Prove that Root 7 is Irrational Number | Is Root 7 an Irrational? [Solved] (2024)

Is root 7an irrational number? A numberthat can be represented in p/q form where q is not equal to 0 is known as arational numberwhereas numbers that cannot be represented in p/q form areknown 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 7is an irrational number.

Prove that Root 7is Irrational Number

The square root of 7 will be an irrational number if the square root of 7 gives avalueafter decimal that isnon-terminating and non-repeating. Before going forward, let us discuss quickly, the root of the number "n". The square root of a number n is denoted by the symbol√n. As wemultiplythe root of a number to itself it gives the original number whose root we have taken. Therefore, √7on multiplication to itself gives the number 7.

We can prove that root 7 is an irrational number, with the help of various methodslike the contradiction method and long division method.

It is known that a decimal number that has a value that does not terminate anddoes not repeat as well, then it is an irrational number.The value of √7is 2.64575131106... It is clear that the value of root 7 is also non-terminating and non-repeating. This satisfies the condition of √7being anirrational number.Hence, √7is an irrational number.

Prove That Root 7is Irrational by Contradiction Method

We canprove that root 7 is irrational also by using the contradiction method.

To prove: We want to prove that root 7is irrational.
Proof:We will start with the contradictory statement of what we have to prove. Let us assumethat square root 7is rational.

Now since it is a rational number, it can be written in the form p/q, wherep, q ∈ Z, and coprime numbers, i.e., GCD (p,q) = 1.

As we know, 7 is a prime number. Using the theory,which says that, if a prime number k divides m², then it also divides m, and vice versa is also true.This implies that since 7 is a factor ofp² then it will also be the factor of p.

Thus we can write p = 7a(where ais someconstant)

Substituting p = 7ain equation (2), we get
(7a)²/7= q²
⇒ (49a²)/7= q²
⇒ 7a²= q²
Rearranging,
⇒ a²= q²/7 ------- (3)

This showsthat 7 will also be the factor of q.
Now, initially, we have assumed thatp and q are the coprimenumbershence only 1 is the number that can evenly divide both of them. But here, we have 7 is the common factor ofp and q, which is contradictory to our initial assumption.

This proves that the assumption of root 7 as a rational number was incorrect. Therefore, the square root of 7is irrational.

Prove That Root 7is Irrational by Long Division Method

There is one more method by which we canprove that root of 7 is irrational and that is the long division method. We find out the value of the root 7 by long division method and check whether we get the non-terminating and non-repeating value after decimal ornot.

The long division method can be appliedusing the following steps:

• Step 1: Add pairs of 0 after 7as 7.00 00 00 and pair the digits starting from the right and find a number whose square is less than or equal to the number 7, it will be our first divisor andquotient. We have 2 square the number and subtract the result from 7, 3is the remainder.
  
• Step 2:Take the next pair of 0 down after the remainder, as 00 is brought down, we get 300 as the next dividend,and double the first quotient to get the partialdivisorof this step. The unit digit the divisor will be the number which on multiplying with the complete divisorthus formed,gives a number equal or less than the new dividend. Here, we get 6at the units place, and 46is our divisor and 6is our quotient. Subtract the result after multiplying 46with 6from 300, and note down the remainder.
  
• Step 3:Take the next pair of 0down after the remainder of the previous step to get the dividend, 2400 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 thisstep. Here, we get 52. The unit digit the divisor will be the number which on multiplying with the complete divisorthus formed,gives a number equal or less than the new dividend. Here, we get 4at the units place, and 524is our divisor and 4is our new quotient. Subtract the result after multiplying 524with 4from 2400, and note down the remainder.
  
• Step 4:Repeat the processuntiltherequired number of digits after the decimal is obtained.

See the following image to seea few of the steps in the long division of the 7.

As we can see the value of root 7 does not terminate after 3 decimal places. It can still be extended further. Hence, this makes √7an irrational number.

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FAQs on Is Root 7an Irrational?

How do you Prove that Root 7is Irrational?

We can prove that root 7 is an irrational number by using various methods like the long division method, and method of contradiction. Also, the square root of 7 will be an irrational number if it gives a value afterthe decimal point that does not terminate and does not repeat. The value of root 7 is2.64575131106...itis clear that it is non-terminating and nonrepeating, hence√7 an irrational number.

Is 2times the Square Root of 7Irrational?

Yes, 2 times the square root of 7 is irrational. To find out whether 2 times the square root of 7 is anirrational number, we multiply both the numbers and check the result. On multiplying 2 with root 7, we get 2× 2.64575131106... =5.29150262212.. which is a non-terminating andnon-repeating term, thereforethe product of the two is an irrational number.

How to Prove Root 7is Irrational by Contradiction?

We canprove it by using the contradiction method where we assume the root 7 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 7 is irrational. To prove root 7 is irrational using contradiction we use the following steps:

• Step 1: Assume that√7 is rational.
• Step 2: Hence,√7 = p/q
• Step 3: Now both sides are squared, simplified and a constant value is substituted.
• Step 4: It is found that 7is a factor of the numerator and the denominator which contradicts the property of a rational number.

Therefore it is proved that root 7 is irrational by the contradiction method.

Is 3 Times the Square Root of 7Irrational?

Yes, 3times the square root of 7is an irrational number. 3times the square root 7is written as 3× √7= 3×2.64575131106... = 7.93725393318... Here, we get the result that is nonterminating as well as nonrepeating. Thus, we can also conclude thatany rational numbermultiplied with root 7will be irrational. Hence, 3times the square root of 7isirrational too.

How to Prove that 1 by Root 7is irrational?

We can prove 1 by root 7 is irrational using various methods, such as directly finding the value of 1 by root 7 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 finding the value of1/√7. The value of 1/√7 is0.377964473.. which extends to infinity and the terms are nonterminating as well as nonrepeating, hence we can clearly say that1/√7is an irrational number.