Real Numbers

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. Some of the examples of real numbers are 23, -12, 6.99, 5/2, π, and so on. In this article, we are going to discuss the definition of real numbers, the properties of real numbers and the examples of real number with complete explanations.

Table of contents:

• Definition
• Set of real numbers
• Chart
• Properties of Real Numbers
  • Commutative
  • Associative
  • Distributive
  • Identity
• Solved Examples
• Practice Questions
• FAQs

Real Numbers Definition

Real numbers can be defined as the union of both 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.

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Set of Real Numbers

The set of real numbers consists of different categories, such as natural and whole numbers, integers, rational and irrational numbers. In the table given below, all the real numbers formulas (i.e.) the representation of the classification of real numbers are defined with examples.

Real Numbers Chart

The chart for the set of real numerals including all the types are given below:

Properties of Real Numbers

The following are the four main properties of real numbers:

• Commutative property
• Associative property
• Distributive property
• 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)

Solved Examples

Example 1:

Find five rational numbers between 1/2 and 3/5.

Solution:

We shall make the denominator same for both the given rational number

(1 × 5)/(2 × 5) = 5/10  and (3 × 2)/(5 × 2) = 6/10

Now, multiply both the numerator and denominator of both the rational number by 6, we have

(5 × 6)/(10 × 6) = 30/60  and (6 × 6)/(10 × 6) = 36/60

Five rational numbers between 1/2 = 30/60 and 3/5 = 36/60 are

31/60, 32/60, 33/60, 34/60, 35/60.  

Example 2:

Write the decimal equivalent of the following:

(i) 1/4  (ii) 5/8 (iii) 3/2

Solution:

(i) 1/4 = (1 × 25)/(4 × 25) = 25/100 = 0.25

(ii) 5/8 = (5 × 125)/(8 × 125) = 625/1000 = 0.625

(iii) 3/2 = (3 × 5)/(2 × 5) = 15/10 = 1.5

Example 3:

What should be multiplied to 1.25 to get the answer 1?

Solution: 1.25 = 125/100 

Now if we multiply this by 100/125, we get

125/100 × 100/125 = 1

Practice Questions

1. Which is the smallest composite number?
2. Prove that any positive odd integer is of the form 6x + 1, 6x + 3, or 6x + 5.
3. Evaluate 2 + 3 × 6 – 5.
4. What is the product of a non-zero rational number and an irrational number?
5. Can every positive integer be represented as 4x + 2 (where x is an integer)?

Real Numbers Class 9 and 10

In real numbers Class 9, the common concepts introduced include representing real numbers on a number line, operations on real numbers, properties of real numbers, and the law of exponents for real numbers. In Class 10, some advanced concepts related to real numbers are included. Apart from what are real numbers, students will also learn about the real numbers formulas and concepts such as Euclid’s Division Lemma, Euclid’s Division Algorithm and the fundamental theorem of arithmetic in class 10.