Relatively prime numbers

Mathematics is the study of numbers. In branch of mathematics, we deal with different types of numbers which are grouped together.
 For example: - Odd numbers, even numbers, prime numbers, composite numbers, etc.

A prime number is one which has factors one and itself.
For example: - 2 is a prime number as it has two factors only that is 1 and itself. Similarly numbers like 3, 5, 7, and 11 are also its examples.

Now we talk about Relatively Prime Numbers. What is Relatively Prime - Relatively Prime Definition says they are those numbers which have the greatest common divisor as one only.

They are also called co prime-numbers. They are those numbers which share no common factors except 1. Let us look at some examples to understand this concept better.

How about number 7 and 15? If we make factors of 7, we see that number 7 is divisible by 1 and 7 only. For number 15, we see that it is divisible by 1, 3, 5 and 15. If we look at the factors of both the numbers, we see that the common factors between them are 1 only.

Hence they can be termed as relatively-prime-numbers. Similarly if we have numbers 8 and 22, we see that number 8 has factors 1, 2, 4 and 8 and number 22 has factors 1, 2 and 11.

If we look at the factors of both the numbers, we see that apart from number 1, 8 and 22 has a common factor 2 as well. Therefore we cannot call them as relatively-prime-numbers.

If we look at the Relatively Prime Numbers List from 1 to 10, we can make groups like 2 and 3, 2 and 5, 2 and 7, 2 and 9, 3 and 4, 3 and 5, 3 and 7, 3 and 8, 3 and 10, 4 and 5, 4 and 7, 4 and 9, 5 and 6, 5 and 7, 5 and 8, 5 and 9, 6 and 7, 7 and 8, 7 and 9, 7 and 10, 8 and 9 and 9 and 10.
They are forms a group of Relatively Prime nos. till 10.

We cannot include 2 and 4 in that group as they have two common factors. Similarly number 6 and 10 are also not in that group as they are divisible by 1 and 2.

Dividing Decimal

Steps for dividing decimals. Let us say we have 31.773 ÷ 5.1. Let us set this up by using long division. We have 31.773 on the inside, being divided by 5.1. One this we have to remember that we have to get rid of the decimal in the outside number. Here we will be dividing decimals with decimals So, in math dividing decimals, we have outside number of 5.1, here we need to get rid of this decimal in the outside number.

Basically we have to push this decimal point as to the right as possible. In this particular example we can only push it over one space to the right. And whatever we do to the outside number same is applicable to the inside number as well. Let us learn how to divide decimals with decimals. So since we moved our outside decimal number on space to the right, we have to do the same thing to the inside decimal. Our 5.1 becomes 51 and our 31.773 becomes 317.73. So 31.773 divided by 5.1 is exactly the same as 317.73 divided by 51.

Now because we have got the rid of the decimal in the outside number, we can just divide this using normal long division. 51 goes into 3 in 0 times, as 51 is bigger than 3, moving to the next digit. 51 goes into 31, in zeros times, again since 51 is bigger. Moving on to the next digit, 51 go into 317 how many times? Well we know that 50 goes in 300, so we can say that 51 goes into 317 about 6 times. Multiplying 6 times 1 gives 6, and 6 times 5 gives is 30. Now subtracting the two numbers we get 11, now bringing down the 7. So it is 117 now, 51 goes into 117 about 2 times. Two times one is 2, and 2 times 5 is 10. So we get 102.

Subtracting 102 from 117 remains with 15. Bringing down 3 gives us 153. Before we go further to solve this problem, we would like to place the decimal point. The decimal point moves straight up, that goes directly between the two numbers that is, 6.2. so now how many time does 51 go into 153? So we know that 50 go 3 times to get 150, so 51 may also similar ways. So 3 times 1 is 3 and 3 times 5 is 15, So 51 times 3 gives us 153.and subtracting 153 from 153 , our remainder becomes 0.  This means that we can stop, so we did took the help with dividing decimals by shifting the decimal point and made division simple and easy.

Rules of Narration

Narration is one of the most important concepts in English grammar. While doing narration, certain rules needs to be followed which are termed as the rules of narration. Let’s have a look at some of the most important rules of narration.
Changing from Direct Speech to Indirect Speech:
When the exact meaning of a speech that is direct speech is conveyed in reported or indirect speech, no inverted commas should be used.
Direct Speech: Hary said, “I always buy Newborn Baby Essentials from online baby store as it is fast and easy.”
InDirect Speech: Hary said that he always buy newborn baby essentials from online baby store as that is fast and easy.

Changing Tenses in Narration:

1. If the reporting verb is in Present tense or future tense, the tense of the verb is not changed.
Direct Speech: The girl says, “Barbie doll is my favorite toy.”
InDirect Speech: The girl says that Barbie doll is her favorite toy.
2. If the reporting verb is in simple present tense, the tense of the verb is changed into simple past tense in the indirect speech.
Direct Speech: Rima said, “I buy Johnson & Johnson products for my baby.”
InDirect Speech: Rima said that she bought Johnson & Johnson products for her baby.
3. If the reporting verb is in present continuous tense the tense of the verb is changed into past continuous tense in the indirect speech.
Direct Speech: Hary said, “Geeta is dancing a folk dance.”
InDirect Speech: Hary said that Geeta was dancing a folk dance.
4. If the reporting verb is in present perfect tense, the tense of the verb is changed into past perfect tense in the indirect speech.
Direct Speech: Maria said, “I have a car.”
InDirect Speech: Maria said that she had a car.
5. If the reporting verb is in simple past, the tense of the verb is changed into past perfect tense in the indirect speech.
Direct Speech: The lady said, “Many people bought baby products from online stores.”
InDirect Speech: The lady said that many people had bought baby products from online stores.
These are some of the most important rules of narration.

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.