Today, the topic which we are going to discuss is the 'General Procedure to Square any number'. Earlier we discussed about the squaring of numbers near base, however, general procedure is another nice formula to do the squaring and is applicable universally. The method or sutra is "Vertically and Crosswise", but here it is used in a different sense; based on a procedure known as 'Dwandwa Yoga' or 'Duplex Combination Process' or 'Duplex'; denoted as (D).
'Duplex' term is used in two different sense; for squaring and for multiplication. And for current formula, it will be used in both the senses. If we are having a single or central digit, then 'Duplex' means squaring that digit (a2 ). Secondly it can be used for even digits number or on numbers having equidistant digits, then 'Duplex' means to double of cross multiplication of the equidistant numbers (2ab). This concept is very important to understand the current formula and will be used in future articles also. Let us see few example to understand it more:
For 1 digit – D(a) = single digit = a2
e.g. D(5) = 52 = 25
For 2 digits – D(ab) = even digits number = twice the product of the digits (2ab)
e.g. D(26) = 2(2)(6) = 24
For 3 digits – D(abc) = product of equidistant digits from center and square of center digits
= twice the product of the outer digits (2ac) + the square of the middle digit (b2 )
e.g. D(734) = 2(7)(4) + 32
= 56 + 9 = 65
For 4 digits – D(abcd) = product of equidistant numbers
= twice the product of the outer digits (2ad) + twice the product of the inner digits (2bc)
e.g. D(1034) = 2(1)(4) + 2(0)(3)
= 8 + 0 = 8
For 5 digits – e.g. D(10345) = product of equidistant digits and square of center digits
= 2(1)(5) + 2(0)(4) + 32
= 10 + 0 + 9 = 19
and so on. This is called Duplex.
Now, let us come to original question i.e. how to square a number. And the square of any number is just the total of its Duplexes.
For Example,
342 = 1156
= D(3) = 9, D(34) = 24, D(4) = 16
Combining these three results in the usual way, we get
= 9 | 24 | 16
Now add these results as
= 9 | 4 | 6
2 1
= 9 | 5 | 6
2
= 11 | 5 | 6
= 1156
562 = 3136
D(5) = 25, D(56) = 60, D(6) = 36
by combining, we get 25 / 60 / 36 = 3136
Equivalent Algebraic Expression is: (10a + b)2 = 100(a2 ) + 10(2ab) + b2 .
This method can also be explained by multiplying a number by itself using the general multiplication method.
Note :- If a number consists of n digits, its square must have 2n or 2n-1 digits.
Following are some more examples:
2632 =
D(2) = 4, D(26) = 24, D(263) = 48, D(63) = 36, D(3) = 9
4 / 24 / 48 / 36 / 9 = 69169
43322 =
D(4) = 16, D(43) = 24, D(433) = 33, D(4332) = 34, D(332) = 21, D(32) = 12, D(2) = 4
16 / 24 / 33 / 34 / 21 / 12 / 4 = 18766224
32472 = 9 / 12 / 28 / 58 / 46 / 56 / 49 = 10543009
463252 = 16 / 48 / 60 / 52 / 73 / 72 / 34 / 20 / 25 = 2146005625
We hope this method will help you in squaring of any number quickly. If you find this difficult, you may use another method which we have discussed earlier( Squaring numbers near base ). Every method will become easy with practice. In our next article, we shall discuss about the cubing of the number.
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