How to obtain the square of a 3 digit numbers

 

You often see a number of questions on obtaining the square of 2 or 3 digit number in the competitive examinations. In this article on Mathematical shortcuts, we will see,  How to obtain the square of a 3 digit numbers.

If you have read my article on Mathematical Shortcut for squaring numbers (If you have not, go through it once, you will like it 🙂 ), you probably know how to obtain the squares  of any number.

No doubt, the shortcut is very useful if you have to obtain the squares till 300.  We will discuss the limitation of this technique later.

 

Using 3 x 3 Multiplication


In this technique, we use the 3 x 3 criss-cross multiplication technique to derive the general formula to obtain the square of any 3-digit number.

If you do not know, please go through the article  Multiply any 3-digit numbers which describe the technique of  criss-cross method of multiplication.

Assume a general 3-digit number, let’s call it abc. Let’s use the criss-cross multiplication to obtain the square of abc2.

The steps are illustrated in the following figure :

 

Square_numbers3_3

 

This can be re-written as shown :

 

Square_numbers3_4

 

Step 1 :  Compute a2, ab, ac, bc and c2 and write as shown in the updated formula.

Step 2 :  Notice the second row of the formula is obtained by repeating the terms ab, ac and bc.

Step 3 :  Now, this is different from obtaining the square of a 2-digit number. Here, just compute b2 and place it below the middle ac term.

Step 4 :  Now, add the terms taking care of the carry-forward in each step to obtain the square of the number.

Let’s understand the steps with an example. Let’s say we want to compute 7132.

The entire steps are illustrated in the following figure.

 

Square_numbers3_5

 

Step 1:  Break the number such that a=7, b=1 and c=3

Step 2:  Compute a2, ab, ac, bc and c2 and write as shown in the updated formula.

Step 3:  Notice the second row of the formula is obtained by repeating the second, third and fourth term

Step 4:  Compute b2 and place it in the 3rd column below ac term.

Step 5:  Now, add the terms taking care of the carry-forward in each step to obtain the square of the number.

 So, 7132 = 508369

Let’s  take up another example. Let’s try to obtain the square of 6422.

The steps are illustrated in the following figure.

Square_numbers3_6

Notice the carry-over in each step.

Easy 🙂 ?

Now, consider an example like 2222. Notice that all the numbers are same.

Square_numbers3_7

 

Notice, how easy it would be to obtain the squares of numbers where all the digits are same.

There is a seperate article on how to obtain the square of  number when all the numbers are same. Please refer  to my detailed article  here.

 

Exercise : Try obtain the squares of the following 3-digit numbers:

  • 342
  • 101
  • 121
  • 111
  • 675

 

Using (a + b)2 Formula


Now, let’s go back to the limitations of the modified formula technique as discussed in Mathematical Shortcut for squaring numbers.

Let us consider the same example. How do we obtain  7132. The steps are illustrated in the following figure.

 

Square_numbers3

 

Step 1 :  Break the number such that a=7, and b=13

Step 2 :  Compute a2, ab and b2 and write as shown in the updated formula

Step 3 :   Notice the second row of the formula is obtained by just repeating the middle term ab of the first row.

Important: Notice the carry-forward in each stage. In the unit’s place we carry forward only 1 instead of 16. This is a very important difference. We keep 2-digits rather than 1-digit as in squaring 2-digit numbers.

In this case, we know both a2 and  b2 in this case. So we can use this technique as well.

 

Let’s take another example. Let’s say we want to compute 2532. In this a=2 and b=53.

 

Square_numbers3_1

 

I wrote 532= 2809 directly. If you do not know how I did it, please refer to my article “How to obtain the square of a number from 51 till 59”.

Let’s consider the square of 4822, you will understand the true limitation, I am talking about.

Let’s break the number such that a=4 and b =82 ( You can also break the number as a=48, b=2).

 

Square_numbers3_2

Compute a2, ab and b2. However, we do not know 822. So we need to compute 822 using the shortcut described in Mathematical Shortcut for squaring any 2-digit number. The rest of the steps are similar.

This method sometimes will prove really time consuming.

When you practice both the techniques, you will realize which technique to use by looking at the number itself.

Exercise : Can you obtain the squares of:

  • 1132
  • 7482
  • 1522
  • 7672

Please contact me if some point in the tutorial is not clear. I would be glad to help.

Kiran Chandrashekhar

Hey, Thanks for dropping by. My name is Kiran Chandrashekhar. I am a full-time software freelancer. I love Maths and Mathematical Shortcuts. Numbers fascinate me. I will be posting articles on Mathematical Shortcuts, Software Tips, Programming Tips in this website. I love teaching students preparing for various competitive examinations. Read my complete story.

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