Prove That Root 11 is an Irrational Number

Is root 11 an irrational number? A number that can be represented in p/q form where q is not equal to 0 is known as a rational number whereas numbers that cannot be represented in p/q form are known as irrational numbers. An irrational number can also be denoted as a number that does not terminate and keeps extending after the decimal point. Now that we know about rational and irrational numbers, let us look at the detailed discussion and prove that root 11 is an irrational number.

The square root of 11 will be an irrational number if the value of the square root of 11 is a number that has values after decimal that is non-terminating and non-repeating. Let us first discuss the root of a number "n", the square root of a number n is represented as √n. On multiplication of the root of a number to itself gives the original number. For example, √11 on multiplication to itself gives the number 11. To prove that root 11 is an irrational number, we use different methods like the contradiction method and long division method.

We can prove that root 11 is an irrational number by various methods. The value of √11 is 3.31662479036... As we know that a decimal number that is non-terminating and non-repeating is also irrational. The value of root 11 is also non-terminating and non-repeating. This satisfies the condition of √11 being an irrational number. Hence, √11 is an irrational number. 

We want to prove that root 11 is irrational. We can also prove that root 11 is irrational also by using the contradiction method.
Proof: Let us assume that square root 11 is rational. Now since it is a rational number, as we have assumed, we can write it in the form p/q, where p, q ∈ Z, and coprime numbers, i.e., GCD (p,q) = 1.

⇒ √11 = p/q
Rearranging the terms,
⇒ p = √11 q ------- (1)

On squaring both sides we get, 
⇒ p² = 11 q² 
Again rearranging the terms,
⇒ p²/11 = q² ------- (2)

As we know, 11 is a prime number. Using the theory, which says that, if a prime number is a factor of a number, it will also be the factor of the given number's square, and vice versa is also true. This implies that since 11 is a factor of p² then it will also be the factor of p.

Thus we can write p = 11a (where a is some constant)

Substituting p = 11a in equation (2), we get 
(11a)²/11 = q²
⇒ (121a²)/11 =  q² 
⇒ 11a²  =  q² 
On rearranging, we get,
⇒ a²  =  q²/11 ------- (3)

This shows that 11 will also be the factor of q.
Now, according to our initial assumptions, p and q are the coprime numbers hence only 1 is the number that can evenly divide both of them. But here, we have 11 as the factor of both p and q, which is contradictory to our initial assumption. This proves that the assumption of root 11 as a rational number was incorrect.
Therefore, the square root of 11 is irrational.

We can also prove that root of 11 is irrational by using the long division method. The value of the root 11 can be obtained by the long division method using the following steps:

• Step 1: Add pairs of 0 after 11 as 11.00 00 00 and pair the digits starting from the right and find a number whose square is less than or equal to the number 11, it will be our first divisor and quotient. We have 3, square the number and subtract the result from 11, 2 is the remainder.
   
• Step 2: Take the next pair of 0 down after the remainder, as 00 is brought down, we get 200 as the next dividend, and double the first quotient to get the partial divisor of this step. The unit digit the divisor will be the number which on multiplying with the complete divisor thus formed, gives a number equal or less than the new dividend. Here, we get 3 at the units place, and 63 is our divisor and 3 is our quotient. Subtract the result after multiplying 63 with 3 from 200, and note down the remainder. 
   
• Step 3: Take the next pair of 0 down after the remainder of the previous step to get the dividend, 1100 is the new dividend. Add the units place of the divisor obtained in the previous step to the divisor itself and get the partial divisor of this step. Here, we get 66. The unit digit the divisor will be the number which on multiplying with the complete divisor thus formed, gives a number equal or less than the new dividend. Here, we get 1 at the units place, and 661 is our divisor and 1 is our new quotient. Subtract the result after multiplying 661 with 1 from 1100, and note down the remainder. 
   
• Step 4: Repeat the process until the required number of digits after the decimal is obtained.

See the following image to see a few of the steps in the long division of the 11. 

As we can see the value of root 11 does not terminate after 3 decimal places. It can still be extended further. Hence, this makes √11 an irrational number.

☛ Also Check:

• Prove that Root 2 is Irrational
• Prove that Root 3 is Irrational
• Prove that Root 5 is Irrational
• Prove that Root 6 is Irrational
• Prove that Root 7 is Irrational

How do you Prove that Root 11 is Irrational?

We can prove that root 11 is irrational by various methods. The square root of 11 will be an irrational number if the value after the decimal point is non-terminating and non-repeating. We have the value of root 11 as 3.31662479036...which is non-terminating and nonrepeating, hence it is an irrational number. We can also prove it by using the contradiction method where we assume the root 11 as the rational number and write it as the ratio of two coprime numbers (p/q) and proceed further if we can find any common factor of the coprime numbers thus assumed, which will prove that the root 11 is irrational. We can also prove it by using the long division method.

Is 2 times the Square Root of 11 Irrational?

Yes, 2 times the square root of 11 is irrational. To find out whether 2 times the square root of 11 is an irrational number, we multiply both the numbers and check the result. On multiplying 2 with root 11, we get 2 × 3.31662479036... = 6.63324958072.. which is a non-terminating and non-repeating term, therefore the product of the two is an irrational number. 

How to Prove Root 11 is Irrational by Contradiction?

To prove root 11 is irrational using contradiction we use the following steps:

• Step 1: Assume that √11 is rational.
• Step 2: Write √11 = p/q
• Step 3: Now both sides are squared, simplified and a constant value is substituted.
• Step 4: It is found that 11 is a factor of the numerator and the denominator which contradicts the property of a rational number.

Therefore it is proved that root 11 is irrational by the contradiction method.

Is 3 Times the Square Root of 11 Irrational?

Yes, 3 times the square root of 11 is an irrational number. 3 times the square root 11 is written as 3 × √11 = 3 ×3.31662479036... = 9.94987437108... Here, we get the result that is non-terminating as well as non-repeating. Thus, we can also conclude that any number multiplied with root 11 will be irrational. Hence, 3 times the square root of 11 is irrational too.

How to Prove that 1 by Root 11 is irrational?

We can prove 1 by root 11 is irrational using various methods, such as directly finding the value of 1 by root 11 and checking whether the result is non-terminating and non-recurring or not, or by using the method of contradiction. Let us prove that by using the method of contradiction. Let us assume that 1/√11 is a rational number. It means that there exist two co-prime integers p and q, such that 1/√11 = p/q, where p, q ∈ Z, and GCD (p,q) = 1.. By squaring both sides, we get, 1/11= p²/q², which can be rewritten as q²/11= p². It states that 11 is a factor of q² and hence a factor of q. Thus 11a = q. Therefore we get,  (11a)²/11= p², on solving we get 11a² = p² if we rearrange this, we get  a² = p²/11, this says that 11 is a factor of p, but since we have assumed that p and q are coprime numbers, they cannot have a common factor other than 1. Hence our assumption was wrong in the beginning. Therefore, 1/√11 is an irrational number.