The Process of Division of Decimal Numbers

The number systems are very interesting to study. Various arithmetic operations can be carried out on them. Multiplication is one of them. The reverse of this operation is division. Division is bit more different from addition and subtraction. The division of natural and whole numbers are almost similar. Any number divided by ‘0’ gives infinity. Infinity is a number which is very large and not defined. Zero on dividing by zero also gives infinity. This question was first raised by the great Indian mathematician Ramanujam.  Zero divided by any number gives zero. Any number divided by the number one gives the same number. This is the same with multiplication. Any number multiplied with one gives the same number. But any number multiplied by zero gives zero.

The process of dividing decimals with whole numbers is similar to the division of two whole numbers. The simple difference is in the placement of the decimal point. This is very important as the placing of decimal point can change the whole number. So, one has to be very careful about this. There can be dividing decimals problems in mathematics and have to be solved carefully. The decimals division is very much similar to the division of natural numbers or the whole numbers. The placement of the decimal point in the final answer is very crucial. Many dividing decimals examples can be used to explain the concept. The following steps to dividing decimals have to be followed to arrive at the final answer.

These are very easy steps.
If a decimal number is divided by a whole number, the division is carried out as usual. The digits present after the decimal point are noted and the decimal point is placed at the same point in the final answer. If the division is carried out between two decimal numbers then the decimal numbers are first converted into whole numbers by multiplying by 10, 100 or thousand and so on depending on how many digits are present after decimal point in the denominator. If the numbers to be divided are 8.68 and 1.2, then the numbers are multiplied by 10. The numbers now become 86.8 and 12. Now the division process is carried out. First ‘868’ is divided by 12. In the final answer decimal point is placed one point from the right. This is how the final answer is got.

Set Theory Rules are as follows

(A∪A) = A,  (A∩A) = A (idempotent law)
(A∪B)’ = A’∩B’, (A ∩ B) ′ = A ′ ∪ B ′ (De Morgan’s law)
A∪ B = B ∪ A, A∩B = B∩A (commutative law)
(A ∩ B) ∩ C = A ∩ (B ∩ C), (A ∪ B) ∪ C = A ∪ (B ∪ C) (associative law)
A ∪ φ = A, A ∩ U = A, A ∪ U = U, A ∩ φ = φ (identity law)
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C), A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)   (distributive law)
(A’)’ = A (involution law)

Basic Set Theory Properties are:
A st is inherently unordered. This means that the order of elements in a st does not make a new st. Change in order of elements inside a st is feasible. For example if st A = {a, x, c} then A = {a, c, x} = {x, c, a} = {x, a, c} = {c, a, x} = {c, x, a}.
Each element in a st is distinct. Multiple repetition of an element makes no difference. For example if st A = {2, 3} then A = {2, 2, 3, 3} = {2, 3, 3} = {2, 2, 3} = {2, 2, 2, 3, 3, 3, 3} = {2, 3} and so on.
Above rules and properties of Set Theory Help you to solve Set Theory Questions easily.
Let us see some Set Theory Problems and Solutions now:

Q.1) prove that (A ′ ∩ B) ′ ∩ (A ∪ B) = A
Sol.) lets begin with LHS of the above problem:
LHS = ((A ′) ′ ∪ B ′) ∩ (A ∪ B) (by De Morgan’s law)
= (A∪ B’) ∩ (A ∪ B) (by involution law)
= A∪ (B∩ B’) (distributive law/property)
= A ∪ φ (complement law)
= A (identity law) = RHS
Hence, proved.

Q.2) in a community if 70 % people can speak English and 60% people can speak French then find the number of people who can speak both languages.
Sol.) Number of people who can speak English = n(E) = 70 %
Number of people who can speak French = n(F) = 60%
Number of people who can speak both languages will be n(E∩F) = ?
Total number of people in the community n (E∪F)= 100%
Now n (E∪F)= n(E) + n(F) - n(E∩F)
100=70 + 60 - n(E∩F)
100=130 - n(E∩F)
n(E∩F)=30%

Articles – a, an, the

An article is a major concept of language learning. In English an article is considered as an adjective that provides some valuable information about a noun in a sentence. In many other different languages an article is considered as a part of speech. An article can be classified based on definite and indefinite characteristics. These are called definite article and indefinite article.

Definite Article

A definite article is that which defines that a noun is particular and has something definitely addressed. Definite article is used for nouns that have been already used in a speech or for nouns that are uniquely specified. In the English language, there is only one definite article ‘the’. Definite article ‘the’ is used for both singulars and plural nouns. For example: Give me a Barbie doll. Give me the Barbie doll. In the second sentence, a specific Barbie doll is referred to while in the first sentence not any particular Barbie doll is referred.
1) Ram said to his wife, “Please pack the gifts for kids that I bought yesterday”. (Here Ram is speaking about those particular gifts for kids that he bought the previous day.)
2) The Taj Mahal is made by Mughal Emperor Shah Jahan. (Here, Taj Mahal is a uniquely specified noun.)

Indefinite Article

An indefinite article is used for a noun that is not particular and uniquely specified. It can be a noun that the speaker is mentioning for the first time. For example: Give me a Chota Bheem DVD. Give me the Chota Bheem DVD. Here, in the first sentence the speaker is talking about no particular Chota Bheem DVD while in the second sentence the reference is definite. In the English language, there are two indefinite articles ‘a’ and ‘an’. The indefinite article ‘an’ is used before words that begin with a vowel sound. At times even a word that starts with a consonant but sounds like a vowel. For example: an hour, an MBA, etc. Also the indefinite article ‘a’ is used before words that begin with a consonant sound even if the word begins with a vowel. For example: a European, etc. These were the basics about article in English language.

Introduction for Solving Number Patterns

Solving number patterns is the main topic under the solving number systems. In this lesson we have to study about the concepts of divisors, factors, multiples and prime numbers. Also in this topic we have to discuss about properties of number patterns. Here we have to discuss about Divisors, Factors ,Multiples,Even,Odd numbers,Prime numbers and Composite numbers. These are the main topics in the solving number patterns.

Solving Divisors, Factors and Multiples Number Patterns:

Divisors:

Step 1: Consider the number 96. Divide 96 by 4. When we divide 96 by 4, the remainder is 0.

Step 2: 4 is the divisor of 96. Similarly if we divide 96 by 2, we will get 0 as the remainder.

Step 3: Again 2 is the divisor 96. By applying the same process, 1,2,3,4,6,8,12,16,24,32,48,96 are also divisors.

Step 4: Thus the divisors of 96 are 1,2,3,4,6,8,12,16,24,32,48,96.

Note:

The divisor in the division algorithm is different from this divisor. In division algorithm the divisor indicates the denominator only.

Example:

Find the divisors of (a) 28, (b) 35

Solution:

(a) Solving The divisors of 28 are 1, 2, 4,7,14, and 28

(b) Solving The divisors of 35 are 1, 5, 7 and 35

Solving even and Odd Number Patterns and Prime and Composite Patterns:

The numbers that are divisible by 2 is called as even numbers.
If we divide the even number by 2, the remainder is always zero.

The numbers that are not divisible by 2 is called as odd number.
If we divide an odd number by 2, the remainder is always 1

Now observe the following:

(a) 2 + 2 = 4 ; 2 + 6 = 8 ; 4 + 8 = 12 ; 10 + 20 = 30

(b) 2 × 2 = 4 ; 2 × 6 = 12 ; 4 × 8 = 32 ; 10 × 20 = 200

From the above examples the following are true.

Even number + even number = even number
Even number × even number = even number
Odd number + odd number = even number
Odd number × odd number = odd number
These are the main rules used for solving number patterns.

Prime numbers and Composite number patterns

From the above, find the numbers that have only 2 divisors?

2, 3, 5 and 7 have only two divisors.

What are the numbers which have more than 2 divisors?

4, 6, 8, 9 and 10 have more than 2 divisors.

The natural numbers that have only two divisors are known as the prime numbers.
The natural numbers that have more than two divisors are known as the composite numbers.

Example:

Solving prime numbers between 20 and 30.

Solution:

The numbers between 20 and 30 are

21 22 23 24 25 26 27 28 29

∴ the prime numbers are 23 and 29

Definition of parabola

Definition of parabola:
A curve in a plane having two ends and one vertex is called a parabola. It is a conic section. That means it is a section of a double cone intercepted by a plane. When a double cone is cut by a plane such that the plane is parallel to any one slant length of the cone, the cross section thus obtained is called as such.

However in co-ordinate geometry we do not study parabola problems as sections of double cone. We use the characteristic property of these to define it and use that to find its equation.

It so happens that in the plane of every parabola there is a line (called the directrix) and a point (called the focus) such that for every point on this the distance from the directrix and the distance from the focus are found to be equal. This is the definition most commonly used in co-ordinate geometry to define these and to derive its equation.

Types of parabola:
Primarily these are of two types and they are horizontal and vertical. In simple terms, the horizontal form of this is x as a function of y. Therefore the equation of a horizontal parabola would be x = f(y) type. On the other hand in a vertical form of this the equation is y as a function of x. Therefore it would look like this: y = f(x).

Another way of classifying these would be: positive and negative parabola. For the horizontal form, if it opens towards the positive side of the x axis, it is called the positive para-bola; where as if it opens on the negative side of the x axis, it is called the negative para-bola. Similarly, for the vertical form, if it opens towards the positive y axis it is called a positive para-bola and if it opens towards the negative side of the y axis, it is called a negative form of same. Below are the pictures of the various types of these:

1. Horizontal positive form of the same:

As you can see in the above picture, this one opens to the right.
X = f(y) = p(y-k)^2 + h
2. Horizontal negative form:

The above form opens to the left.
X = f(y) = -p(y-k)^2 + h

3. Vertical positive form of the same:

A vertical positive form opens up (as can be seen in the picture above).
Y = f(x) = a(x-h)^2 + k

4. Vertical negative form:

Lastly the vertical negative form would open down.
Y = f(x) = -a(x-h)^2 + k