The method to represent and work with numbers is known as the number system. A number system is a system of writing to represent numbers. It is the mathematical notation used to represent numbers of a given set by using digits or other symbols. It has arithmetic operations to perform division, multiplication, addition, and subtraction between numbers. Some important number systems are as follows:

• Decimal Number System
• Binary Number System
• Octal Number System
• Hexadecimal Number System

Numbers and Digits 

Numbers are the counts or measurements used in mathematics, numerals are used to define numbers. A numeral can be defined as a symbol used for counting, for instance, there are 55 books in the library, where 56 is the numeral which is a combination of digits 5 and 6. A digit is a single numeral, the combination of digits form numerals. In the decimal number system, there are 10 digits, they are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

What are Naturals Numbers?

In the Number system, Natural numbers are the numbers that start from 1 and count up to Infinity. For example – (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,12, 13, 14, 15, 16, 17, 18, 19, 20, 21..…….. ∞) are Natural Numbers.

What are Rational Numbers?  

A rational number is a number that can be represented in a p/q form such that q is not equal to 0. The set of rational numbers Include positive, negative numbers, and zero and it is denoted by Q. Rational number can also be expressed as a fraction.

When a number is expressed in a p/q form or in fraction form where both the numerator and the denominator part are integers, then the number is known as a rational number.  

Some examples of rational numbers are: 1/2, -2/7, 7/10, -7/10, 14/99

The number “0” is also a rational number, as we can represent it in many forms such as 0/1, 0/2, 0/3, etc. But, 1/0, 2/0, 3/0, etc. are not rational numbers because they give us infinite values.

Note: Rational numbers can also be expressed in decimal form.  

Types of Rational Numbers

There are different types of rational numbers and they are:

• Natural Numbers: All natural numbers are rational numbers because they can be written as in p/q form. Like 2 can be expressed in 2/1 (p/q) form.

Example – 1, 2, 3, 4, 5 …. etc.

• Terminating Decimals: Rational numbers can also be expressed in decimal form because decimal numbers can be represented in p/q form. For example, 1.1 can be written as 1.1 = 11/10. Thus all terminating decimals are rational numbers.

Example – ( 0.45, 0.7120, 0.9778 etc. )

• Non-Terminating Decimals: Non-Terminating decimals having repeated numbers after the decimal point such as 0.2222….., 0,12121212…. are also rational numbers. Since 0.222… can be written as 1/2, therefore it is a rational number. 

Example – ( 0.22222….., 0.121212…..  etc. )

• Fractions: When a number is expressed in a p/q form or in fraction form where both the numerator and the denominator part are integers, then it is a rational number.  

Example – 3/4, 2/7, 7/10, -7/10, 14/99 (all of them are in p/q form) 

• Whole Numbers: All whole numbers are rational numbers because the whole numbers can be expressed in p/q fraction form. 

Example – 0 is a rational number because it can be written as a fin 0/1, 0/-2,… etc.

How to identify rational numbers?

There are some conditions to check whether a number is a rational number or not. They are:

1. Always it is represented in the p/q form, where q≠0. For example – 3/4 , 2/7,  7/10, -7/10, 0/1 etc.

2. A rational number can be further simplified and represented in decimal form. For example – 0.9, -0.875, 3.25, -2.0 etc.

Sample Questions

Question 1. Is 2.2 is a rational number?

Answer:

Yes 2.2 is a rational number because 2.2 can also be written as p/q form that is 22/10.  

Question 2. Identify whether Mixed Fraction 1 5/4 is a Rational Number or Not?

Answer:

The Simplest Form of Mixed fraction 1 5/4 is 9/4. It is in p/q form. Thus, 9/4 is a Rational Number.

Question 3. How to Identify a Rational Number?

Answer: 

When a number is expressed in the p/q form where p, q are integers and q is non-zero then it called as a Rational Number.

Question 4. Is 12 a Rational Number?

Answer:

Yes 13 is a Rational Number because it can be expressed in 12/1 that is p/q form.

Question 5. Is 0 a rational number?

Answer:

Yes, 0 is a rational number because it is an integer and It can be written in p/q form such as 0/1, 0/2, where b is a non-zero integer. Hence, 0 is a rational number.

Question 6. Is 1 a Rational Number?

Answer:

Yes, 1 is a Rational Number because it can be expressed in 1/1 that is p/q form.

Question 7. Which of the following rational numbers are positive?

(i) (-4)/3         (ii) 9/8          (iii) (-14)/(-9)       (iv) (-21)/13

Answer:

9/8 is the only positive rational number.