How to obtain the square of any 4-digit number

 

I can tell you with lot of confidence, that you donot often find many questions where you need to obtain the square of a 4-digit number.  Squaring a  2 or 3 digit number is probably the most commonly asked question in any competitive examinations. However, obtaining the square of a 4-digit is very similar to obtaining square of a 2 or 3-digit number

There is a detailed article on Mathematical Shortcut for squaring numbers and Squaring any 3-digit number. Please go through it, if you haven’t already.

Assume a general 4-digit number and call it abcd. Let’s use the criss-cross multiplication to obtain the square of abcd. The steps are illustrated in the following figure:

Square_numbers4_1

 

This can be re-written as shown :

 

Square_numbers4_2


Step 1 :  
Break the number such that you have a, b, c and d.

Step 2 :  Compute a2, b2, c2, d2, ab, ac, ad, bc, bd and cd.

Step 3 :  Notice the first row.  You have 7 columns. You start with a2, ab, ac and ad.  Next, leave bc. Place bd, cd, and d2. That completes the first row.

Step 4 :  Now the second row is obtained by repeating the first row elements. We take the middle 5 terms, leaving out the end terms aand d2.

Step 5 :  The third row, have 3-elements. So you b2 and cas the end elements and bc as the middle term.

Step 6 :  In the forth row, you just repeat the middle term bc from the third row.

So, far for populating the elements. It might look difficult. However, if you practice the steps on few random 4-digit numbers, you will understand the pattern.

Add up all the elements taking care of the carry-forward in each addition.

 

Let’s understand the steps with an example. Let’s use this technique to obtain the square of 12342.

The steps are illustrated in the following figure.

 

Square_numbers4_3


Let’s consider another example. Let’s use this technique to obtain the square of 34782.  The steps are illustrated in the following figure.

Square_numbers4_4

 
Now, consider an example like 22222. Notice that all the numbers are same. There is a separate article on how to obtain the squares of any number if all the digits are same. You can go through the article here.

 

Square_numbers4_5

 

Notice, how easy it is to obtain the square of such a big number. Very easy right 🙂

Try to obtain the squares of the following 4-digit numbers:

  • 3499
  • 1012
  • 1111
  • 5678

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.

7 Comments

  1. Hey Megan,

    I am glad you liked it. I would be interested to know about your project and like to contribute to it as well.

  2. A lot of thanks for all your work on this web page. My daughter takes pleasure in making time for investigation and it’s obvious why. Many of us know all regarding the compelling form you create priceless things on your blog and therefore improve response from other individuals about this matter so our own child is without question being taught a whole lot. Have fun with the rest of the new year. You are conducting a first class job.

  3. Hi! I am a grade 10 student. Thanks for posting such a nice method sir. But I used to square 3-digit, 4-digit and even 5-digit numbers using another trick, which is far quicker than your method. This trick is based on the identity ->
    A^2 = (A-d) (A-d) + d^2

    • Hi Sunny,
      I am glad you liked it. I use the technique that you mentioned. However, you need to be very good with multiplication to use that shortcut. I will write about it soon.

  4. You have an error that confused me till I recognized it was an error. That is in the squaring of a three digit number. The example of 21 squared — the the left it should be
    a=2
    b=1
    But you have something different. When I saw your next example then I could see that there was an error in the previous example. No biggy but it is an error. I like your site!! I have some discoveries that you might be interested in. If you are interested then let me know.

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