Squaring made easy

Squaring Numbers in a Simple and Fun way

Squaring numbers is a tedious job if you do not know vedic maths shortcuts and other tricks and tips. Imagine having to waste your precious seconds in carrying out actual multiplications to find out squares during an exam. It can prove disastrous.

It is always advisable to learn various time saving tricks and alternate methods to find out percentages, fractions, squares, cubes, etc. because these will be useful in Quantitative Aptitude as well as Logical Reasoning sections of any competitive examination you take.

The idea of this post is contributed by Piyush Goel and it’s further edited by me to share with you a trend that he observed in squares of numbers.

If we talk about Square, Square is the result of multiplying a number by itself. So, square of 3 is 3×3=9. Similarly, square of 10 is 10×10=100. Since you know what is a square, observe this:

If we have to square 11, instead of 11×11=121,
we can simply put 1 (1*2)(1^2) and get 121.
Similarly, we can write the square of 12 as: 1(2*2)(2^2) and get 144. In the same fashion we can get the square of 13 as 1(3*2)(3^2)=169 and the square of 14 as 1(4*2)(4^2)=1|8|16 =196(after carrying over the 1 of 16 and adding it to 8).

Look at (2,4,6,8,10,12,14,16,18,20….difference of 2) & (1,4,9,16,25,36,49,64,81…….. difference is 3,5,7,9,11,13,15,17,19….) and difference of 3,5,7,9,11 is 2 so there is True Symmetry of 2.

11^2 = 1 2 1
12^2 = 1 4 4
13^2 = 1 6 9
14^2 = 1 8 16 = 100 + 80 + 16 = 196
15^2 = 110 25 = 100 + 100 + 25 =225
16^2 = 112 36 = 100 + 120 + 36= 256
17^2 = 114 49 = 100 + 140 + 49= 289
18^2 = 116 64 = 100 + 160 + 64= 324
19^2 = 118 81 = 100 + 180 + 81= 361
20^2 = 111^2 = 1 20 100 = 100 + 200 + 100 = 400
21^2 = 111^2 = 1 22 121 = 100 + 220 +121 = 441
22^2 = 112^2 = 1 24 144 = 100 + 240 +240 = 484
23^2 = 113^2 = 1 26 169 = 100 + 260+169 = 529
24^2 = 114^2 = 1 28 196 = 100 + 280+196 =576
25^2 = 115^2 = 1 30 225 = 100 +300 +225 =625
26^2 = 116^2 = 132 256 = 100 + 320 +256 =676
27^2 = 117^2 = 134 289 = 100 + 340 + 289 =729
28^2 = 118^2 =136 324 = 100 + 360 + 324 = 784
29^2 = 119^2 = 138 361 = 100 + 380 + 361 = 841
30^2 = 120^2 = 1 40 400 = 100 + 400 +400 =900
31^2 = 121^2= 121^2 = 1 42(111^2) =1 42 (1 22 121) =100 + 420 + 441 =961
41^2 = 131^2 = 1 62 (1 21^2) = 1 62 (1 42) (1 11^2) = 1 62 (1 42) (1 22 121) = 1 62 (961) = 100 + 620 + 961 = 1681

This is time saving and the best part is we do not have to mug up additional formulae or large algorithms of processes.

Please share your feedback and other interesting ways that you know.

About The Author :
Piyush Goel, born on 10th February, 1967. He has a Diploma in Mechanical Engineering and a Diploma in Vastu Shastra. Always wanting to do something new, Mr. Goel has written 15 Spiritual and World Famous Books with his own hands in Mirror Image in different ways. He is now known as “Mirror Image Man”. He also has World’s First Hand-written Needle Book “Madhushala” to his credit.

Edited by Preeti Patawari

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Calculate Square Root Quickly Without Calculator

In this post, we will learn how to find the square root of numbers which are not perfect squares. The answer we get using this quick calculation technique gives us an approximate answer. However, approximation becomes a necessity when we are attempting questions in a competitive exam, where time is short and options are given. Most of the time we’re not required to get exact answer.

Prerequisite to Use this Method

The prerequisite of using this method is you should remember the squares of as many numbers as possible. I would recommend that one should memorize square of numbers from 1 to 50. Later this can be extended t0 100.

It will be a wonderful idea to first learn the short cut method of finding the square of any number. Other awesome short cut method which you should consider knowing before moving forward is Herons method of finding roots.

Square Root of any number which is not a Perfect Square

Square Root = Sq. Root of Nearest Perfect Square + {difference of the given number from the nearest perfect square / 2 x (Sq. Root of Nearest Perfect Square)}

For example,

Short cut to find the Square Root of 47.
Square Root of 47
= 7 + (47-49) / 2 x 7 (since the perfect square closest to 47 is 49; we will take square root of 49 i.e. 7 for calculations)
= 7 – 2/2×7
=7 – 1/7
= 6.86 (approx).
The exact answer is 6.85565

Short cut to find the Square Root of 174
Square Root of 174
= 13 + (174-169) / 2 x 13 (since the perfect square closest to 174 is 169; we will take square root of 169 i.e. 13 for calculations)
= 13 + 5 / 26
= 13.19 (approx).
The exact answer is 13.1909

Short cut to find the Square Root of 650
Square Root of 650
= 25 + (650-625) / 2 x 25 (since the perfect square closest to 650 is 625; we will take square root of 625 i.e. 25 for calculations)
= 25 + 25/50
= 25.50 (approx).
The exact answer is 25.4950

This method has got a limitation. You can use it only till the point you remember the square
If you’ve questions related to the above shortcut method or anything else related to mathematics, please post on QM’s Q & A platform. For anything else you can comment below.

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How to Quickly Calculate Square of Three Digit Numbers?

Learn to Quickly Calculate Square of Any Three Digit Number

The method of squaring any 3 digit number is an extension of my last post on finding square of any two digit number. To understand and appreciate squaring of 3 digit numbers you should be well versed with shortcut of squaring any 2 digit number. Let us start learning this with the help of an example.

What is the square of 384?

Step 1: To begin with ignore the 3 of 384. You are left with only 84, a two digit number. Using the method of squaring 2 digit numbers, find the square of 84. We get the answer as

Square of 8  |   twice of 8 X 4   |  square of 4

64            64           16

7056 (consolidating the result obtained above)

Step 2: This step is new and different from what we’ve learned in the previous post for squaring 2 digit numbers. Watch carefully.

We have to multiply the first and last digits of our original number and double it. Essentially, that is multiplying together 3 and 4 and then doubling it. Hence we get 24.

Add this number directly to the two left hand digits of our number obtained from the first step.

7056

Add 24 to 70. 70+24=94. So 7056 gets converted to 9456.

Step 3: In the first step we left out the first digit of our number and squared the last two digits. Now we will forget about the unit’s digit 4 and square the first two digits i.e. 38 as before just omitting to square the last digit 8.

Square 38 as a regular 2 digit number, except that you omit the 8 squared.

Square of 3 | twice of 3 X 8

9       |   48

Step 4: Consolidating this with the result obtained in step 2,

9   |  48   |  9456

14        7       456

Hence the answer is 147456.

I’ve shared this method of squaring 3 digit numbers as an extension to the shortcut of squaring 2 digit numbers. Initially you might feel that the traditional method is quicker than having to memorize and execute these steps. However, this method can prove to be quicker than the useful one only if you master this technique with lot of practice.

Do you think this method will help you in reducing the time to calculate square of 3 digit numbers?

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How to Quickly Calculate Square of Any Two Digit Number?

In this post I’ve tried to improvise on the method of squaring presented in one of the earlier posts on Quickermaths.com itself. This trick of squaring any two digit number with ease is inspired by squaring techniques from the book – The Trachtenberg Speed System of Basic Mathematics

I would like to explain this method of squaring any 2 digits number with the help of an illustration.

What is the square of 32?

Step 1. In finding the last two digits of the answer, we shall find the square of the last digit of the number. Square the right-digit digit, which is 2 in this case. Hence we get 04

_ _04

Step 2. We shall now need to use the cross product. This is what we get when we multiply the two digits of the given number together. Multiply the two digits of the number together and double it: 3 times 2 is 6, doubled is 12: We write 12 as 2 and carry over 1 to the next step.

_ 24

Step 3: In finding the first two digits of the answer we shall still need to square the first digit of the number. That means we square the left hand figure of the number. Here square of 3 will be 9. Add 1 which is carried over from last step. Hence we get 9 + 1 =10

1024

That’s the answer.

This method can also be compared with another shortcut to find the square of any number posted by me on Quickermaths.com in the past.

Let’s try another example by squaring 64.

Square of 64 =     Square of 6 | double of cross product of both given digits 4 & 6| square of 4

Square of 64 =      36 |2x6x4 | 16

=      36 | 48 | 16

Collapsing the numbers

=      36 | 48 + 1 | 6

=      36 + 4 | 9 | 6

=      40 | 9 | 6

Hence the answer is 4096.

Have you come across any other squaring method like this?

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Shortcut to Find Square of 2-Digit Numbers with Unit Digit as 1

Special shortcut methods of squaring 2 digit numbers

In previous articles we’ve discussed special shortcut Vedic Math Techniques to find the square of any number ending in 5 and square of 2 digit numbers ending in 9.  In this article we will discuss –

Square of 2-digit numbers whose unit digit is 1

Let us take a 2 digit number in its generic form. Any two digit number whose unit digit is 1, say a1 can be expressed as 10a+1, where a is the digit in ten’s place

Square of a1= a2 | 2xa | 1

Here, ‘|’ is used as separator.

That means for the left most part of the answer, a is squared, hence first part will be a2. The middle part will be twice of a and the last or the right most part will always be 1.

Let us see a few examples.

(21)2= 2 squared | 2 . 2 |1 = 441
(31)2= 3 squared | 2 . 3 |1 = 961
(41)2= 4 squared | 2 . 4 |1 = 1681
(51)2= 5 squared | 2 . 5 |1 = 2601 (Here the square of 5 is 25 but since the product of 2.5 is 10 we write down 0 and add 1 to 25). Similarly,
(91)2= 9 squared | 2 . 9 |1 = 8281

Square of 3-digit numbers ending in 1

Now let us try to extend the above shortcut method to 3 digit numbers as well. Let us straight away start with an example 131.

Like earlier separate the given number in 2 halves, left hand side will have digits other than 1 and right hand side will have 1 as usual.

Hence, the answer is

(131)2= 13 squared | 2 . 13 | 1 =

= 169 | 26 | 1

= 17161

Let’s take another example of squaring a three digit number ending in 1.

261 = 26 squared | 2×26 | 1

= 676 | 52 | 1

= 68121

From the above illustrations you must have noticed that higher the initial number to be squared higher is the square in first part. Hence to extend this method to large 3 digit numbers one should be proficient in squaring any 2 digit number.

Try it yourself

21, 71, 91, 161, 231 and 321

Answers: 441, 5041, 8281, 25921, 53361 and 103041

This article is based on a suggestion given by a Quickermaths.com reader Vishal Mishra. If you also have some suggestions you can email it to me or post it here as comment. 

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How to Find Square of Numbers Ending in 9

Squaring any number ending in 9

We can easily calculate the square of any number ending in 9 using the method described in this post. Let us understand this method with the help of an example –

Finding the square of 39

Firstly add 1 to the number. The number now ends in zero and is easy to square.
40^2 = (4*4*10*10) = 1600. This is our subtotal.

In the next step, add 40 plus 39 (the number we squared plus the number we want to square) Read More

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Trick to Find Square of Numbers from 51 to 59

I’ll share with you one simple method of finding the square of numbers between 50 and 60. Like many other Vedic Mathematics methods, in this method also, we will get the answer in two parts. Since the numbers are in 50s and square of 50 is 2500, we will just use 25 in our calculations, ignoring the zeros.

  1. To get the first part of the answer, add the digit at the units place to 25 and write the sum
  2. To get the second part, calculate the square of units place digit and write it

It’ll be easier to understand this with an example. Read More

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Squaring any 2-digit number

A guest post by Maria Rainier

Shortcut to Squaring Any 2-Digit Number

What do you do when your calculator has been confiscated and the world is depending on you to square a two-digit number within a minute? Don’t panic – just follow three simple steps that require basic addition and multiplication, and you’ll be able to solve the problem in no time. If you practice enough, you’ll even be able to complete each step mentally, rendering scratch paper unnecessary. This will save you time on drills and strengthen your skills so you can tackle other challenges. Eventually, you’ll be able to solve multi-step squaring problems without ever breaking a sweat – or a pencil. Read More

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Perfect Square Puzzles

Squaring Puzzles- Find below 2 interesting puzzles related to square of some number. Hope you will like them.

Puzzle 1

The square of 13 is 169.  Take the last digit of the square, 9, and place it in the middle, making 196.  This is the square of 14, the next number above 13.

What are the next numbers which also have this property?

Puzzle 2

The following multiplication example uses every digit from 0 to 9 at least once.  Letters have been substituted for the digits.  Can you replace the letters and make the original multiplication problem?

B O G
x     B O G
_______________
L Y L E
G G U L
T U O O
___________________
U N I T O E

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Shortcut to Find Square of a Number

Today I will discuss a very simple method of finding square of numbers between 26 to 74 mentally. In the subsequent post we will cover higher numbers. So keep watching this space to learn squaring any number within your mind

Square (also called perfect square) is an integer that is the square of an integer; in other words, it is the product of some integer with itself. So, for example, 9 is a square number, since it can be written as 3 × 3.

How to find the square of any number?

To apply this method you should know squares of 1 to 25 by heart. You can refer to this table to learn the same. Read More

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