Mathematics is nothing but a game of numbers. A number is an arithmetical value that can either be an object, word or symbol representing a quantity that has multiple implications in counting, measurements, labelling etc. Numbers can either be integers, whole numbers, natural numbers, real numbers. or complex numbers. Real numbers are further categorized into rational and irrational numbers. Rational numbers are those numbers that are integers and can be expressed in the form of x/y where both numerator and denominator are integers whereas irrational numbers are those numbers which cannot be expressed in a fraction. In this article, we will discuss rational numbers, irrational numbers, Rational and irrational numbers examples, the difference between irrational and rational numbers etc.
Rational Numbers
The term ratio came from the word ratio which means the comparison of any two quantities and represented in the simpler form of a fraction. A number is considered as a rational number if it can be expressed in the form of a/b where both a (numerator) and b(denominator) are integers. The denominator of a rational number is a natural number(a non-zero number). Integers, fractions including mixed fraction, recurring decimals, finite decimals etc all come under the category of rational numbers.
Irrational Numbers
A number is considered as an irrational number if it cannot be able to simply further to any fraction of a natural number and an integer. The decimal expansion of irrational numbers is neither finite nor recurring. Irrational numbers include surds and special numbers such as π. The most common form of an irrational number is pi (π). A surd is a non-perfect square or cube which cannot be simplified further to remove square root or cube root.
Rational and Irrational Numbers Examples
Some of the examples of rational numbers
Number 4 can be written in the form of 4/1 where 4 and 1 both are integers.
0.25 can also be written as 1/4, or 25/100 and all terminating decimals are rational numbers.
√64 is a rational number, as it can be simplified further to 8, which is also the quotient of 8/1.
0.888888 is a rational number because it is recurring in nature.
Some of the Examples of Irrational Numbers
3/0 is an irrational number, with the denominator equals to zero.
π is an irrational number that has value 3.142 and it is non-recurring and non terminating in nature.
√3 is an irrational number, as it cannot be able to simplify further.
0.21211211 is an irrational number as it is non-recurring and non terminating in nature.
The important difference between rational numbers and irrational numbers are given below in the tabulated form.
What are the Important Differences Between Rational and Irrational Numbers?
How to Classify Rational and Irrational Numbers?
Let us now study how to identify rational and irrational numbers on the basis of the below examples.
As we know, the rational numbers can be expressed in fraction and it includes all integers, fractions, and repeating decimals.
Rational numbers can be identified with the below conditions:
It is expressed in the form of a/b, where b≠0.
The ratio of a/b can further be simplified and illustrated in the decimal form.
Irrational numbers are those numbers that are not rational numbers. Irrational numbers can be represented in the decimal form but not in fractions which implies that the irrational numbers cannot be expressed as the ratio of two integers.
rational numbers have infinite non-repeating digits after the decimal point.
Below are Some Examples of Rational and Irrational Numbers.
[Image will be Uploaded Soon]
Solved Examples
1. Find any 4 Rational Numbers Between 2/5 and 1/2.
Solution: To find the 5 rational numbers between -⅖ and ½, we will first make the denominator similar.
Hence, -⅖ = (2 * 10) /(5 * 10) = 20/50
And, ½ = (1* 25) /(2* 25) = 25/50
4 rational number between ⅖ and ½. = 5 rational numbers between 20/50 and 25/50.
Hence, the 4 rational numbers between -⅖ and ½ are 21/50,22/50, 23/50, and 24/50.
2. Which of the Numbers Given Below is not an Irrational Number?
\[\sqrt{7}\] , \[\sqrt{5}\] , \[\sqrt{16}\] , \[\sqrt{11}\]
Solution: 16 is a perfect square i.e. \[\sqrt{16}\] = 4 which is a rational number
As we know the square root of prime numbers are irrational numbers. 7, 5 , and 11 are prime numbers. Hence, the only number which is not an irrational number is \[\sqrt{16}\].
Quiz Time
1. Which of the Following Numbers is Irrational?
21/99
\[\sqrt{100}\]
\[\sqrt{36/3}\]
2/94
2. Is the Square Root of 225
Rational number
Irrational number
3. Which of the Following Numbers is Rational?
¼
3.7
.25
1.2314
4. 9.0 is a
Rational number
Irrational number
Facts
Hippassus introduced irrational numbers when attempting to write the square root of 2 in the fraction form ( using geometry, it is thought). He proved that the square root of 2 cannot be written in the fraction form, so it is irrational.
FAQs on Difference Between Rational and Irrational Numbers for JEE Main 2024
1. How are rational and irrational numbers defined for JEE Main, and what is the key distinction?
For JEE Main, a rational number is any number that can be expressed as a fraction p/q, where p and q are integers and q is not zero. An irrational number cannot be expressed in this form. The key distinction lies in their decimal representation: rational numbers have either terminating or repeating (recurring) decimal expansions, while irrational numbers have non-terminating and non-repeating decimal expansions. Examples include 5/2 (rational) and π or √2 (irrational).
2. What are the key properties of decimal expansions that help distinguish between rational and irrational numbers in JEE Main problems?
The nature of a number's decimal expansion is a critical identifier for JEE Main questions:
- Terminating Decimals: Numbers like 0.75 (3/4) or 2.125 (17/8) are always rational.
- Non-Terminating but Repeating Decimals: Numbers with a repeating pattern, such as 0.333... (1/3) or 5.141414... (509/99), are always rational.
- Non-Terminating and Non-Repeating Decimals: Numbers that continue infinitely without any repeating pattern, like π (3.14159...) or e (2.71828...), are always irrational.
3. What are the closure properties for rational and irrational numbers under arithmetic operations, and what exceptions must a JEE aspirant know?
Understanding how numbers behave under operations is crucial for problem-solving. A key JEE Main concept is that the result of an operation between a non-zero rational number and an irrational number is always irrational. However, operations between two irrational numbers can yield either a rational or an irrational result. For example:
- √3 + (-√3) = 0 (Rational)
- √3 + √3 = 2√3 (Irrational)
- √5 × √5 = 5 (Rational)
- √5 × √3 = √15 (Irrational)
4. Are the numbers π (pi) and e (Euler's number) rational or irrational, and why is this significant in JEE Main topics?
Both π and e are transcendental irrational numbers. This means they are not the root of any non-zero polynomial equation with rational coefficients. Their irrational nature is fundamental in higher-level mathematics relevant to JEE Main. For instance, the irrationality of π is crucial in understanding the continuous nature of trigonometric functions, while e is the base of the natural logarithm and is central to problems in calculus involving exponential growth and decay.
5. How does the concept of rational and irrational extend to algebraic expressions, and why is this important for solving JEE Main problems?
The concept extends to functions and expressions. A rational expression is a ratio of two polynomials, P(x)/Q(x), where Q(x) ≠ 0. An irrational expression involves roots of variables, like √(x² + 4). This distinction is critical in JEE Main for determining the domain of a function, identifying points of discontinuity, and selecting the correct method for integration or differentiation, as the rules applied to rational and irrational functions can differ significantly.
6. What is a common conceptual trap in JEE Main questions involving operations with irrational numbers?
A common trap is assuming that the sum, difference, or product of two irrational numbers is always irrational. As shown with examples like √2 + (-√2) = 0, this is false. Another frequent mistake involves simplification. An expression like (√3 + 1) / (√3 - 1) might look complex, but upon rationalising the denominator, it simplifies to 2 + √3, which is still irrational. Students must perform the complete simplification before concluding the nature of the number.
7. What is the density property of rational and irrational numbers, and what does it imply for the real number line?
The density property states that between any two distinct real numbers, there exist infinitely many rational numbers and infinitely many irrational numbers. This implies that both the set of rational numbers (Q) and the set of irrational numbers (R-Q) are dense on the real number line. For JEE Main aspirants, this means you can never identify two 'adjacent' rational or irrational numbers; there is always another one in between.
