103 LearnersLast updated on July 4th, 2025
Irrational Numbers
Irrational numbers are not simple fractions because they have non-repeating, non-terminating decimal expansions. Common examples include π (pi) and √2. Irrational numbers are often used in number theory, geometry, and calculus.
What are Irrational Numbers?
Irrational numbers are not simple fraction in the form of (p/q), where p and q are whole numbers, and q 0. Their decimal representations are non-terminating and non-repeating, meaning they go on forever without creating a repeating pattern. Irrational numbers are applied in fields like mathematics, science, and engineering.
There are many properties of irrational numbers. Some of them are mentioned below:
- Cannot be Expressed as a Fraction
Irrational numbers cannot be expressed as a simple fraction (p/q).
- Non-Terminating and Non-Repeating Decimals
Irrational numbers have infinite decimal expansions that are non-repeating. For example, π = 3.14159……
- Irrational + Rational = Irrational
The sum of an irrational number and a rational number is always irrational. For example, the sum of 4 (rational) and 2 (irrational) is 4 + 2, which is irrational.
- Irrational x Rational (Non-Zero) = Irrational
The product of an irrational number and a non-zero rational number is always irrational. E.g., the product of 3 (non-zero rational) and 2 (irrational) is 32 (irrational).
Differences Between Rational and Irrational Numbers
Rational and irrational numbers have many differences between them. Some of them are given below.
|
Rational Numbers |
Irrational Numbers |
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Numbers that can be expressed in the form of a simple fraction (p/q). |
Numbers which cannot be expressed as a simple fraction in the form of p/q. |
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It includes integers, fractions, finite decimals, and repeating decimals. |
It includes surds, transcendental numbers (like and e), and logarithms of non-rational bases or arguments. |
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Between any two rational numbers, there always exists another rational number. |
Between any two irrational numbers, there exist both rational and irrational numbers. |
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Used in money calculations, measurements, and fractions. |
Used in geometry, physics, and nature. |
Symbols to Represent Irrational Numbers
We use symbols to represent irrational numbers. The most common ones used are as follows:
- I: Some texts use this to represent the set of irrational numbers.
- R\Q: The most formal way to express irrational numbers, which means all real numbers except rational numbers.
- ∉Q: This symbol is used to convey that the number is not rational.
Real-life Applications of Irrational Numbers
Irrational number is an important mathematical concept that has wide applications in many real-life situations. Some of them are given below.
Engineering and Construction:
Irrational numbers are used to achieve accurate and precise calculations in construction and engineering. For example, we use the golden ratio to get aesthetically pleasing structures. Similarly, we use 2 in diagonal measurements in square-based designs.
Geometry:
In geometry, irrational numbers are used in calculating area, circumference of circular objects. is an irrational number, which is a non-terminating number. An approximation of 3.14159 or 3.142 for is used for calculations.
Physics and Scientific Research:
There are uses of irrational numbers in physics equations. For example, the number e (Euler’s number) is fundamental in modeling exponential growth and decay, like radioactive decay, population growth, and compound interest calculations. Irrational numbers are also used in wave mechanics and quantum physics.
Common Mistakes and How to Avoid Them in Irrational Numbers
Students tend to make mistakes while understanding the concept of irrational numbers. Let us see some common mistakes and how to avoid them in irrational numbers:
Mistake 1
Confusion Between Rational and Irrational Numbers
To differentiate rational from irrational numbers, check if the number is a non-repeating, non-terminating decimal expansion. If it is, then it’s an irrational number.
Mistake 2
Rounding Off Irrational Numbers Incorrectly
Students should remember that numbers like 3.14 or e 2.718 are only approximations. Therefore, we must always indicate that they are approximations and use the symbol.
Mistake 3
Assuming Square Roots of Non-Perfect Squares are Rational
Students tend to assume that if one square root is rational, then all square roots are rational. This mistake can be avoided by remembering the fact that only perfect squares have rational square roots, and non-perfect squares have irrational square roots.
Mistake 4
Believing That Irrational Number can be Expressed as Terminating or Repeating Decimals
Always remember that irrational numbers cannot be expressed with terminating or repeating decimals. If the decimals terminate or repeat, then the number is rational.
Mistake 5
Incorrectly Adding or Multiplying Irrational Numbers
Students assume that the product or sum of two irrational numbers is always irrational. This is not necessarily true. For example, 2 + (-2) = 0, which is rational.
Solved Examples of Irrational Numbers
Problem 1
Simplify √50 to its simplest form.
5√2
Explanation
Factor the radicand:
50 = 25 × 2
Separate the radical:
√50 = √25 * 2 = √25 x √2
Simplify:
√25 = 5
√50 = 5√2.
Problem 2
Simplify √18 + √8
5√2
Explanation
Simplify each radical:
√18 = √9 * 2 = 3√2
√8 = √4 * 2 = 2√2
Combine like terms:
3√2 + 2√2 = (3 + 2)√2 = 5√2
Problem 3
Multiply √2 and √3
√6
Explanation
Multiply the radicands:
√2 x √3 = √2 * 3 = √6.
Problem 4
Expand (√2 + 3)2
5 + 2√6
Explanation
Apply the binomial square formula:
(a + b)2 = a2 + 2ab + b2
Compute each term:
a2 = (√2)2 = 2
b2 = (√3)2 = 3
2ab = 2(√2 x √3) = 2√6
Combine:
2 + 3 + 2√6 = 5 + 2√6
Problem 5
Approximate to three decimal places.
3.142
Explanation
Known approximation:
3.14159…
Round to three decimals:
3.142.
FAQs on Irrational Numbers
1.What are irrational numbers?
2.What are some examples of irrational numbers?
3.Is 0 an irrational number?
4.Is (Pi) an irrational number?
5.Can irrational numbers be negative?
6.How can children in Philippines use numbers in everyday life to understand Irrational Numbers?
7.What are some fun ways kids in Philippines can practice Irrational Numbers with numbers?
8.What role do numbers and Irrational Numbers play in helping children in Philippines develop problem-solving skills?
9.How can families in Philippines create number-rich environments to improve Irrational Numbers skills?
Hiralee Lalitkumar Makwana
About the Author
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
Fun Fact
: She loves to read number jokes and games.
