In the previous article, we have discussed about the method of multiplication by using the base value. In this article, we shall learn the squaring of numbers by using base value. Squaring numbers near base is much easier as there is no possibility of different cases that we discussed earlier for multiplication, like

1. One number is above the base and the other number is below it

2. Numbers near different bases like multiplier is near to different base and multiplicand is near to different base.

So it is comparatively simpler. Here, we can use the sub sutra

*.*

**“whatever the extent of its deficiency, lessen it still further to that extent; and also set up the square of that deficiency”**In this, first corollary is

**”. This method will work for any type of squaring. There is another method by taking the sutra "**

*“All from 9 and the last from 10***" but that we will discuss later.**

*Vertically and Crosswise*Suppose we have to find the square of 8. The following will be the steps for it:

1. We shall take the nearest power of 10 (10 itself in this case) as our base.

2. 8 is '2' lesser than 10, so we shall decrease 2 from 8 (i.e. 8 - 2 = 6). This will become the left side of answer.

3. And, for right part of answer, we write down the square of that deficiency i.e. 2 x 2 = 4

4. Thus 8 x 8 = 64

In exactly the same manner, we say

**7**= (7-3) | 3

^{2}^{2}

= 4 | 9

=

**49**

**9**= (9-1) | 1

^{2}^{2}

= 8 | 1

=

**81**

**6**= (6-4) | 4

^{2}^{2}

= 2 | 6 (Here, since right side is 2 digit number, so '1' will be carried to its left)

1

= 3 | 6

=

**36**

Now, if numbers are above base value, approach will be almost same. The only difference will be that instead of reducing the number from its deficiency, we increase the number by the surplus. For example,square of 13:

Here working base is 10.

**13**= (13+3) | 3

^{2}^{2}

= 16 | 9

=

**169**

See few more examples:

**14**= (14+4) | 4

^{2}^{2}

= 18 | 16 (Carry over 1)

= 19 | 6

=

**196**

**15**= (15+5) | 5

^{2}^{2}

= 20 | 25 (Carry over 2)

= 22 | 5

=

**225**

**19**= (19+9) | 9

^{2}^{2}

= 28 | 81 (Carry over 8)

= 36 | 1

=

**361**

And then, extending the same rule to numbers of two or more digits, we proceed further as:

**98**= (98-2) | 2

^{2}^{2}

= 96 | 04

=

**9604**

**93**= (93-7) | 7

^{2}^{2}

= 86 | 49

=

**8649**

**106**= (106+6) | 6

^{2}^{2}

= 112 | 36

=

**11236**

**986**= (986-14) | 14

^{2}^{2}

= 972 | 196

=

**972196**

**9996**= (9996-4) | 4

^{2}^{2}

= 9992 | 0016

=

**99920016**

**Note:**Number of digits in right side part of the answer should always be equal to the zeros in the base value. Extra should be carried forward to left side answer. If number of digits in right side answer are less than the zeros, then it should be prefixed by zeros.

The

**Algebraic Expressions**are as follows:

*(a*

__+__b)^{2}= a^{2}__+__2ab + b^{2 }

Thus,

**97**= (100-3)

^{2}^{2}

= 10000 - 600 + 9

=

**9409**

**107**= (100+7)

^{2}^{2}

= 10000 + 1400 + 49

=

**11449**

Another case arise here that if numbers are not near the base value (i.e. where base value is not power of 10). In that case we follow the same method as we discussed in our previous article. For example:

**29**. Here working base is 30. So,

^{2}**29**

^{2}29 -1

29 -1

-----------

28 | 1

= 28 x 30 | 1 (multiply the left part of answer with the base)

= 840 | 1

= 840 + 1

=

**841**

Another example to understand it more:

**786**. Here working base is 800.

^{2}**786**

^{2}786 -14

786 -14

-------------

772 | 196

= 772 x 800 | 196

= 617600 | 196

= 617600 + 196

=

**617796**

So this is all about squaring the numbers by using the base values. Isn't that quite simple and interesting approach.. In next article, we shall discuss about Vertical and Crosswise multiplication.

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