[Maths Class Notes] on Value of Root 5 Pdf for Exam

The Square Root of 5 ([ sqrt{5}])

The square root of five is a positive, real number and gives the prime number 5 when it is multiplied by itself. To differentiate it from the negative numbers that tend to hold the same properties, it is also referred to as the “principal square root of 5”.

The value of square root 5 is 2.2360. 

Different Methods To Find the Square Root of 5 

1. The Average Method

The steps that are to be followed for finding [ sqrt{5}] by this method are as follows: 

Step 1: Find out two perfect square numbers that happen to be very close to the given number, which, in this case, is 5, on either side. 

Now, the immediate perfect square lesser than 5 is 4 and the immediate perfect square which is greater than the given number is 9. 

Step 2: Write down the square roots of the perfect squares that were found in the first step. 

So, the square root of 4 would be 2, and that of 9 would be 3. 

Step 3: Further, it can be inferred that the square root of 5 tends to lie somewhere between 2 and 3, i.e., the square roots from the second step. 

Step 4: Divide the given number whose square root can be determined by any of the numbers from the second step. 

So, 5 can be divided by either 2 or 3. If we divide 5/2, then we get 2.5 as the answer. 

Step 5: Now, the average of the quotient and divisor from the last step is to be calculated. 

Average =[frac{2+2.5}{2} = frac{4.5}{2} ]

If you wish to find an accurate and precise value of the square root of 5 through this step, you can find the average of the answer from the previous step and keep finding it till you get the exact value. 

[ sqrt{5}] = 2.25

2. The Number Line Method

The steps that are to be followed for finding  [ sqrt{5}] by this method are: 

Step 1: Start by the construction of a number line with minimum units that are equal to the square root of the perfect square that is on the immediate right of the given number. So, in this case, construct a number line with a minimum of 3 units to the right and 3 towards the left of the starting point which is to be referred to as zero as 9 is the immediate perfect square greater than 5 and its square root is 3. 

Step 2: Represent the given number as the sum of two perfect squares and determine the square roots of those two numbers. So, 5 can be expressed as 4+1; the square root of 4 is 2 and that of 1 is 1. 

Step 3: Now, label the point that represents 2 as “A” and from there, draw a perpendicular to the number line AB, whose length should be 1 unit. 

Step 4: Join the tip of the line in step 3 to the reference point “O”. The measure of OB will give you the value of the root of 5.  (the length of this hypotenuse is about 2.3-2.4 which is the value of  [ sqrt{5}]) 

3. The Long Division Method 

The steps that are to be followed for finding  [ sqrt{5}] by this method are: 

Step 1: Determine the square root of 5 upto 4 decimal places; that gives us 5.00000000. The digits after the decimal point are then to be grouped in pairs. 

Step 2: Take a perfect square with a value lesser than that of 5; choosing 4, we get 2 as its square root. 

Step 3: Write 2 in the place of both the quotient and the dividend and write the number 4 below 5. Subtract 4 from 5 to get 1 as the difference. 

Step 4: Carry down the first pair of zeroes in the dividend and place a decimal point in the quotient. Now, add 2 to the divisor to get 4 as the sum. Take a number that succeeds 4 to get a 2 digit number, so that when the latter is multiplied by the number taken, the product is less than 100. 

Step 5: Continue the above process from step 4 to get the 2.2360 as the final answer in the quotient. 

Fun Facts:

  • Square roots of negative numbers are not real numbers. They are imaginary numbers. Hence, the knowledge of complex numbers is required for understanding their square roots.

  • It is believed that the symbol for square root is derived from the first letter of the word ‘radix’ in Greek and Latin which means a ‘root’ or ‘base’.


The square root of 5 can be written as [ sqrt{5}] in the radical form and as (5)½ or (5)0.5 in the exponential form. The square root of 5 can be rounded up to five decimal places is 2.23607. It is the positive solution of the equation x2 = 5.

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