Content Writer
Irrational numbers are those numbers that cannot be expressed as a ratio between two integer values. The value of getting an irrational number is non-terminating, and there is no pattern in the values of a number after the decimal.
- An irrational numbers can have a decimal expansion that will never end.
- It will never repeat after so many iterations.
- The number cannot be expressed in terms of integers.
- It is expressed in terms of R\Q.
- When the ratio of two line segments is irrational, then those line segments are called incommensurable.
- It was discovered by Pythagorean philosopher Hippasus in the 5th century BC.
- Euler numbers and golden ratios are common examples of irrational numbers.
Key Terms: Irrational Numbers, Real Numbers, Fraction, Non-terminating, Ratio, Integer, Decimal, Euler Number, Golden Ratio, Rational Numbers, Addition, Subtraction, Multiplication
What are Irrational Numbers?
[Click Here for Sample Questions]
Irrational numbers are the set of real numbers that cannot be expressed in the form of a fraction p/q where p q are integral values. The denominator is said to be not equal to zero (q ≠ 0).
- The square root of any number which is not a perfect square will always be an irrational number.
- The expansion used for these numbers is neither terminating nor repeating.
- The reason for this is that real numbers are not countable.
- Real numbers form the majority of irrational numbers.
- They are expressed in terms of positional notation.
- These types of numbers can be expressed on a number line.
Example of What are Irrational Numbers?Example: √2 is an irrational number. If we are going to calculate the value of √2, it will be 1.4121356230951, and these numbers will go till infinity and will not be repeated. So, these numbers are considered irrational numbers. |
Real Numbers
Read More:
| Chapter Related Concepts | ||
|---|---|---|
| Cosine Rule | Arithmetic | Root 2 is an Irrational Number |
| Fraction | Degree of polynomial | Number Systems |
Irrational Numbers Symbol
[Click Here for Sample Questions]
The symbol used for the representation of the irrational numbers is “P”. P represents the group of numbers that complement rational numbers, as irrational numbers are negatively defined.
- Irrational numbers are represented as a set difference between real numbers minus rational nationals.
- P specifies the relationship between real and rational numbers.
- It can mathematically be represented as:
P = R- Q or R\Q
- It means union of P and Q will form a real numbers which can be represented as:
P U Q’ = R
Properties of Irrational Numbers
[Click Here for Sample Questions]
The important properties of irrational numbers are as follows:
- Irrational numbers consist of non-terminating and non-recurring decimals.
- Their LCM (Least common multiple) may or may not exist.
- Addition, subtraction, multiplication, and division of irrational numbers may or may not be rational numbers.
When a rational number and an irrational number are added, subtracted, multiplied and divided from each other, their result will only be considered as an irrational number.
List of Irrational Numbers
[Click Here for Sample Questions]
Pi, Euler’s number, and Golden Ratio are some of the popular irrational numbers. The number obtained in the case of square root and cube root is also irrational. But the result is not true for all numbers.
- The square root of any type of prime number is irrational.
List of some common Irrational Numbers is as follows:
| Irrational Numbers | Value of Number |
|---|---|
| π | 3.14159265.... |
| √2 | 1.414213562... |
| √3 | 1.73205080... |
| √5 | 2.23606797.... |
| √7 | 2.64575131.... |
| √13 | 3.605551275... |
| -√3/2 | -0.866025.... |
| ∛47 | 3.60882608 |
| e | 2.7182818..... |
Sum and Product of Irrational Numbers
[Click Here for Sample Questions]
The sum and product of irrational numbers is explained as follows:
Product of two Irrational Numbers
Statement: The product of two irrational numbers is either rational or irrational.
Proof: This can be explained with the help of an example:
- Suppose we have an irrational number √7.
- When √7 is multiplied by √7, we will get 7 as a result.
- This is a rational number.
- Similarly, when e is multiplied by e, we will get e2 as a result.
- This is an irrational number.
Sum of two Irrational Numbers
Statement: The sum of two irrational numbers is either rational or irrational.
Proof: This can be explained with the help of an example:
- Suppose we have an irrational number √7 + 2.
- When √7 + 2 is added with √7, we will get 2√7 + 2 as a result.
- This is a irrational number.
- Similarly, when √7 + 2 is added with (-√7), we will get 2 as a result.
- This is an rational number.
Irrational Number Proof
[Click Here for Sample Questions]
Here’s a step-by-step process to prove a non-perfect number square which is an irrational number.
Statement: To prove √2 is an irrational number
Suppose: Let √2 is a rational number. Then, by definition of rational number, we can write that
√2 = p/q …. (1)
- Where p and q are co-prime integral values and q ≠ 0.
- Squaring both sides of eq (1)
- 2 = p2 / q2
- p2 = 2 * q2 …..(2)
- From the theorem which states that,
“If p is a prime number given and a2 is divisible by p, (where ‘a’ is any positive integral value), then it can be said that p also divides a”.
- From the above statement, if 2 is a prime factor of p2, then 2 is also a prime factor of p.
- So, p = 2 * k (where k is an integer)
- Substituting the value of p in eq (2),
- (2k)2 = 2q2
- → q2 = 2k2
This concludes that 2 is a prime factor of q2 also. Now, according to the initial assumption, p and q are co primes but the result obtained above denies this assumption, which is that p and q have 2 as a common factor other than 1.
- This kind of contradiction arose with the incorrect assumption that we made as “√2 is a rational number”.
- That’s why from the above result, we can say that “√2 is an irrational number”.
Irrational Number
Read More:
| Class 9 Maths Related Concept | ||
|---|---|---|
| Real-Valued Function | Descending Order | Trigonometric Functions |
| Roots of Polynomial | Number Lines | Division of rational expression |
Uses of Irrational Numbers
[Click Here for Sample Questions]
Some important uses of irrational numbers are as follows:
- They cannot be converted into different forms of number systems like hexadecimal, octal, binary, and so on.
- Natural logarithms having base e are considered irrational numbers.
- Many engineering works and civil constructions are achieved by using irrational numbers.
- Major types of irrational numbers are Pi, Euler's number, Golden ratio, and many others.
- "e", which is an Euler's number, is used to derive many physics formulas and prove many proofs.
Things to remember
- Irrational numbers are types of numbers that can be expressed in terms of P/Q.
- These numbers are not closed under the multiplication process, unlike the set of rational numbers.
- We can prove √2 is irrational by a simple procedure of proof by contradiction.
- The addition or multiplication of two irrational numbers may result in a rational number.
- The product of two irrational numbers can result in a rational or an irrational number.
Sample Questions
Ques. How to identify an irrational number? (2 Marks)
Ans. As irrational numbers are the real numbers that cannot be expressed in the form of p/q where p and q are integers and q is not equal to zero. So, we have to check whether the number is in the same form or not. Examples - √5 and √3 are rational numbers. The numbers which can be expressed in the form of p/q are considered rational numbers.
Ques. Compare √6 and √5? (2 Marks)
Ans. Given two irrational numbers, √6 and √5,
We know that, if p and q are two numbers and q is greater than p, and q2 is greater than p2.
Then, (√6)2 = 6
(√5)2 = 5
As 6 > 5, so we can say that √6 is greater than √5.
Ques. Insert any two irrational numbers between √13 and √19? (2 Marks)
Ans. Given two irrational numbers, √13 and √19
In the next step, we will find the squares of both -
(√13)2 = 13
(√19)2 = 19
Since, there are 14, 15, 16, 17, 18 lies between 13 and 19. So, we can write two of them.
Hence two irrational numbers between √13 and √19 are √14, √17.
Ques. Find an irrational number between √6 and 6?(2 Marks)
Ans. Given √6 and 6,
A real number between √6 and 6 = ½ √6 + 1
But, 1 is a rational number and ½ √6 is an irrational number. The sum of a rational number and an irrational number always results in an irrational number.
So, ½ √6 + 1 is the irrational number that lies between √6 and 6.
Ques. Write the irrational numbers 4√6, √3, and 3√7 in ascending and descending orders? (3 Marks)
Ans. Given irrational numbers 4√6, √3, and 3√7,
The order of the irrational numbers is 4, 2, 3
The LCM of (4, 2, 3) = 12.
Change 4√6 = (4*3) √63 = 12√216
√3 = (2*6) √36 = 12√729
3√7 = (3*4) √74 = 12√2401
216<729<2401
Therefore, ascending order is 4√6, √3, and 3√7, and descending order is 3√7, √3, 4√6.
Ques. What is the difference between rational and irrational numbers? (5 Marks)
Ans. The difference between rational and irrational numbers
| S.No | Rational numbers | Irrational numbers |
|---|---|---|
| 1. | Rational numbers are those which can be expressed as a ratio of two numbers p and q where p and q are any integer and q is not equal to zero is called rational numbers. | Irrational numbers are those which cannot be expressed as a ratio of two numbers p and q where p and q are any integer and q is not equal to zero is called rational numbers. |
| 2. | These numbers are finite or recurring. | These numbers are non-repeating and non-recurring. |
| 3. | In this, both the numerator and denominator are integral values in which the denominator is equal to zero. | These numbers cannot be written in fractional form. So, there is no involvement of numerator and denominator. |
| 4. | Rational numbers include perfect squares such as 4, 9, 16, 25, 36 etc and so on. | Irrational numbers include surds instead of perfect squares such as √2, √6, √3, etc and so on. |
| 5. | Example - 3/2 = 1.5, 3.7676, 6, 9.31, 0.6666, etc and so on. | Example - √5, √11, e (Euler's number), π (pi), etc and so on. |
Ques. The square root of a perfect square is an irrational number. Is this statement true or false? (2 Marks)
Ans. The above statement is false with respect to the irrational numbers. The correct fact is that the square root of the perfect square is a rational number, for instance, √64 = 8, √36 = 6. Irrational numbers are the square roots of those numbers that are not perfect squares, for instance, √2 and √6 respectively.
Ques. Is Pi(π) a rational or an irrational number, explain why? (2 Marks)
Ans. π is an irrational number as it is non-terminating and non-repeating. However, to make calculations easier, pi is rounded off to 3.14 and is also represented in fraction form as 22/7.
Ques. Rani is playing "Roll a dice-Number game" with his friend. Rani takes a turn and rolls a dice. She gets 5. If She gets 5, she is supposed to collect all the irrational numbers from his friend. Help Rani to collect all the irrational numbers without missing even one. {e, -5, √9,√13, π, -2/8} ? (3 Marks)
Ans. From the set of the given numbers:
- -5 is an integer. √9 is a perfect square. -2/8 has a recurring terminating decimal value.
- These numbers are rational numbers.
- The irrational numbers are e, √13, π.
- Therefore, Rani collected all the irrational numbers and those are e, √13 and π.
Ques. Jadeja has a box with four irrational numbers. Jadeja wants only one irrational number which is closest to 3 and should not exceed 3. Help Jadeja to find out the right one. The irrational numbers in the box are √3, √6, √10, √5? (2 Marks)
Ans. First, determine the value of these irrational numbers given in the list.
So the value are given as follows:
- √3 = 1.732020..,
- √6 = 2.449489..,
- √10 = 3.162277..
- √5 = 2.236067...
- Thus, √6 = 2.449489... comes closest to 3.
- Therefore, √6 is the closest number to 3.
Read More:
Comments