Introduction to Rational Numbers and Irrational Numbers Class 10 Notes | EduRev

Mathematics (Maths) Class 10

Class 10 : Introduction to Rational Numbers and Irrational Numbers Class 10 Notes | EduRev

The document Introduction to Rational Numbers and Irrational Numbers Class 10 Notes | EduRev is a part of the Class 10 Course Mathematics (Maths) Class 10.
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Real numbers are simply the combination of rational and irrational numbers, in the number system. In general, all the arithmetic operations can be performed on these numbers and they can be represented in the number line, also. At the same time, the imaginary numbers are the un-real numbers, which cannot be expressed in the number line and is commonly used to represent a complex number. The concepts related to real numerals are explained here in detail, along with examples and practice questions. The key concept in the number system is included in this article.

Examples

23

-12

6.99

5/2

π(3.14)

Real Numbers Definition

Real numbers can be defined as the union of both the rational and irrational numbers. They can be both positive or negative and are denoted by the symbol “R”. All the natural numbers, decimals and fractions come under this category. See the figure, given below, which shows the classification of real numerals.
Introduction to Rational Numbers and Irrational Numbers Class 10 Notes | EduRev

Set of Real Numbers

The set of real numbers consist of different categories, such as natural and whole numbers, integers, rational and irrational numbers. In the table given below, all these numbers are defined with examples.

Category

Definition

Example

Natural Numbers

Contain all counting numbers which start from 1.


N = {1,2,3,4,……}

All numbers such as 1, 2, 3, 4,5,6,…..…

Whole Numbers

Collection of zero and natural number.


W = {0,1,2,3,…..}

All numbers including 0 such as 0, 1, 2, 3, 4,5,6,…..…

Integers

The collective result of whole numbers and negative of all natural numbers.

Includes: -infinity (-∞),……..-4, -3, -2, -1, 0, 1, 2, 3, 4, ……+infinity (+∞)

Rational Numbers

Numbers that can be written in the form of p/q, where q≠0.

Examples of rational numbers are ½, 5/4 and 12/6 etc.

Irrational Numbers

All the numbers which are not rational and cannot be written in the form of p/q.

Irrational numbers are non-terminating and non-repeating in nature like √2

Real Numbers Chart

The chart for the set of real numerals including all the types are given below:
Introduction to Rational Numbers and Irrational Numbers Class 10 Notes | EduRev

Properties of Real Numbers
There are four main properties which include commutative property, associative property, distributive property and identity property. Consider “m, n and r” are three real numbers. Then the above properties can be described using m, n, and r as shown below:
Commutative Property
If m and n are the numbers, then the general form will be m + n = n + m for addition and m.n = n.m for multiplication.

  • Addition: m + n = n + m. For example, 5 + 3 = 3 + 5, 2 + 4 = 4 + 2
  • Multiplication: m × n = n × m. For example, 5 × 3 = 3 × 5, 2 × 4 = 4 × 2

Associative Property
If m, n and r are the numbers. The general form will be m + (n + r) = (m + n) + r for addition(mn) r = m (nr) for multiplication.

  • Addition: The general form will be m + (n + r) = (m + n) + r. An example of additive associative property is 10 + (3 + 2) = (10 + 3) + 2.
  • Multiplication: (mn) r = m (nr). An example of a multiplicative associative property is (2 × 3) 4 = 2 (3 × 4).

Distributive Property
For three numbers m, n, and r, which are real in nature, the distributive property is represented as:
m (n + r) = mn + mr and (m + n) r = mr + nr.

  • Example of distributive property is: 5(2 + 3) = 5 × 2 + 5 × 3. Here, both sides will yield 25.

Identity Property

There are additive and multiplicative identities.

  • For addition: m + 0 = m. (0 is the additive identity)
  • For multiplication: m × 1 = 1 × m = m. (1 is the multiplicative identity)
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