Prove that Root 5 is Irrational Number

Is root 5 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 take a look at the detailed discussion and prove that root 5 is irrational. 

Problem statement: Prove that Root 5 is Irrational Number
Given: The number 5
Proof: On calculating the value of √5,  we get the value √5 = 2.23606797749979...As discussed above a decimal number that does not terminate after the decimal point is also an irrational number. The value obtained for the root of 5 does not terminate and keeps extending further after the decimal point. This satisfies the condition of √5 being an irrational number. 
Hence, √5 is an irrational number. 

The square root of 5 is commonly also called "root 5". The root of a number "n" is represented as √n. Thus, we define the root of a number as the number that on multiplication to itself gives the original number. For example, √5 on multiplication to itself gives the number 5. In order to prove that root 5 is an irrational number, we use different methods like the contradiction method and long division method.

Given: Number 5
To Prove: Root 5 is irrational
Proof: Let us assume that square root 5 is rational. Thus we can write, √5 = p/q, where p, q are the integers, and q is not equal to 0. The integers p and q are coprime numbers thus, HCF (p,q) = 1.

√5 = p/q
⇒ p = √5 q ------- (1)

On squaring both sides we get, 
⇒ p² = 5 q² 
⇒ p²/5 = q² ------- (2)

Assuming if p was a prime number and p divides a², then p divides a, where a is any positive integer.
Hence, 5 is a factor of p².
This implies that 5 is a factor of p.
Thus we can write p = 5a (where a is a constant)

Substituting p = 5a in (2), we get 
(5a)²/5 = q²
⇒ 25a²/5 =  q² 
⇒ 5a²  =  q² 
⇒ a²  =  q²/5 ------- (3)

Hence 5 is a factor of q (from 3)
(2) indicates that 5 is a factor of p and (3) indicates that 5 is a factor of q. This contradicts our assumption that √5 = p/q. 
Therefore, the square root of 5 is irrational.

The value of the root 5 can be obtained by the long division method using the following steps:

• Step 1: First we write 5 as 5 00 00 00 and pair digits starting from one's place.
• Step 2: Now find a number whose square results in a number less than 5. 
• Step 3: The number obtained is 2. The quotient becomes 2 and the remainder obtained is 1.
• Step 4: Now the 00 is brought down and the quotient obtained in step 3 is doubled, i.e; (2×2 = 4). 
• Step 5: The unit digit with 4 should be 2 as 42×2 = 84 as it is less than 100. On subtracting 84 from 100 we get 16.
• Step 6: 00 is brought down and the quotient obtained in step 5 is doubled. Thus, 22 is doubled and 44 is obtained as the starting of a new divisor. 
• Step 7: The unit digit with 44 should be 3 as 443×3 = 1329 as it is less than 1600. On subtracting 1329 from 1600 we get 271.
• Step 8: Now bring the 00 down and the quotient obtained in step 7 is doubled. Thus, 223 is doubled and 446 is obtained as the starting of a new divisor. 
• Step 9: The unit digit with 446 should be 6 as 4466×6 = 26796 as it is less than 27100. On subtracting 26796 from 27100 we get 304.
• Step 10: The long division is continued further in order to obtain the required number of digits after the decimal.

As we can see the value of root 5 does not terminate after 3 decimal places. It can still be extended further. Hence, this proves √5 an irrational number.

☛ Also Check:

• Prove that Root 2 is Irrational
• Prove that Root 3 is Irrational
• Prove that Root 6 is Irrational
• Prove that Root 7 is Irrational
• Prove that Root 11 is Irrational

How do you Prove that Root 5 is Irrational?

Root 5 cannot be expressed in p/q form where p, q are integers and q is not equal to 0. On determining the square root of 5 we can find that it gives a value √5 = 2.23606797749979... which does not terminate after the decimal but only extends further. There are two ways to prove that root 5 is irrational, one by using the method of contradiction and the other by using the method of long division. 

Is 2 times the square root of 5 Irrational?

Yes, 2 times the square root of 5 is an irrational number as 2 × √5 = 2 × 2.23606797749979 = 4.472135954999579...A rational number multiplied with an irrational number is irrational. 2 is a rational number which is multiplied with root 5 which is irrational. Thus, 2 times the square root of 5 is irrational.

How to Prove Root 5 is Irrational by Contradiction?

In order to prove root 5 is irrational using contradiction we use the following steps:

• Step 1: Assume that √5 is rational.
• Step 2: Write √5 = p/q
• Step 3: Now both sides are squared, simplified and a constant value is substituted.
• Step 4: It is found that 5 is a factor of the numerator and the denominator which contradicts the property of a rational number.

Is 3 Times the Square Root of 5 Irrational?

Yes, 3 times the square root of 5 is an irrational number as it can be written as 3 × √5 = 3 × 2.23606797749979 = 6.708203932499369... A rational number multiplied with an irrational number is root 5. We can see that any rational number multiplied with root 5 will be irrational. Thus, 3 times the square root of 5 is irrational too.

How to Prove that 1 by Root 5 is irrational?

When we rationalize 1/√5, we get 1/√5 × √5/√5 = √5/5. As we know, that √5 is an irrational number, and dividing an irrational number by a rational number also gives an irrational number, thus we can conclude that 1/√5 is irrational.