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The representation of rational numbers is similar to the representation of negative or positive numbers on a number line. Rational numbers are integers p and q expressed in p /q where q> 0.
- Rational numbers can be positive, negative, or zero.
- The representation of rational numbers in the number line depends on the kind of fraction.
- Any rational numbers can be displayed on the number line.
- The centre of the number line is called the origin (O).
- Positive numbers are shown to the right of zero (so-called origin).
- Negative numbers are positioned to the left of the origin.
- A number line can be used to represent any infinite number of rational numbers.
| Table of Content |
Key Terms: Representation of Rational Numbers, Rational Numbers, Numbers, Number line, Fraction, Positive Numbers, Negative Numbers, Proper Fraction, Improper Rational, Integers, Whole Numbers
Representation of Rational Numbers on a Number Line
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To represent a rational number on a number line, we need to draw a number line with point O as the centre of the number line. The Centre of the number line is also called the reference line.
- The numbers that lie on the left side of O are negative numbers.
- Numbers lying on the right-hand side of O are positive numbers.
- A similar number line can be used to represent negative numbers.
- In positive numbers, the direction of movement is the right-hand side of the origin.
- In negative numbers, the direction of movement is the left-hand side of the origin.
Example of Representation of Rational Numbers on a Number LineExample: Let’s consider ¼ To draw a rational number ¼,
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Representation of Rational Numbers on a Number Line
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Rational Number
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Rational Number are number that can be expressed in terms of x/y where y ≠ 0. These numbers can be represented in terms of decimal form. The fraction is divided into two categories namely numerator or denominator.
- Real numbers are broadly divided into two types, which are rational and irrational numbers.
- All of these numbers can be represented on the number line.
- It is derived from the word ratio.
- A number line is a straight line where all real positive and negative numbers are arranged.
- In addition, the point of reference in the number line is zero.
Example of Rational NumberExample: ¼, 3/7, etc. are all rational numbers. |
Representation of Rational Numbers on a Number Line
Proper Fraction
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A fraction is called a proper fraction if its number is less than a denominator. The value of the appropriate fraction is always less than 1. In all these fractions, the numerator is less than the denominator.
- Its value lie in the range between 0 and 1.
Example of Proper FractionExample: ¼ and 3/9 are proper fraction. |
Improper Fraction
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Improper fraction is the type of fraction where the number is greater or equal to the denominator. The value of the appropriate fraction is always greater than 1.
- In all these fractions, the numerator is greater than the denominator.
Example of Improper FractionExample: 3/1, 5/2, -7/3, etc are improper fraction |
Read More:
| Class 10 Mathematics Related Concept | ||
|---|---|---|
| Degree of Polynomial | Operations on Rational Numbers | Prime Numbers |
| Rational Numbers NCERT Solutions | Degree of polynomial | Number System |
Things to Remember
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- Representation of rational numbers on the number line is a simple process.
- The process is plotting a fraction on the number line.
- If we look at counting numbers, they are all rational numbers with denominator 1.
- Every positive number lies to the right of O, and every negative number lies to the left of O.
- Rational numbers are closed under addition, subtraction, and multiplication functions.
- A proper fraction's upper value is a number that is less than a denominator.
- Improper fractions have a denominator value less than the numerator value.
- When rational numbers are arranged in a number line, the positive integer numbers are placed between 0 and 1.
- The improper fractions must be first converted to a mixed fraction and then represented on the number line.
Sample Questions
Ques. Represent -½ and ½ on the number line. (3 marks)
Ans. To draw -½ and ½ on the number line, we need to follow these steps-
- Draw a point A and A’ on the number line.
- Point A will be marked at 1 and point A’ will be marked at -1.
- Between AA’, point O will be the center of it.
- Now, OA and OA’ will be at an equal distance.
- Now, again divide OA into two parts - OP and PA. Similarly, divide OA’ as OP’ and P’A’.
- As we can see, OP = OP’, which means OP = ½, OP’ = -½.
Ques. Represent 3/4 and -5/6 on the number line. (3 marks)
Ans. To draw 3/4 and -5/6 on the number line,
Let us first draw a line AA’ with point O as a center of it.
- Draw OA on the right side of point O and OA’ on the left side of point O.
- Now, divide OA into three equal parts. i.e. OP will be the point that shows ? on the number line.
- Similarly, divide OA’ into three equal parts, and OP’ will be marked as -? on the left side of the number line.
- So, OP=OP’, both are at equal distance.
- Therefore, 3/4 and -5/6 are drawn in the number line below.
Ques. Represent 5/9 on the number line. (3 marks)
Ans. As positive integers lie between 0 to 1. That means, first, we need to draw a number line.
- The origin of the number line will be marked from 0.
- Now, from 0 to 1, divide the segment into nine equal parts which are equal to the denominator value.
- Then, mark the 5th value on the numerator, i.e. 5/9.
Ques. Represent 1/11 on the number line. (3 marks)
Ans. As positive integers lie between 0 to 1. That means, first, we need to draw a number line.
- The origin of the number line will be marked from 0.
- Now, from 0 to 1, divide the segment into eleven equal parts which are equal to the denominator value.
- Then, mark the 1st value on the numerator, i.e. 1/11 on the number line.
Ques. Represent 2/6 on the number line. (3 marks)
Ans. To draw 2/6 on the number line, follow these steps mentioned below -
- Draw a line segment from point 0 to 1.
- From point 0 to 1, divide it into six equal parts, equal to the denominator value.
- Now, mark the 2nd value on the line segment as 2/6, which lies between 0 to 1.
Ques. The square root of a perfect square is an irrational number. Is this statement true or false. (2 Marks)
Ans. The above statement is false with respect to the irrational numbers. The correct fact is that the square root of the perfect square is a rational number, for instance, √64 = 8, √36 = 6. Irrational numbers are the square roots of those numbers that are not perfect squares, for instance, √2 and √6 respectively.
Ques. What is the difference between rational and irrational numbers. (5 Marks)
Ans. The difference is as follows:
| Rational numbers | Irrational numbers |
|---|---|
| Rational numbers are those which can be expressed as a ratio of two numbers p and q where p and q are any integer and q is not equal to zero is called rational numbers. | Irrational numbers are those which cannot be expressed as a ratio of two numbers p and q where p and q are any integer and q is not equal to zero is called rational numbers. |
| These numbers are finite or recurring. | These numbers are non-repeating and non-recurring. |
| In this, both the numerator and denominator are integral values in which the denominator is equal to zero. | These numbers cannot be written in fractional form. So, there is no involvement of numerator and denominator. |
| Rational numbers include perfect squares such as 4, 9, 16, 25, 36 etc and so on. | Irrational numbers include surds instead of perfect squares such as √2, √6, √3, etc and so on. |
| Example - 3/2 = 1.5, 3.7676, 6, 9.31, 0.6666, etc and so on. | Example - √5, √11, e (Euler's number), π (pi), etc and so on. |
Ques. Write the irrational numbers 4√6, √3, and 3√7 in ascending and descending orders. (3 Marks)
Ans. Given irrational numbers 4√6, √3, and 3√7,
The order of the irrational numbers is 4, 2, 3
The LCM of (4, 2, 3) = 12.
Change 4√6 = (4*3) √63 = 12√216
√3 = (2*6) √36 = 12√729
3√7 = (3*4) √74 = 12√2401
216<729<2401
Therefore, ascending order is 4√6, √3, and 3√7, and descending order is 3√7, √3, 4√6.
Ques. Find an irrational number between √6 and 6. (2 marks)
Ans. Given √6 and 6,
A real number between √6 and 6 = ½ √6 + 1
But, 1 is a rational number and ½ √6 is an irrational number. The sum of a rational number and an irrational number always results in an irrational number.
So, ½ √6 + 1 is the irrational number that lies between √6 and 6.
Ques. Find out the total number of rational numbers between 2/17 and 5/17. (2 marks)
Ans. Here the denominators are equal, i.e., 17.
Now we can consider 2 and 5 as integers and 5 are greater than 2.
The rational numbers are, 3/17, 4/17
Ques. Find out the rational numbers between 4/27 and 9/27. (2 marks)
Ans. The denominators are the same, 27. Thus, the numerators are considered. There are 4 numbers between 4 and 9, thus there are 4 rational numbers between these two fractions, and they are:
5/27, 6/27, 7/27, and 8/27.
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