Rational, Irrational & Real Numbers
Rational and irrational numbers together form the set of real numbers.
Questions based on their identification, decimal expansion, square roots
and classification are frequently useful in competitive examinations.
This topic explains how these number sets are related and how to identify
them quickly.
1. Rational Numbers
A number that can be expressed in the form
p/q, where p and q are integers and
q ≠ 0, is called a rational number.
Rational Number = p/q, where p, q ∈ Z and q ≠ 0
Examples:
- 3/5
- −7/4
- 11/3
- 5 = 5/1
- 0 = 0/1
- −8 = −8/1
Important: Every integer is a rational number because
any integer n can be written as n/1.
2. Why Can the Denominator Not Be Zero?
In the expression p/q, the denominator q must not be zero because
division by zero is undefined.
Therefore:
7/0 is not a rational number.
The condition q ≠ 0 is an essential part of the
definition of a rational number.
3. Positive and Negative Rational Numbers
A rational number greater than zero is called a
positive rational number.
Examples: 2/3, 7/5, 4
A rational number less than zero is called a
negative rational number.
Examples: −2/3, −7/5, −4
Zero is a rational number but is neither positive nor negative.
4. Equivalent Rational Numbers
Multiplying or dividing the numerator and denominator of a rational
number by the same non-zero integer does not change its value.
2/3 = 4/6 = 6/9 = 8/12
These are called equivalent rational numbers.
5. Standard Form of a Rational Number
A rational number p/q is said to be in standard or simplest form when:
- p and q have no common factor other than 1, and
- the denominator is positive.
Example:
−18/24 = −3/4
Thus, −3/4 is the simplest form.
6. Decimal Expansion of Rational Numbers
The decimal expansion of a rational number is either:
- Terminating, or
- Non-terminating recurring/repeating.
Terminating Decimal
A decimal expansion that ends after a finite number of decimal places
is called a terminating decimal.
Examples:
- 1/2 = 0.5
- 3/4 = 0.75
- 7/8 = 0.875
- 13/20 = 0.65
Non-Terminating Recurring Decimal
A decimal expansion that continues indefinitely but repeats a digit
or a group of digits in a fixed pattern is called a non-terminating
recurring decimal.
Examples:
- 1/3 = 0.333...
- 2/3 = 0.666...
- 1/11 = 0.090909...
Exam Rule: Every terminating decimal and every
non-terminating recurring decimal represents a rational number.
7. Test for Terminating Decimal Expansion
First express the rational number p/q in its lowest terms.
If the prime factorisation of the denominator q contains
only the prime factors 2 and/or 5, its decimal
expansion terminates.
q = 2m × 5n
where m and n are non-negative integers.
If the denominator in lowest terms contains any prime factor other
than 2 or 5, the decimal expansion is non-terminating recurring.
Very Important: Always reduce the fraction to its
lowest terms before applying this test.
Example 1
Determine whether 7/40 has a terminating decimal expansion.
40 = 23 × 5
The denominator contains only 2 and 5.
Therefore, 7/40 has a terminating decimal expansion.
Example 2
Determine whether 7/12 has a terminating decimal expansion.
12 = 22 × 3
The denominator contains the prime factor 3.
Therefore, its decimal expansion is
non-terminating recurring.
Example 3: Why Lowest Form Matters
Consider 6/15.
Although 15 contains the factor 3:
6/15 = 2/5
The denominator in lowest form is 5. Therefore,
6/15 has a terminating decimal expansion.
8. Number of Decimal Places in a Terminating Decimal
Suppose a rational number in lowest terms has denominator:
2m × 5n
Then its terminating decimal expansion has at most
max(m, n) decimal places.
Example:
7/200
200 = 23 × 52
max(3, 2) = 3
Indeed:
7/200 = 0.035, which has 3 decimal places.
9. Rational Number Between Two Rational Numbers
There are infinitely many rational numbers between any two distinct
rational numbers.
One simple way to find a rational number between a and b is:
(a + b)/2
Example: Find a rational number between 2 and 3.
(2 + 3)/2 = 5/2 = 2.5
Therefore, 2.5 is a rational number between 2 and 3.
Between any two distinct rational numbers, there are
infinitely many rational numbers.
10. Operations on Rational Numbers
Rational numbers are closed under:
- Addition
- Subtraction
- Multiplication
They are also closed under division
provided the divisor is not zero.
If a and b are rational numbers, then a + b, a − b and ab are
rational. The quotient a/b is rational whenever b ≠ 0.
11. Irrational Numbers
A real number that cannot be expressed in the form
p/q, where p and q are integers and q ≠ 0, is called an
irrational number.
Its decimal expansion is:
Non-terminating and non-recurring
Common examples:
Do not confuse:
0.333... is non-terminating but recurring, so it is
rational.
An irrational decimal must be both
non-terminating and non-recurring.
12. Square Roots and Irrational Numbers
The square root of a positive integer that is
not a perfect square is irrational.
Examples:
- √2 is irrational.
- √3 is irrational.
- √10 is irrational.
- √17 is irrational.
However, the square root of a perfect square is rational.
- √4 = 2
- √9 = 3
- √25 = 5
- √64 = 8
Exam Shortcut: Do not assume that every square root
is irrational. First check whether the number under the square root
is a perfect square.
13. Real Numbers
The collection of all rational and irrational numbers is called the
set of real numbers.
Real Numbers = Rational Numbers ∪ Irrational Numbers
The set of real numbers is generally represented by R.
Examples of real numbers include:
Every rational number and every irrational number is a
real number.
14. Classification of Real Numbers
The important number sets studied so far can be arranged as follows:
N ⊂ W ⊂ Z ⊂ Q ⊂ R
where:
- N = Natural Numbers
- W = Whole Numbers
- Z = Integers
- Q = Rational Numbers
- R = Real Numbers
Irrational numbers are also part of R, but they are not part of Q.
Number-System Hierarchy:
Every natural number is a whole number.
Every whole number is an integer.
Every integer is a rational number.
Every rational number is a real number.
However, the reverse statements are not always true.
15. Rational and Irrational Numbers are Disjoint
A real number cannot be both rational and irrational at the same time.
Therefore:
Q ∩ (Irrational Numbers) = ∅
For example, √2 is irrational and therefore cannot be rational.
Similarly, 3/4 is rational and therefore cannot be irrational.
16. Every Real Number Has a Position on the Number Line
Every real number corresponds to a unique point on the number line,
and every point on the number line represents a real number.
This includes rational numbers as well as irrational numbers such
as √2 and π.
The real-number line contains both rational and irrational numbers.
17. Important Results Involving Rational and Irrational Numbers
Rational + Rational
The sum of two rational numbers is always rational.
Example:
2/3 + 1/6 = 5/6
Rational − Rational
The difference of two rational numbers is always rational.
Rational × Rational
The product of two rational numbers is always rational.
Rational ÷ Rational
The quotient of two rational numbers is rational provided the
divisor is non-zero.
Rational + Irrational
The sum of a rational number and an irrational number is
always irrational.
Example:
3 + √2 is irrational.
Rational − Irrational
The difference between a rational number and an irrational number
is always irrational.
Non-Zero Rational × Irrational
The product of a non-zero rational number and an
irrational number is always irrational.
Example:
5√2 is irrational.
Why “non-zero” is important:
0 × √2 = 0, and 0 is rational.
Irrational + Irrational
The sum of two irrational numbers may be rational or irrational.
Rational result:
√2 + (−√2) = 0
Irrational result:
√2 + √3
Irrational − Irrational
The difference of two irrational numbers may also be rational or
irrational.
Example:
√5 − √5 = 0, which is rational.
Irrational × Irrational
The product of two irrational numbers may be rational or irrational.
Rational result:
√2 × √2 = 2
Irrational result:
√2 × √3 = √6
Important Exam Trap:
Do not assume that the sum, difference or product of two irrational
numbers must always be irrational. Depending on the numbers involved,
the result may be rational or irrational.
18. Identifying Rational and Irrational Numbers Quickly
| Number/Type |
Classification |
| Integer |
Rational |
| Fraction p/q, q ≠ 0 |
Rational |
| Terminating decimal |
Rational |
| Non-terminating recurring decimal |
Rational |
| Non-terminating non-recurring decimal |
Irrational |
| √n, when n is a perfect square |
Rational |
| √n, when n is a positive integer but not a perfect square |
Irrational |
| π |
Irrational |
19. Converting a Terminating Decimal into a Rational Number
A terminating decimal can easily be written as a fraction.
Example: Convert 0.375 into a fraction.
0.375 = 375/1000
Dividing numerator and denominator by 125:
0.375 = 3/8
Hence, 0.375 is rational.
20. Converting a Recurring Decimal into a Rational Number
Every recurring decimal can be expressed as a fraction.
Example: Convert 0.333... into a fraction.
Let:
x = 0.333...
Multiplying by 10:
10x = 3.333...
Subtracting the first equation:
10x − x = 3.333... − 0.333...
9x = 3
x = 1/3
Therefore, 0.333... is rational.
21. Recurring Decimal with More Than One Repeating Digit
Example: Convert 0.272727... into a fraction.
Let:
x = 0.272727...
Since two digits repeat, multiply by 100:
100x = 27.272727...
Subtract:
100x − x = 27
99x = 27
x = 27/99
x = 3/11
22. Useful Shortcut for Pure Recurring Decimals
For a decimal in which the repeating part begins immediately after
the decimal point:
Numerator: Repeating digits
Denominator: As many 9s as there are repeating digits
Examples:
- 0.777... = 7/9
- 0.454545... = 45/99 = 5/11
- 0.123123123... = 123/999 = 41/333
This shortcut applies directly when the repeating block starts
immediately after the decimal point.
23. Solved Examples
Example 1
Is 0.125 rational or irrational?
Solution:
0.125 is a terminating decimal.
0.125 = 125/1000 = 1/8
Therefore, 0.125 is rational.
Example 2
Is 0.101001000100001... rational or irrational?
Solution:
The decimal neither terminates nor repeats in a fixed recurring
pattern.
Therefore, it is irrational.
Example 3
Determine whether 13/160 has a terminating decimal expansion.
Solution:
160 = 25 × 5
The denominator contains only the prime factors 2 and 5.
Therefore, 13/160 has a terminating decimal expansion.
Example 4
Determine whether 11/30 has a terminating or recurring decimal expansion.
Solution:
30 = 2 × 3 × 5
The denominator contains the prime factor 3 in addition to 2 and 5.
Therefore, the decimal expansion is
non-terminating recurring.
Example 5
Classify √81 and √10.
Solution:
√81 = 9, which is an integer and therefore rational.
10 is not a perfect square, so √10 is irrational.
Example 6
Is 5 + √3 rational or irrational?
Solution:
5 is rational and √3 is irrational.
The sum of a rational number and an irrational number is irrational.
Therefore, 5 + √3 is irrational.
Example 7
Is √7 × √7 rational or irrational?
Solution:
√7 × √7 = 7
7 is an integer and hence rational.
Therefore, the product is rational.
Example 8
Find one rational number between 3/5 and 4/5.
Solution:
Taking their average:
[(3/5) + (4/5)] / 2
= (7/5) / 2
= 7/10
Therefore, 7/10 is one rational number between 3/5 and 4/5.
24. Common Exam Traps
Trap 1: A non-terminating decimal is not necessarily
irrational. If it repeats, it is rational.
Trap 2: Every integer is rational because n = n/1.
Trap 3: Zero is rational because 0 can be written
as 0/1.
Trap 4: Not every square root is irrational.
√49 = 7 is rational.
Trap 5: Reduce p/q to lowest terms before applying
the denominator test for terminating decimals.
Trap 6: The sum or product of two irrational numbers
need not be irrational.
Trap 7: The product of a non-zero rational number
and an irrational number is irrational, but multiplying an irrational
number by 0 gives the rational number 0.
25. Quick Revision
- A rational number can be written as p/q, where q ≠ 0.
- Every integer is rational.
- Zero is rational.
- A terminating decimal is rational.
- A non-terminating recurring decimal is rational.
- A non-terminating non-recurring decimal is irrational.
- In lowest terms, denominator 2m × 5n gives a terminating decimal.
- Any other prime factor in the reduced denominator gives a non-terminating recurring decimal.
- Between two distinct rational numbers, infinitely many rational numbers exist.
- √n is irrational when positive integer n is not a perfect square.
- √n is rational when n is a perfect square.
- π is irrational.
- Rational and irrational numbers together form the real numbers.
- N ⊂ W ⊂ Z ⊂ Q ⊂ R.
- A rational number plus an irrational number is irrational.
- A non-zero rational number multiplied by an irrational number is irrational.
- The sum or product of two irrational numbers may be rational or irrational.
- Every real number corresponds to a point on the real-number line.
Practice these verified previous-year questions on rational, irrational
and real numbers. Try each question before revealing its answer and explanation.
Practice these exam-oriented questions on rational, irrational and real
numbers. Try to solve each question before revealing the answer and explanation.