Trapezoidal numerical integration.

Trapezoidal Numerical Integration: When we cannot find a workable anti-derivative for a function f that we have to integrate, we partition the interval of integration, replace f by a closely fitting polynomial on each sub interval  integrate the polynomials and add the result to approximate the integral of f. The higher the degrees of the polynomials for a given partition, the better the results are.  For a given degree, the finer the partition, the better the results, until we reach the limits imposed by round-off and truncation errors.

The polynomials do not need to be of high degree to be effective. Even line segments (graphs of polynomials of degree 1) give good approximations if we use enough of them  .To see why, suppose we partition the domain [a, b] of f into n subintervals of length delta x = h = (b – a)/n and join the corresponding points on the curve with line segments.

The vertical lines from the ends of the segments to the partition points create a collection of trapezoids that approximate the region between the curve and the x-axis. We add the areas of the trapezoids, counting area above the x-axis as positive and area below the axis as negative.
T = ½ (y0 + y1) h + ½ (y1 + y2)h + ….. + ½ (yn-2 + yn-1) h + ½ (yn-1 + yn)h= h (1/2y0 + y1 + y2 + ….. + yn-1 + ½ yn) = h/2 (y0 + 2y1 + 2y2 + …. + 2yn-1 + yn).Where, Y0 = f (a), y1 = f(x1), Yn-1 = f(x n-1), yn = f(b).

The trapezoidal rule says: use T to estimate the integral of f from a to b. Numerical Integration in R As we have seen , the definition of the Riemann integral is not very efficient way to prove that a function is Riemann integral . However once it is known that a function f is Riemann integral on some interval [a, b] a modification of the definition makes it possible to evaluate the integral of simple limit.

Volume of sphere

Introduction to Sphere : A tennis ball and a fully blown football  are some familiar objects which bring to our mind the concept of a sphere .A sphere is a three dimensional geometrical object  which can be defined as follow The set of all points in space which are equidistant from a fixed point , is called a sphere.

The fixed point is called the center of the sphere and the constant distance is called its radius. A line segment through the center of a sphere, and with the end points on the sphere . All diameters of a sphere  are of constant length , being equal to twice the radius of the sphere .Thus , if d is the length of a diameter of a sphere   of radius r  then d= 2r . The length of diameter is also called the diameter of sphere .The solid sphere is the region in sphere, bounded by sphere .

Also every point whose distance from the center is less than or equal to the radius is a point of the solid sphere .A sphere can also be considered as a solid obtained on rotating a circle about its diameter.

Volume of a sphere: The volume of a sphere (v) of radius r is given by v = 4/3 ∏r3 cubic units .Let us take an example how to volume of sphere.

Find the sphere volume  of radius 7 cm We know that formula for volume of a sphere of radius r is given by v = 4/3 ∏r3  cubic units here r = 7 cm  therefore v = 4/3 x 22/7 x 7 x 7 x 7 cm3 another example of volume of a sphere.

Calculate volume of a sphere whose surface area is 154 square cm. Let the radius of the sphere be r cm .then ,
Surface area = 154 cm2 => 4∏r2  = 154 => 4 x 22/7 x r2 = 154 => r2 = 154 x 7 / 4 x 22 = 49/4 => r = 7/2 cm , so let v be the volume of sphere .
We will use volume of a sphere equation

That is , v= 4/3∏r3   = 4/3 x 22/7 x 7/2 x 7/2 x 7/2 cm3 = 179.66cm3.

Let us take more example volume of a sphere. A sphere of diameter 6 cm is draped in a right circular cylinder vessel partly filled with water .The diameter of the cylindrical vessel is 12 cm. If the sphere is completely submerged in water, by how much will the level of water rise in the cylindrical vessel?

We have radius of sphere = 3cm, volume of a sphere formula = 4/3 ∏ r3  cm3 = 4/3 ∏ (3)³ cm³ = 36cm³, radius of cylindrical vessel = 6cm.
Suppose of water level rises by h cm and radius 6 cm = (∏x 62 x h) cm 3 = 36∏h cm3 , clearly volume of the water displaced by the sphere is equal to the volume of  the  sphere =36∏h = > h=1 cm , hence water level rises by 1 cm