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 “All from 9 and the last from 10”. This method will work for any type of squaring. There is another method by taking the sutra "Vertically and Crosswise" but that we will discuss later.
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
72 = (7-3) | 32
= 4 | 9
= 49
92 = (9-1) | 12
= 8 | 1
= 81
62 = (6-4) | 42
= 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.
132 = (13+3) | 32
= 16 | 9
= 169
See few more examples:
142 = (14+4) | 42
= 18 | 16 (Carry over 1)
= 19 | 6
= 196
152 = (15+5) | 52
= 20 | 25 (Carry over 2)
= 22 | 5
= 225
192 = (19+9) | 92
= 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:
982 = (98-2) | 22
= 96 | 04
= 9604
932 = (93-7) | 72
= 86 | 49
= 8649
1062 = (106+6) | 62
= 112 | 36
= 11236
9862 = (986-14) | 142
= 972 | 196
= 972196
99962 = (9996-4) | 42
= 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 = a2 + 2ab + b2
Thus,
972 = (100-3)2
= 10000 - 600 + 9
= 9409
1072 = (100+7)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: 292 . Here working base is 30. So,
292
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: 7862 . Here working base is 800.
7862
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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