Real numbers are a mixture of rational, irrational, integers, non-integers, whole, negative, zero, and natural numbers including positive and negative. In this article, you will learn about real numbers in mathematics, their properties, and solved & unsolved examples.

What are Real Numbers?

Real numbers are a fundamental concept in mathematics, encompassing both rational and irrational numbers. They represent a continuum of values that can be used to measure quantities in the real world, such as distance, time, temperature, and many other measurable attributes.

Types of Real Numbers

• Rational Numbers: These are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. Examples include:
  • Integers: …, -3, -2, -1, 0, 1, 2, 3, …
  • Fractions: 1/2, 3/4, -7/8, …
• Irrational Numbers: These cannot be expressed as a simple fraction. They have non-repeating, non-terminating decimal expansions. Examples include:
  • Square roots of non-perfect squares: √2, √3
  • Mathematical constants: π (pi), e (Euler’s number)

Properties of Real Numbers

Some properties of real numbers are:

• Order: Real numbers can be ordered on a number line. For any two real numbers aaa and bbb, one of the following is true: a<b, a=b, or a>b.
• Density: Between any two distinct real numbers, there is always another real number. This property implies that the real numbers form a dense set.
• Completeness: Every non-empty set of real numbers that is bounded above has a least upper bound (also known as the supremum).

Worksheet: Real Numbers (Solved)

Example 1: Identify real numbers from the following:

√4, -3, 2/5, √(-9), π, 0

Solution:

√4 = 2 (real, rational)

-3 (real, integer)

2/5 = 0.4 (real, rational)

√(-9) (not real, it’s imaginary)

π (real, irrational)

0 (real, integer as well as rational)

Real numbers: √4, -3, 2/5, π, 0

Example 2: Order the following from least to greatest: -1.5, √2, -π/2, 0, 3/4

Solution:

• -1.5
• √2 ≈ 1.41
• -π/2 ≈ -1.57
• 0
• 3/4 = 0.75

Ordered: -π/2 < -1.5 < 0 < 3/4 < √2

Example 3: Evaluate: (√9 + 2³) ÷ (1.5 – 0.5)

Solution:

(√9 + 2³) ÷ (1.5 – 0.5)

= (3 + 8) ÷ 1

= 11 ÷ 1

= 11

Example 4: Classify the following as rational or irrational:

a) 3.14

b) π

c) √25

d) 22/7

Solution:

a) 3.14 – Rational (can be expressed as 314/100)

b) π – Irrational

c) √25 = 5 – Rational

d) 22/7 – Rational

Example 5 : If 126+ 256 =x + 256, then by using the commutative property, obtain the value of x.

Solution:

Given; 126+ 256 =x + 256

According to the commutative property we can state that; p + q = q + p (in terms of addition)

126+ 256 =x + 256

Therefore x=126

Example 6: 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 7: 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 8 :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

Example 9: Simplify: (√8 + √2) * (√8 – √2)

Solution:

(√8 + √2) × (√8 – √2)

= (√8)² – (√2)² (using the difference of squares formula: (a+b)(a-b) = a²-b²)

= 8 – 2

= 6

Example 10: Solve for x: 2x² – 5x – 3 = 0

Solution:

Using the quadratic formula: x = [-b ± √(b² – 4ac)] / (2a)

Where a = 2, b = -5, and c = -3

x = [5 ± √((-5)² – 4(2)(-3))] / (2(2))

⇒ x = [5 ± √(25 + 24)] / 4

⇒ x = (5 ± √49) / 4

⇒ x = (5 ± 7) / 4

So, x = (5 + 7) / 4 = 12 / 4 = 3

or x = (5 – 7) / 4 = -2 / 4 = -1/2

The solutions are x = 3 or x = -1/2, both of which are real numbers.

Unsolved Worksheet: Real Numbers

Q1: What is the H.C.F of the smallest composite number and the smallest prime number? (CBSE 2018)

Q2: The product of a non-zero rational and an irrational number is always .

Q3: After how many decimal places the decimal expansion of the rational number 14587/1250 will terminate?

Q4: “The product of three consecutive positive integers is divisible by 6”. True or False. Justify.

Q5: Can two numbers have 18 as their HCF and 380 as their LCM? Give reason.

Q6: Find the largest positive integer that will divide 398, 436 and 542 leaving remainders 7, 11 and 15 respectively.

Q7: Find the greatest number of six digits exactly divisible by 24, 15 and 36.

Q8: Three sets of English, Hindi and Mathematics books have to be stacked in such a way that all the books are stored topic wise and the height of each stack is the same. The number of English books is 96, the number of Hindi books is 240 and the number of Mathematics books is 336. Assuming that the books are of same thickness, determine the number of stacks of English, Hindi and Mathematics books.

Q9: Two brands of chocolates are available in packs of 24 and 15 respectively. If I need to buy an equal number of chocolates of both kinds, what is the least number of boxes of each kind I would need to buy?

Q10: Given √2 is irrational, prove that 5 + 3√2 is an irrational number. (CBSE 2018)

Q11: Using Euclid’s Division Algorithm, find the HCF of 1260 and 7344. (CBSE 2019)

Q12: Find HCF and LCM of 404 and 96 and verify that HCF × LCM = Product of the two given numbers. (CBSE 2018)

Read More,

• Real Numbers
• Rational and Irrational Numbers
• Rational Numbers
• Irrational Numbers
• Properties of Real Numbers

FAQs: Real Numbers

What are 10 examples of real numbers?

Some examples of real numbers are 3 (a whole number), -1 (an integer), 1/2 (a rational number), √2 (an irrational number), π (an irrational number), 2.5 (a decimal number), etc.

What are 5 types of real numbers?

There are 5 classifications of real numbers: rational, irrational, integer, whole, and natural/counting.

What are the first 10 real numbers?

These are the sets of all positive counting numbers such as 1, 2, 3, 4, 5, 6, 7, 8, 9, …….. ∞. Real numbers are numbers that have both rational and irrational numbers. Rational numbers are integers (-2, 0, 1), fractions (1/2, 2.5), and irrational numbers (√3, 22/7 )