In mathematics, numbers are divided into various types based on their characteristics. Among the most important and foundational classifications are rational and irrational numbers. These numbers play a critical role not just in mathematics but also in our daily lives—like calculating money, measurements, distances, and more. By learning and understanding the concept of rational and irrational numbers, you can create the foundational knowledge required for many advanced topics.
Table of Contents
Rational numbers are numbers that can be expressed in the form of a fraction, that is, p/q, where:
p and q are integers
q ≠ 0
This includes:
Integers (like 3 and -5)
Fractions (like ½, -¾)
Terminating decimals (like 0.75)
Repeating decimals (like 0.666…)
Irrational numbers are numbers that cannot be written as a simple fraction. These numbers have non-terminating and non-repeating decimal expansions.
In simple terms:
You can’t express irrational numbers as p/q.
They go on forever in decimal form without any pattern.
Can be expressed as fractions
Decimal form is either terminating or repeating
Closed under addition, subtraction, multiplication, and division (except division by zero)
Can be positive, negative, or zero
Rational Number |
Reason |
5 |
Can be written as 5/1 |
-3/4 |
In fractional form, both numerator and denominator are integers |
0.25 |
Terminating decimal, equal to 1/4 |
0.333... |
Repeating decimal, equal to 1/3 |
Cannot be written as fractions
The decimal form is infinite and non-repeating
Not closed under basic arithmetic operations
Always non-repeating and non-terminating
Irrational Number |
Why? |
√2 |
Decimal = 1.4142135…, does not terminate or repeat |
π (Pi) |
Approx. 3.14159…, no repeating pattern |
√3, √5 |
Cannot be simplified to fractions |
e (Euler's number) |
Approx. 2.718…, also non-repeating |
Feature |
Rational Numbers |
Irrational Numbers |
Can be written as p/q |
Yes |
No |
Decimal form |
Terminating or repeating |
Non-terminating and non-repeating |
Examples |
1/2, 5, -3, 0.75 |
√2, π, e |
Countability |
Countable |
Uncountable |
All rational and irrational numbers together form the set of real numbers. On the number line:
Rational numbers have fixed locations (e.g., 1/2, -3)
Irrational numbers fall between rationals, but never exactly hit a fraction mark
For instance:
√2 lies between 1.41 and 1.42
π lies between 3.14 and 3.15
Banking & Finance: Rational numbers are used in banking and finance for calculating interest rates, profit margins, percentages, etc.
Time Measurement: Time is usually represented using rational numbers, as it is divided into hours, minutes, seconds, etc. For example, ¾ of a day.
Grocery Calculations: Calculations of weight is done using rational numbers, like 2.5 kg, ₹3.75.
Architecture: We use π to calculate the area of curves or circles.
Physics: Wave functions, natural constants like 'e'
Engineering: Diagonal of squares, irrational root calculations
Concept |
Formula |
Rational Number |
p/q where q ≠ 0 |
Decimal to Fraction |
Convert recurring part to algebraic form |
Irrational Number Check |
Check if decimal is non-repeating and non-terminating |
Q1. Is 0.272727... rational or irrational?
Solution: Yes, 0.272727 is a rational number, as it is a repeating decimal and can be written as a fraction, 27/99 = 3/11. The shortcut method of converting the repeating decimal 0.272727... into a fraction is given below:
1. Let a = 0.272727...
2. Multiply both sides by 100: 100x = 27.272727...
3. Subtract: 100x - x = 27.272727... - 0.272727...
4. Result: 99x = 27
5. Solve: x = 27 / 99
6. Simplify: x = 3 / 11
Include boxed final answer: 0.272727... = 3/11
Q2. Is √16 a rational number?
Solution: Yes, √16 is a rational number because it is equal to 4 and can be represented in the form p/q=4/1.
Q3. Is π + 1 rational?
Solution: No, π is not a rational number because π is an irrational number, so π + 1 is also irrational.
Q4. Is the sum of √2 + √3 irrational?
Solution: √2 and √3 are both irrational. Their sum is also irrational.
Identify whether the following numbers are rational or irrational:
√9
2.375
√7
0.101001000100001...
Write 0.888... as a rational number
Find 3 irrational numbers between 1 and 2
Classify the number: 0.333… + √5
Is 22/7 a rational approximation of π? Why?
Understanding rational and irrational numbers gives you insight into the very structure of the number system. Whether you’re dealing with money, measurements, or scientific data, these concepts appear everywhere. Once you grasp the definitions and learn to spot examples, you'll be able to categorize any number quickly and confidently. Keep practicing with real-life examples and problems to build strong number sense!
1. Can a number be both rational and irrational?
Answer: No. A number can only be one or the other.
2. Is 0 a rational number?
Answer: Yes. 0 = 0/1, which is a valid rational number.
3. Why is √2 irrational?
Answer: Because it cannot be expressed as a fraction, and its decimal goes on without repeating.
4. Is every real number either rational or irrational?
Answer: Yes. Real numbers are divided into rational and irrational.
5. Is 1.01001000100001… irrational?
Answer. Yes. It’s a non-repeating and non-terminating decimal.
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