Rational Numbers

Rational Numbers


What is a Rational Number?

 

The definition of a rational number is “A number which can be expressed in the form of p/q or fraction where p and q are integers and q ≠ 0 is a Rational Number.

 

 

Examples of Rational Numbers

 

All the integers and decimal numberes can be expressed in the form of p/q or fraction of two integers. For example 0 = 0/1, 1 = 2/2, -2 = -4/2, 0.5 = ½, 1.25 = 5/4, 0.3333… = 1/3 etc. Therefore, 0, 1, -2, 0.5, 1.25, 0.3333… etc. are some examples of rational numbers



Rational numbers include all the integers on a number line along with the fractions between integers. Look at the number line given below.


Rational Number Line

The set of rational numbers is denoted by Q.

Q = {… -1, …, -¾, …, -½, …, -¼, …, 0, …, ¼, …, ½, …, ¾, …, 1, …}

 

Note:

-      There are infinitely many rational numbers between any two integers.

-      There are infinitely many rational numbers between any two rational numbers.

Integers

Integers


What is Integer?

 

The definition of integer is “Any counting number whether it is negative(-), positive(+) or zero(0) is an integer.”

 

The set of integers is an infinite set from both the sides and it is denoted by Z.

Z = {…… -5, -4, - 3, -2, -1, 0, 1, 2, 3, 4, 5 ......}

Prime or Composite?

Prime or Composite


What are Prime and Composite Numbers?

 

Answer:

A number which has only two factors is called a Prime Number and a number having more than two factors is called a Composite Number.

For example,

2 has only two factors 1 and 2, so 2 is a prime number.

3 has only two factors 1 and 3, so 3 is a prime number.

5 has only two factors 1 and 5, so 5 is a prime number.

etc. But,

4 has three factors 1, 2, and 4, so 4 is a composite number.

6 has four factors 1, 2, 3, and 6, so 6 is a composite number.

8 has four factors 1, 2, 4, and 8, so 8 is a composite number.

etc.

Composite Number | Questions and Answers

Composite Number | Questions and Answers


Questions and Answers

 

What is composite number? Define with examples.

 

Answer: The definition of composite number: "A number which has more than two factors is called a composite number." Or, a number having more factors other than 1 and the number itself is called a composite number.

 

For example:

4 has a factor 2 other than 1 and 4, so 4 is a composite number.

6 has factors 2 and 3 other than 1 and 6, so 6 is a composite number.

8 has factors 2 and 4 other than 1 and 8, so 8 is a composite number.

9 has a factor 3 other than 1 and 3, so 9 is a composite number.

10 has factors 2 and 5 other than 1 and 10, so 10 is a composite number.

12 has factors 2, 3, 4, and 6 other than 1 and 12, so 12 is a composite number.

14 has factors 2 and 7 other than 1 and 14, so 14 is a composite number.

15 has factors 3 and 5 other than 1 and 15, so 15 is a composite number.

etc.

Prime Number | Questions and Answers

Prime Number Questions and Answers


Questions and Answers

 

Define prime number.

 

Answer: The definition of prime number: “A number which has only two factors is called a prime number. The two factors are 1 and the number itself.”

 

For example:

2 has its factors 1 and the number 2 itself. So, 2 is a prime number.

3 has its factors 1 and the number 3 itself. So, 3 is a prime number.

5 has its factors 1 and the number 5 itself. So, 5 is a prime number.

7 has its factors 1 and the number 7 itself. So, 7 is a prime number.

11 has its factors 1 and the number 11 itself. So, 11 is a prime number.

13 has its factors 1 and the number 13 itself. So, 13 is a prime number.

17 has its factors 1 and the number 17 itself. So, 17 is a prime number.

19 has its factors 1 and the number 19 itself. So, 19 is a prime number.

etc.

Transformation: Enlargement And Reduction

Enlargement And Reduction


Enlargement (or reduction) is a transformation in which the size of an object is changed without changing its original shape. If the size of the object increase, we call it an enlargement and if the size of an object decrease, we call it a reduction.

The enlargement is made with the help of a fixed point called centre of enlargement and by the fixed ratio called scale factor i.e. the ratio of the corresponding sides of the image and object. 


As enlargement changes the size of the object, it will have no real sense in case of a point.

Transformation: Translation

Transformation: Translation


The Translation is a transformation in which each point of the given object is displaced through definite distance and direction. The displacement is defined by a translation vector (a, b).


Translation: Examples


The following are the properties of translation:


1.   The object and the image under the translation are congruent.

2.   The lines joining any point of the object with its corresponding image are parallel and equal.

Transformation: Rotation

Transformation: Rotation


Rotation is a transformation in which each point on the object is rotated through an angle about a fixed point. The fixed point is called the centre of rotation and the angle is called the angle of rotation.

There are two types of rotations on the basis of directions:


1. Anticlockwise or Positive (+) rotation

2. Clockwise or Negative (–) rotation

Transformation: Reflection

Transformation: Reflection


Reflection is a transformation that has a mirror image about a line as a two-way mirror. A line that plays the role of a two-way mirror to give the image of the given objects is called the axis of reflection.


Reflection : Examples


The following are the properties of reflection:


a.   The object and the image under the reflection are congruent.

b.   The object and the image will be at equal distances from the axis of reflection.

c.   The line joining the object and its image will be perpendicular to the axis of reflection.