Squares of two digit numbers using Duplex by Bhārati Kṛṣṇa Tīrtha

From Vidyalayawiki

In this chapter we will learn to find squares of any random number. To find out squares of numbers we will use the

द्वन्द्व योग

" Dwandwa yoga "

" Duplex combination process "[1]

Duplex

'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). Let us see few example to understand it more. We denote the Duplex by the symbol D.

1.Duplex of a single digit number is its square.

D(4) = 42 = 16 ; D(7) = 72 = 49 ; D(6) = 62 = 36

2. Duplex of a double digit number is equal to twice the product of both the numbers as shown in the below diagram.

Duplex - double digit
Duplex - double digit

D (3 2) = 2 (3 X 2) = 12

D (4 5) = 2 (4 X 5) = 40

D (8 0) = 2 (8 X 0) = 0

3. Duplex of a triple digit number is equal to twice the product of the first and last digit + square of the middle digit as shown in the below diagram.

For a triple digit number we make pair of the first and the last digit and take it as a double digit number and the middle digit as a single digit number. Find their duplexes and add them.

Duplex three digit
Duplex three digit

D (1 3 5 ) = 2(1 X 5) + 32 = 10 + 9 = 19

D (4 0 7) = 2(4 X 7) + 02 = 56 + 0 = 56

D (2 1 3) = 2(2 X 3) + 12 = 12 + 1 = 13

4. Duplex of a four digit number is twice the the product of first and last digit + twice the the product of second and third digit as shown in the below diagram.

Four digit number
Duplex Four digit

D (1 2 3 4) = 2(1 X 4) + 2(2 X 3) = 8 + 12 = 20

D (4 2 5 7) = 2(4 X 7) + 2(2 X 5) = 56 + 20 = 76

D (3 9 1 5) = 2(3 X 5) + 2(9 X 1) = 30 + 18 = 48

5. Duplex of a five digit number is twice the the product of first and last digit + twice the the product of second and fourth digit + square of the third digit as shown in the below diagram.

Duplex five digits
Duplex five digit

D (1 2 3 4 5) = 2(1 X 5) + 2(2 X 4) + 32 = 10 + 16 + 9 = 35

D (4 2 5 7 8) = 2(4 X 8) + 2(2 X 7) + 52 = 64+ 28 + 25 = 117

D (3 9 1 5 6) = 2(3 X 6) + 2(9 X 5) + 12 = 36 + 90 + 1 = 127

The below diagrams show the method to make pairs of bigger numbers.

Duplex six digit
Duplex six digit
Duplex seven digit
Duplex seven digit

Squares of any Number using Duplex

To find the square of any number we use the " Dwandwa yoga " with " Ūrdhvatiryagbhyām "

द्वन्द्व योग

" Dwandwa yoga "

" Duplex combination process "

+ ऊर्ध्वतिर्यग्भ्याम्

" Ūrdhvatiryagbhyām "

" Vertical and crosswise "

the detailed steps will be explained through the below examples.

Squares of Two digit Numbers

Example: 232

This is a double digit number . Starting from the right we get answer in three parts.

Left Hand Side (LHS) Middle Right Hand Side (RHS)
Duplex of the left most digit (2)

D(2) = 22 = 4

Duplex of both the digits of 23

D(23) = 2(2 X 3) = 12

Duplex of the right most digit (3)

D(3) = 32 = 9

4 12 9
4 Put 2 and carry over 1 9
4 + Carry over (1) 2 9
5 2 9

Answer : 232 = 529

Example: 672

Left Hand Side (LHS) Middle Right Hand Side (RHS)
Duplex of the left most digit (6)

D(6) = 62 = 36

Duplex of both the digits of 67

D(67) = 2(6 X 7) = 84

Duplex of the right most digit (7)

D(7) = 72 = 49

36 84 49
36 84 Put 9 and carry over 4
36 84 + Carry over (4) 9
36 88 9
36 Put 8 and carry over 8 9
36 + Carry over (8) 8 9
44 8 9

Answer : 672 = 4489

See Also

द्वैध उपयोग से दो अंकों की संख्याओं का वर्ग - भारती कृष्ण तीर्थ

References

  1. Singhal, Vandana (2007). Vedic Mathematics For All Ages - A Beginners' Guide. Delhi: Motilal Banarsidass. pp. 221–226. ISBN 978-81-208-3230-5.