In this blog, we will explore irrational numbers, which cannot be expressed as fractions of two integers. Unlike rational numbers, irrational numbers have non-terminating, non-repeating decimal expansions, making them unique and important in mathematics.
We'll also cover key concepts such as their definition, properties, and real-world examples, like π and √3. Additionally, we’ll delve into the sum and product of irrational numbers with examples.
What are Irrational Numbers?
An irrational number is a number that cannot be expressed as a fraction of two integers (i.e., it cannot be written in the form p/q, where both p and q are integers and q≠0). Irrational numbers have non-terminating and non-repeating decimal expansions, meaning their decimal goes on forever without forming a repeating pattern.
Examples of Irrational Numbers
Let’s take a look at some examples to better understand. These examples are also termed the 4 famous irrational numbers.
π (Pi): Pi is the ratio of a circle's circumference to its diameter and is approximately 3.14159..., but the decimal continues without repeating. Pi is irrational because it cannot be expressed exactly as a fraction.
√2 (Square root of 2): The square root of 2 is approximately 1.41421356..., and the decimal never repeats or terminates. It cannot be written as a fraction of two integers, making it irrational.
e (Euler’s number): Euler's number, used in mathematics, is approximately 2.71828... and has a non-terminating, non-repeating decimal expansion, making it an irrational number.
Golden Ratio Phi Φ (pronounced fye): The Golden Ratio is approximately 1.6180339887... and is irrational because its decimal does not repeat or terminate.
Properties of Irrational Numbers
Irrational numbers are a subset of real numbers. Below are the key properties of irrational numbers:
When an irrational number is added to a rational number, the result is always an irrational number. For instance, let’s say x is irrational, y is rational, and adding these numbers together (x + y) results in an irrational number z.
Multiplying any irrational number by any nonzero rational number will yield an irrational product. For example, if x and y are numbers and their product xy = z is rational, then by rearranging, x = z/y, which contradicts the assumption that x is irrational. Therefore, the product of x and y must be irrational.
The least common multiple (LCM) of two irrational numbers may or may not exist, depending on the numbers involved.
The sum or product of two irrational numbers can sometimes be a rational number. For instance, if we take √3 (which is irrational) and multiply it by √3, we get 3, a rational number. This shows that multiplying two irrational numbers can result in a rational number.
Unlike rational numbers, the set of irrational numbers is not closed under multiplication, meaning multiplying two irrational numbers does not always produce another irrational number.
How to Identify Irrational Numbers?
Irrational numbers are numbers that cannot be expressed as fractions of two integers (p/q), where q ≠ 0. These numbers have decimal expansions that never terminate or repeat. Here's how to identify irrational numbers:
If the decimal form of a number never ends and never repeats a pattern, it is an irrational number.
The square roots of numbers that are not perfect squares are irrational.
Operations like adding, subtracting, multiplying, or dividing irrational numbers with rational numbers can result in irrational numbers, depending on the result.
If you encounter a fraction and its decimal form is non-terminating and non-repeating, it’s an irrational number.
Solve NCERT Maths Question Paper for Class 9: NCERT Solutions for Class 9 Maths Chapter 1 – Number Systems
Difference Between Rational and Irrational Numbers
The key difference between rational and irrational numbers is that rational numbers are expressed in p/q, whereas irrational numbers are not. Here is a brief overview of the difference.
Read More: Rational Numbers in Standard Form - Conversion and Solved Examples
Sum and Product of Two Irrational Numbers
Let’s take a closer look at how the sum and the product of two irrational numbers can result in either a rational or an irrational number. We’ll also explore this with examples of both scenarios—where the sum is rational and where it is irrational.
Sum of Two Irrational Numbers: The sum of two irrational numbers is not always irrational. Let us take a look at the examples:
Example 1:
Consider √3 (an irrational number) and 1 (a rational number).
When we add √3+1, the result is irrational, as the decimal of √3 is non-terminating and non-repeating.
Therefore, the sum of two irrational numbers generally results in an irrational number, though there can be exceptions.
Example 2:
Let’s consider 5+√2 and -√2
5 + √2 is an irrational number because √2 is irrational.
−√2 is also irrational.
Now, add them together: (5+√2) + (−√2)= 5 + √2 − √2 = 5
The sum is 5, which is a rational number.
Product of Two Irrational Numbers: Similarly, the product of two irrational numbers is not always irrational.
Example 1: Consider √2 and √2. Their product is √2 × √2 = 2, which is rational.
Example 2: Consider the irrational numbers √2 and √3. When we multiply them, √2 x √3 = √6. Thus, the product √6 is an irrational number.
Also Read: Prove that the sum of a rational number and an irrational number is always irrational
Conclusion
Understanding irrational numbers and their properties is essential for building a strong foundation in mathematics. These concepts help in solving more complex mathematical problems. Let’s take a look at the key takeaways.
Irrational numbers cannot be written as fractions and have non-terminating, non-repeating decimals.
Examples of irrational numbers include π, √3, and e.
The sum of two irrational numbers can sometimes be rational, as shown in the example above.
The product of two irrational numbers is usually irrational but can be rational.
Rational numbers can be written as fractions p/q, while irrational numbers cannot.
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