How to Manually Calculate the Square Root of a Number

By Matrixkavi

First of all, let me assure you that this is the easiest thing to do in mathematics ( at least for me). Talking straight about the subject, there are basically two main methods to find out the square root of a number :

a) Method of factorisation

b) method of division

and here goes explanation for the two methods:

Method of factorisation

This method is convenient to find the square root of numbers whose roots are exactly whole numbers. Such numbers are called exact(whole) numbers. This is a good method for beginners in mathematics, but as the numbers get larger, finding out the square roots become cumbersome. The method could be explained in simple words as follows :

We resolve the given number into product of prime factors and choose a number out of every pair of recurring ones.

For example, 900= 2x2x3x3x5x5

so, square root of 900 is given by 2x3x5, (that makes it 30).

Method of division

As said earlier, the method stated above can be cumbersome if the number is too large, or if the number cannot be factorised into exactly into pairs of whole numbers. To find the square root, we just adopt a simple method as described below:

We separate the digits of a given number whose square root is to be found out into pairs( each pair called period) beginning from the right. If the given number consists of an even number of digits, then, the digits of this number will give exact number of pairs. If the number consists of odd number of digits, don't worry. The leftmost digit should be left as a single, while pairing up the rest of the numbers. The periods have great importance because the number of digits in the resulting square root will be equal to the number of periods. For example, square root 0f 2121387 will have 4 digits. ( Hope this piece of information will help you a lot if you are going to write any objective type test in future!).

An example for the square root is as shown below. The figure is self explanatory.

vote up vote downshare print flag

Was this Hub ...?

Science and Space: Indian space history crosses three decades of success

Science and Space: what is the need to clone a hard drive?

CommentsLoading...

Go to last comment

Teresa McGurk 3 years ago

Em -- I know I have a cold, and am maybe not thinking straight today, but can you explain why square roots are useful? I've always been embarrassed that my knowledge of math is so lame. And in the last example, reading down the left column, where did the number 23 come from?

NO -- on second thoughts, I'll just slink off into the sunset -- I don't think my brain can handle this without a stiff drink. (And I don't drink.)

Matrixkavi Hub Author 3 years ago

okay let me explain it to you... I just grouped the digits into pairs starting from the right... then I got a single 1 on the left. the greatest square near that number is 1, whose root is 1. so i wrote 1 on the left and also below the number.. note the number after the paranthesis on the right.. i could write a 1 there

then i performed subtraction. i took the next pair down, ie,. 73. Then i just added the '1' on the left and the '1' after "(" to get the 2 in second step. Next, i looked for the possible combinations of numbers (0-9) with 2 such that i get a number near 73.. for example let the number be "y" ... then (2x10+y) x y should give a number less than or equal to ( but close to) the number 73. I got y in the second step as 3., ie, 23x3=69<73.. took the difference as before to get 4

then i wrote down the next pair down. As said above, i added 23 and 3 together to get 26 on the left.. looked for possible combinations only to find out that "y" in this case is 1 ie., (26x10+y)xy<444 can only be true only if u give y=1.. And i performed the subtraction to get 1 83 and wrote down 89, making the next level left integer = 261+1= 262.. proceeding to the next level, i got (262x10 +y) x y= 18389 for y=7..

thus square root is 1317 obtained on the right

maven101 Level 5 Commenter 3 years ago

And I always thought that to get a square root ( and I agree with McGurk, who needs it ), you simply multiplied a number by itself.... I found that a quick process of elimination got me to the answer fast enough... now, don't get me started on string theory......Larry

Teresa McGurk 3 years ago

Ah. Thank you so much for taking the time to explain. I do appreciate it.

ssas 3 years ago

asas

Het 3 years ago

Thank you so much Matrixkavi . I was trying to understand this method since days. And now its like very easy for me to find sq roots of any num.

Thankyou,

Het

Matrixkavi Hub Author 3 years ago

Thank you Hets... By the way, the above stated method is not easy to implement in programming. To find out square root(x) of a given number (N), it would be better if you implement the following formula in a loop

xnew = (x+N/x)/2

Initially, assign xnew=N/2 and then perform the check. If it is wrong, put x= calculated value of xnew. Continue the steps till the LHS and RHS becomes equal.. the final answer is obtained as x

Matrixkavi Hub Author 3 years ago

correction: assign x=N/2 at the start of execution and find out xnew. If xnew < x, assign x=xnew and recalculate using the formula

Swati Rana (India) 2 years ago

well finding out square roots has been the most frightening thing for me since school days. but the way u explained it is quite good. i hope i'll be able to solve maths problems now.. thanks :-)

Matrixkavi Hub Author 2 years ago

thank u swati