Answer :
Nikhilam Sutra is one of the techniques from Vedic Mathematics used to simplify multiplication, particularly when numbers are close to powers of 10, such as 10, 100, etc. Let's explore using the Nikhilam method to solve the specified multiplication problems.
Multiply 6 x 9:
- Find the nearest base: 10
- Write each number as (base - deviation). For 6 and 9, we have:
- 6 is 10 - 4
- 9 is 10 - 1
- First step: Multiply deviations: -4 x -1 = 4
- Second step: Cross-subtract: Either 6 - 1 or 9 - 4 gives 5
- Combine: 54
Multiply 7 x 9:
- Nearest base: 10
- Represent: 7 is 10 - 3, 9 is 10 - 1
- Multiply deviations: -3 x -1 = 3
- Cross-subtract: 7 - 1 or 9 - 3 gives 6
- Result: 63
Multiply 16 x 8:
- Nearest base: 10
- Represent: 16 is 10 + 6, 8 is 10 - 2
- Multiply deviations: 6 x -2 = -12
- Cross-add: 16 - 2 gives 14, adjust for negative: 14 becomes 13 (borrow 1)
- Combine: 128
Multiply 19 x 5:
- Nearest base: 10
- Represent: 19 is 10 + 9, 5 is 10 - 5
- Multiply deviations: 9 x -5 = -45
- Cross-add: 19 - 5 gives 14, adjust for negative: 14 becomes 13 (borrow 1 for 100 being 50 less, making -45 into -50, 500 turns to 450)
- Combine: 95
Multiply 89 x 90:
- Nearest base: 100
- Represent: 89 is 100 - 11, 90 is 100 - 10
- Multiply deviations: -11 x -10 = 110
- Cross-subtract: 89 - 10 or 90 - 11 gives 79
- Result: 8010 (since 110 is greater than base: subtract 1, turning it into 8000 + 110)
Multiply 90 x 69:
- Nearest base: 100
- Represent: 90 is 100 - 10, 69 is 100 - 31
- Multiply deviations: -10 x -31 = 310
- Cross-subtract: 90 - 31 or 69 - 10 gives 59
- Result: 6210 (310 adjustment not needed as it's below base)
Each step follows a similar pattern of identifying the nearest base, expressing numbers as deviations from the base, and then applying the sutra's methodology to calculate the product efficiently.
