The Last Frontier: Mental Square Roots!!


Now, that is impressive! Mental calculation of square roots?!? I can barely calculate square roots on paper!! How would I be able to calculate square roots on my mind?

Well, I can assure you that it is possible, and it is easier than you think. You can impress everybody at your job when in the need of calculating a square root, you provide the answer even before the others have found the calculator. Let’s see how it works.

The Technique

This technique works for numbers of 4 cyphers or less. In this post we are going to concentrate only in calculating the integer part of the result. We are not going to calculate the decimal of the root. It happens that most of the times, this is what really matters.

You will need to have mastered the mental technique of calculating square numbers. If you do not know how to do it mentally, read this post and master that technique first.

In order to explain the technique, the best way is to use an example. Let’s calculate the square root of 3456.

  1. If the number has less than 3 cyphers, then, you can use the following table to calculate the square root. Simply, select the number of the table whose square is closer and lower than the number you want to calculate the square root for. The table has an additional column, called Last Digit, that will be required below.
Number Square Last Digit
0 0 0
1 1 1
2 4 4
3 9 9
4 16 6
5 25 5
6 36 6
7 49 9
8 64 4
9 81 1
    For our example, the number has 4 cyphers, hence the previous table is not required.
  1. Next, we take the first two cyphers of the number (in this case, it is 34). Then we calculate the number whose square is closer to that number, but that is smaller or equal than it. In our example

5^2 = 25 and 6^2 = 36

hence the first cypher will be 5

  1. For the last two cyphers (in our case, those are 56) we use the Last Digit column of the previous table. Since, the last number of the two cyphers is a 6, this is the Last Digit. By checking on the table of basic squares for the numbers that have the Last Digit column equal to 6, the candidates of the table could be either 4 or 6 because both of them have a Last Digit of 6. At this point, we know that the square root is either 54 or 56 (that is, the combination of the number calculated in the previous step (5) with any of the candidates calculated here.

the second cypher is either 4 or 6

  1. In order to identify the correct one (either 54 or 56) we start by calculating the square of 56, which is 3136. Since this number is smaller than 3456, it means that 56 is the correct square root. If 56^2 were bigger than 3456, then it would mean that the result would have been 54.

The results is 56



The Workout

Perform the following workout a few minutes every day until you find yourself comfortable with the technique.

For the workout you will need the following equipment:

  1. A random number generator program. You can use for instance, this one for Android, this one for iOS or this one for web computer.
  2. A timer. You can use any timer that comes with your cell phone.

Mental Math workout : square root

  • Start the random numbers generator and configure it to generate numbers between 0 and 9999 (two cyphers random number). Generate 10 random numbers and write them down on a paper.
  • Start a timer to control your time. Your goal is to perform 10 squared roots in 1 minute.
  • Calculate the squared root of the 10 numbers as quickly as possible, using the technique explained above.
  • When calculated the squared root of a number, write the result on the paper and go too the next number. Do not check now if you have correctly calculated the square root.
  • When the 10 numbers are done, stop the timer.
  • Then, check how many squared roots were correct. For each wrong answer, add 20 seconds more to the total time.
  • Write down on your Training Sheet the total time it took you to calculate the squared roots.
  • Repeat the whole process five times.
  • Next day, try to beat the smaller total time.


This chapter concludes with the basics of mental math. Together with the previous posts about mental calculation of addition, subtraction, multiplication, division and square, they all provide a full overview of the basic mental calculation techniques. This is the minimum you need to be a master of mental math.

In future posts, I will show you additional tricks and rules that you can apply for specific calculus situations, all of the based on those 6 basic mathematical skills.


photo credit: bdu via photopin cc

photo credit: mattemagnus via photopin cc

One Response to The Last Frontier: Mental Square Roots!!

  1. Robert Loggins 17/10/2014 at 19:27 #

    You might want to check your math. The square root of 3456 is 58.78………..and NOT 56 as you say in your example

Deja un comentario

Powered by WordPress. Designed by Woo Themes