## Saturday, July 25, 2009

### Vedic Mathematics Lesson 10: Multiplication Special Case 3

This is the last of the special cases that can be derived as a corollary from the general method we have dealt with in previous lessons. This special case deals with the squaring of numbers.

You can find previous posts about Vedic Mathematics below:

Introduction to Vedic Mathematics
A Spectacular Illustration of Vedic Mathematics
10's Complements
Multiplication Part 1
Multiplication Part 2
Multiplication Part 3
Multiplication Part 4
Multiplication Part 5
Multiplication Special Case 1
Multiplication Special Case 2

The upasutra (sub-rule) that applies to this case reads Yavadhunam Thavadhunikritya Varga Cha Yojayetu. Literally, this means Whatever The Extent Of The Deficiency, Lessen It Further To That Extent; And Also Set Up The Square Of That Deficiency.

We will illustrate the application of this corollary by working out a simple example. Let us compute the square of 94 using the method explained above. From our previous lessons, we know that the figure below can be drawn to calculate the square of 94:

94 - 6
94 - 6
-----------
88 | 36

This directly gives us the answer to the problem as 8836.

Now let us see how the special case is applicable in this case. We see that the left hand side of the answer is equivalent to the number to be squared reduced by the deficiency of that number from the base. This is obviously the application of Whatever The Extent Of The Deficiency, Lessen It Further To That Extent. The right hand side is simply the square of the deficiency. This is the application of And Also Set Up The Square Of That Deficiency.

The algebraic explanation of this corollary can be derived directly from the algebraic explanation of the general method that has been provided in previous lessons. Now we will apply this special case to solve a few more problems.

89 x 89 = 7921 (the right hand side of the answer is 121 (the square of 11), so we carry over the 1 to the left hand side, which started out as 89 - 11 = 78)
99 x 99 = 9801 (the right hand side of the answer has been padded with zeroes to the left so that it has the same number of digits as the number of zeroes in the base)
997 x 997 = 994009 (once again, the right hand side of the answer has been padded with the appropriate number of zeroes)
880 x 880 = 774400 (the left hand side of the answer has been augmented by the carryover of 14 to the left hand side from the original right hand side of 14400)

Now, let us examine how to modify the upa-sutra slightly to deal with squares of numbers that are a little greater than the base rather than being a little less than the base. Let us deal with the case of 105 x 105. We get the figure below when we use the method we have learnt in previous lessons:

105 + 5
105 + 5
-----------
110 | 25

The general rule thus becomes quite clear: instead of reducing the number by the deficiency, if there is an excess, increase the number by the excess to get the left hand side of the answer. As before, use the square of the excess as the right hand side of the answer.

Notice that the square of 105 could have been derived simply by using the technique explained in the previous lesson for squaring of numbers ending in 5. But, the application of the technique explained in this chapter to the same problem serves to illustrate that there are several tools to perform the same function, and it is up to us to choose the method that is most suitable and easy to apply to the given situation.

Let us now solidify our understanding of the technique by applying it to a few more problems:

13 x 13 = 169 (3 is the excess of 13 over 10, 16 is 13 + the excess, and 9 is the square of the excess)
1003 x 1003 = 1006009 (the right hand side has been padded with zeroes to make it 3 digits long, the same length as the number of zeroes in our base, 1000)
1100 x 1100 = 1210000 (the original left hand side of 1200 has been added to the carryover of 10 from the right hand side to get the final answer)
111 x 111 = 12321 (the original left hand side of 122 has been added to the carryover of 1 from the right hand side to get the final answer)

That is all there is to it. This has been a short lesson, but hopefully, this has been a useful addition to our arsenal of mental arithmetic techniques. As is always the case, when we start accumulating more and more tools for our toolbox, it is important to understand the functions, strengths and weaknesses of each tool, and use the most appropriate tool for the job that confronts us at any given point. Practicing is the key to understanding the tools at our disposal. Happy computing, and good luck!

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