Square 2+ Digit Numbers Mentally

For my first Instructable, I posted my first Instructable about a fun trick I had learned back in junior high for taking the cube roots of large numbers (6 or more digits) mentally. It was a well liked and well received 'ible, getting featured and making the popularity section in spite of being of being about math, so thank you very much for the warm welcome everyone!

The Instructable brought about many requests for additional mental math skills, if I happened to know any, and concerns about the practicality of mental cube root extraction. Of the comments, however, one in particular intrigued me. A lone user suggested I apply the concepts of Vedic Mathematics to my mental math repertoire. Having never heard of vedic mathematics, I went and found what I could off the internet. What I found was fascinating, and I will share with you a trick from Vedic Math today. So today, I will teach you how to square 2 digit numbers mentally, enjoy!

For more information on Vedic mathematics, including the background and history of Vedic Math, check this out.

Step 1: A Note on Practice

This instructable will hopefully teach you a few strategies to quickly determine the squares of numbers. However, to get the best results, you will have to practice a fair bit, and be pretty good at multiplying numbers in your head.

So, if you find you can't solve mental squares as quickly as you'd like, one of the best ways to improve is to write out a whole bunch of multiplication problems on a sheet of paper, things like 25x46, 17x64, 7x920, and so on. Then, mentally solve them (don't write anything down, no calculators!) and finally, check your answers with a calculator. The more mental math problems you do, the better you'll be at it.

It should be noted that 2-digit integers will be used for the most part in this Instructable. However, the techniques presented here can be applied to decimals and numbers of arbitrary length. With practice, it is possible to mentally calculate squares of decimals and 3 or more digit numbers. The only limit is your ability to remember numbers and multiply numbers mentally quickly.

Comments

[abhishek727]

Handy method. Here is something you'd find useful.

whenever you have a number which ends in 5 you can use this. multiply the number preceding 5 with its successor. say you have 75 then multiply the preceding  number 7 with it's successor 8 ( for 85 it would be 8*9), Write down the result and just write 25 to the end of this result. 

So 75*75 = 7*8 = 56 now write 25 to the end of this digit = 56 25 this is the ans 5625.

The idea is to treat the number as 2 distinct numbers 1st part is the number preceding 5 and the second part is the digit 5.

so for 125 * 125 we get 12 & 5
12*13 = 156 and write 25 to the end = 156 25  ---> 15625

[linkmaster03]

Very cool instructable. Good job dude, I'd love to see more of these. From what I see, you have a couple more in your profile which I am heading to check out right now.

[mahi16]

Method for checking.
1. we get the digit sum of a no. by "adding across" the no. For instance, the digit sum 0f 13022 is 8.
2. we always reduce the digit sum to a single figure if it is not already a single fig.
3.In "adding across" a no. we may drop out 9's. Thus if we happen to notice two digits that add up to 9, such as 2 and 9, if we ignore both of them; so the digit sum of 99019 = 1 at a glance.(If we add up 9,s we get the same result)
4. because :nine don't count" in the process, as we saw in3 step, a digit sun of 9 is the same as a digit sum of zero. The digit sum of 441,e.g = 0.

You may use this for Multiplication, Cubes , Squares etc.

you may also check weather your squared no, is correct or not.
I will explain it:-
take an example for 207.
square(207)= 42849
now add digits of LHS and RHS separately. we get
Sq(2+7) = 18
Sq(0) = 18
0=2+7
0=9
0=0

thus your calculation is correct.

take another example.
Sq(897) = 804609
sq(6) = 18
36 = 180
0=0.

[purduecer]

Right you are, sir. After many a school-related delay, I posted an instructable on the digital root topic, which I encourage you to check out if you find the time. http://www.instructables.com/id/Mental_Math_Digital_Root_Extraction/

Thanks,
Purduecer

[mahi16]

Great work .

[telittleone]

Very nice. At first it does seem a bit complicated, but its much easier than multiplying the numbers mentally in your head.

[PKM]

For the people wondering how/why this works (and what on earth you are adding the square of the deficiency for)...

Ever wondered why, for instance, 9*11 is one less than 10*10? 4*6 is one less than 5*5? 164*166 is one less than 165*165? (It is, trust me).

This is a specific instance of the identity (x+a)(x-a) == x² - a², called completing the square, and the Vedic squares method is another application of this. If you add a² to each side you get x² == (x+a) * (x-a) + a².

For example, if x=67 and a=3,

x * x = 67 * 67

(x+a)*(x-a) + a*a = (70 * 64) + 3*3
= 4480 + 9
= 4489
= 67²

[purduecer]

yes, thank you very much for clarifying this point (I appreciate it)