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.

Step 2: Strategies for Computing Squares

Alright, hopefully at this point we've either practiced or ignored the previous step and clicked ahead anyway. Suppose we're given a number, such as 45. How can we square the number mentally?

Turns out there are a couple different ways:
1) Memorize every 2-digit number, and simply recite the square from memory (rote memorization).
2) Multiply the number by itself mentally (45*45)
3) Split 45 into components and apply the identity (a+b)² = a²+2ab+b²
4) Apply a high speed Vedic Mathematics technique for squaring numbers

As can be seen from above, the first method isn't general or practical. Even if you knew the squares of every 2 digit number, that's only 99 numbers (I'm including 1-9 here, I'm aware they are not 2-digits), and in the broader realm of math, won't get you very far.

Method 2 seems like a reasonable approach, though. However, the numbers you have to hold in memory are not very nice. At one time, you must remember that 5*45 is 225 and 40*45 is 1800 and add them in your head. For many numbers, this proves to be quite difficult.

Method 3 is a decent method, decidedly better than the first 2. We could express 45² in the following way:
45²= (40+5)²=40²+2*40*5+5² = 1600 + 400 + 25 = 2025

But, better than any of those is the 4th method, which I will show you next.

Step 3: Mental Squares: the Vedic Way

The technique used here is the same one that appears in the wikipedia page (you did read the wikipedia page, didn't you?). It can be summarized using the following phrase:

"Whatever the extent of the deficiency, lessen it still further by that extent, and add to it the square of the deficiency"

That's a little cryptic, so below is the technique expressed in plain English:
For a given number, choose an arbitrary base. Now, determine the difference between the number and the base (this is the "deficiency"). Add the difference to the number, and multiply this new number by the base. Finally add the square of the difference to the number, which gives you the final answer.

That can be a little hard to digest, so we will apply the technique using the example of 45² from step 2.

1) Given 45², choose 50 to be the base. Now, the deficiency is 45 - 50 = -5.
2) Add the deficiency to the number, so 45 + (-5) = 40.
3) Multiply the new number by the base: 40 * 50 = 4*5*100 = 2000
4) Add to the product the square of the deficiency: 2000 + (-5)² = 2000 + 25 = 2025

This technique, when practiced, allows for rapid mental square computation. More examples in the following steps.

Step 4: Prove It!

For most of you, using the trick and assuming it works is enough to satisfy you. Still, some of the mathematical purists out there need a proof, so they can take security in knowing this technique is general. A wise user by the name of PKM posted the same proof in the comments below, but I added a step, so that people who don't read comments could have the proof as well.

So, we start with the following algebraic expression, which always holds true (foil out the right side and this can be seen):
x²-a² = (x+a)(x-a)

Now, simply isolate the x² term:
x² = (x+a)(x-a) + a²

The above expression is the mathematical form of the vedic sutra.

Step 5: A Few More Examples

Additional examples using the vedic method:

Example 1: Compute 67²
Choose a base (70).
Now, determine the deficiency (67 - 70 = -3), and add it to the number (67 + -3 = 64).
Multiply the base by the sum (64*70) = 7*64*10 = 448*10 = 4480
Finally, add the square of the deficiency to the product: 4480 +9 = 4489
And voila! There you go :-)

Example 2: Compute 12.5²
Choose a base (10)
Determine the deficiency (12.5 - 10 = 2.5) and add it to the number (12.5 + 2.5 = 15)
Multiply the base and the sum (15 * 10 = 150)
Add the square of the deficiency to the product: 150 + 2.5² = 150 + 6.25 = 156.25

Example 2 illustrates 2 important points. First, the method works even for non-integers, and second, the deficiency need not be negative.

In looking at the method, some of you may be wondering if the choice of base matters. Not at all. Suppose we redid example 1 using a base of 60 instead of 70:

Example 3: Compute 67² (using 60 as the base instead of 70)
Choose a base (60).
Compute the deficiency (67 - 60 = 7) and add it to the number (67 + 7 = 74)
Multiply the base and the sum (60 * 74 = 6*74*10 = 444*10 = 4440)
Add the square of the deficiency to the product: 4440 + 7² = 4440 + 49 = 4489

Your choice of base need not be a multiple of 10 either, but using multiples of 10 generally simplifies the math, so they are used most frequently.

Step 6: Additional Practice

A few more problems for practice (answers in step 6)

1) 21²
2) 37²
3) 93²
4) 44²
5) 62²
6) 55²
7) 10.2²
8) 115²

And finally, for the mental masterminds in the audience:
9) 467²
10) 783²