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Real Number System
Introduction to Number System | Mathematics (Maths) Class 9

Classification of Real Numbers


Numbers

In real life, we use Hindu Arabic numerals - a system which consists of the symbols 0 to 9.

This system of reading and writing numerals is called, “Base ten system” or “Decimal number system”.


Natural Numbers
Counting numbers are called natural numbers. These numbers start with the smallest number 1 and go on without end. The set of all natural numbers is denoted by the symbol ‘N’.
N  =  { 1, 2, 3, 4, 5, .......} is the set of all natural numbers.


Whole Numbers
Natural numbers together with zero (0) are called whole numbers. These numbers start with the smallest number 0 and go on without end.
The set of all whole numbers is denoted by the symbol ‘W’.
W  =  { 0, 1, 2, 3, 4, 5, .......} is the set of all whole numbers.


Integers
The whole numbers and negative numbers together are called integers.
The set of all integers is denoted by Z.
Z  =  {... - 2, - 1, 0, 1, 2, ...,} is the set of all integers


Fractions
A fraction is a part or parts of a whole.

Fig: FractionsFig: Fractions

In a fraction, the number above the line is called the numerator and the number below the line is called the denominator.

Decimal Numbers
A number in which we have "point" is called a decimal number. 
A decimal number has two parts namely an integral part and a decimal part.
Examples:
1)  Let us consider the decimal number 0.6
0.6 can be written as 0 + 0.6 
Here, integral part = 0 and decimal part = 6
2)  Let us consider the decimal number 7.2
7.2 can be written as 7 + 0.2 
Here, integral part = 7 and decimal part = 2
In a decimal number, the digits to the left of the decimal point is the integral part. 
The digits to the right of the decimal point are the decimal part. 
The value of all the decimal parts is less than 1.  

Rational Numbers

Introduction to Number System | Mathematics (Maths) Class 9

Both "p" and "q" must be integers and q≠ 0

So, any number in the form of a fraction can be treated as a rational number. 

Examples: 
5, 2.3, 0.02, 5/6
Because all these numbers can be written as fractions.  
5 = 5/1 
2.3 = 23/10
0.02 = 2/100 = 1/50
5/6 (This is already a fraction)
Apart from the above examples, sometimes we will have recurring decimals like 1.262626..........
1.262626........ is a non-terminating recurring decimal. 
All these recurring decimals can be converted into fractions and they are also rational numbers.

Important note:
All the fractions and decimal numbers will come under this category. 
Hence, all the fractions and decimal numbers to be considered as rational numbers.

Irrational Numbers
A number that can not be converted into a fraction is called irrational numbers.
Examples: 

Introduction to Number System | Mathematics (Maths) Class 9

All the above non-terminating numbers can not be converted into fractions because they do not have repeated patterns.
When we are trying to find the square root of a number that is not a perfect square, we get this non-repeating and non-terminating decimal.
And these non-recurring decimals can never be converted into fractions and they are called irrational numbers. 

Decimal Expansions of Real Numbers 

There are three types of decimal expansions of real numbers.
1. Terminating
2. Non-Terminating but Repeating
3. Non-Terminating and Non-Repeating

1. Terminating: The remainder becomes zero.
Let us take examples to know it.
Example 1:
Expansion of 7/4
Introduction to Number System | Mathematics (Maths) Class 9After performing some steps, we get the remainder as 0.
The remainder is 0, and the decimal expansion ends at 5. So it means the expansion is terminating.
Example 2: 
Expansion of 32/5
Introduction to Number System | Mathematics (Maths) Class 9Again we found that after some steps the remainder becomes zero with the decimal expansion as 6.4.
Similarly, the expansion of 32/5 is terminating.
Example 3: 
Expansion of 578/25
Introduction to Number System | Mathematics (Maths) Class 9Decimal expansion of 578/25 = 23.12
Therefore, the expansion of 578/52 is terminating.

2. The remainder never becomes zero: There are two cases:
(a) Non-Terminating but Repeating.
(b) Non-Terminating and Non-Repeating
Example: Expansion of 10/3
Introduction to Number System | Mathematics (Maths) Class 9Decimal expansion of 10/3 = 3.333333... Introduction to Number System | Mathematics (Maths) Class 9 
The expansion of 10/3 does not end that is not terminating, and number 3 is repeating, so it is a non-terminating but recurring expansion.
Example: Expansion of 1/7
Introduction to Number System | Mathematics (Maths) Class 9Decimal expansion of 1/7 = 1.142857...
The expansion of 1/7 does not end that is not terminating, and numbers 142857 are repeating, so it is a non terminating but recurring expansion.
1/7 = Introduction to Number System | Mathematics (Maths) Class 9
Example:
π is a non-terminating, non-repeating decimal. π = 3.141 592 653 589 793 238 462 643 383 279 ...
e is a non-terminating, non-repeating decimal. e = 2.718 281 828 459 045 235 360 287 471 352 ...

The document Introduction to Number System | Mathematics (Maths) Class 9 is a part of the Class 9 Course Mathematics (Maths) Class 9.
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FAQs on Introduction to Number System - Mathematics (Maths) Class 9

1. What is the real number system?
Ans. The real number system is a set of numbers that includes all rational and irrational numbers. It consists of numbers that can be expressed as terminating or repeating decimals, as well as numbers that cannot be expressed in this form, such as square roots of non-perfect squares.
2. What are decimal expansions of real numbers?
Ans. Decimal expansions of real numbers refer to the representation of real numbers in decimal form. It involves expressing a real number as a decimal, either terminating or repeating. For example, the decimal expansion of the rational number 1/4 is 0.25, while the decimal expansion of the irrational number √2 is non-terminating and non-repeating.
3. How are real numbers related to the number system?
Ans. Real numbers are a fundamental part of the number system. They encompass all rational and irrational numbers, providing a complete representation of quantities in mathematics. The real number system includes integers, fractions, decimals, and irrational numbers, allowing for precise calculations and measurements in various mathematical contexts.
4. What is the significance of understanding the real number system?
Ans. Understanding the real number system is essential in various fields of mathematics and everyday life. It enables accurate measurement and comparison of quantities, facilitates solving equations and inequalities, and forms the foundation for calculus, analysis, and other advanced mathematical concepts. Real numbers also find applications in physics, engineering, finance, and many other disciplines.
5. Can all real numbers be expressed as a decimal expansion?
Ans. No, not all real numbers can be expressed as a decimal expansion. Rational numbers can always be represented as either a terminating or repeating decimal. However, irrational numbers, such as the square root of 2 or pi, have non-terminating and non-repeating decimal expansions. These numbers cannot be expressed exactly in decimal form and require approximation.
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