Ekadhikena Purvena (एकाधिकेन पूर्वेण)
Sanskrit: एकाधिकेन पूर्वेण
English Translation: “By one more than the previous one.”
This sutra is part of Vedic Mathematics, which simplifies calculations by focusing on logical shortcuts. It’s especially useful for:
- Squaring numbers ending in 5.
- Approximating reciprocals of certain numbers.
- Simplifying multiplication in special cases.
Example 1: Squaring Numbers Ending in 5
Let’s calculate 35².
Steps:
1. Refer to the sutra: ‘By one more than the previous one.’
– The number before 5 is 3.
2. Add 1 to this number: 3 + 1 = 4.
3. Multiply the original number (3) by the result: 3 × 4 = 12.
4. Append 25 to this product: 12 becomes 1225.
Result: 35² = 1225.
Why this works:
The square of any number ending in 5 can be split into two parts:
1. The first part is the product of the number preceding 5 and the ‘next’ number.
2. The second part is always 25.
Example 2: Squaring Larger Numbers Ending in 5
Let’s calculate 105².
Steps:
1. Refer to the sutra: ‘By one more than the previous one.’
– The number before 5 is 10.
2. Add 1 to this number: 10 + 1 = 11.
3. Multiply the original number (10) by the result: 10 × 11 = 110.
4. Append 25 to this product: 110 becomes 11025.
Result: 105² = 11025.
Why this works: The sutra leverages the structure of numbers ending in 5 to break down the squaring process into an easy-to-follow multiplication step.
Example 3: Approximating Reciprocals
Let’s approximate 1/19.
Steps:
1. Refer to the sutra: ‘By one more than the previous one.’
– For 19, the ‘one more’ number is 20.
2. Start by dividing 1 by 20: 1 ÷ 20 = 0.05.
3. Refine the answer iteratively:
– Multiply 19 × 0.05 = 0.95.
– Subtract this from 1: 1 – 0.95 = 0.05.
– Add this correction to the initial result, giving 0.0526.
Result: 1/19 ≈ 0.0526.
Why this works: The sutra helps estimate reciprocals by using the ‘one more than the number’ approach, reducing complex calculations into manageable steps.
Detailed Example: Multiplication of Numbers Close to a Power of 10
Let’s calculate 98 × 97.
Steps:
1. Recognize the base:
– Both 98 and 97 are close to 100, which is a base power of 10.
2. Express the numbers as deviations from the base:
– 98 = 100 – 2
– 97 = 100 – 3
– These differences (-2 and -3) will be used later.
3. Subtract one deviation from the other number:
– Subtract -2 from 97: 97 – 2 = 95.
4. Multiply the result by the base:
– Multiply 95 by 100: 95 × 100 = 9500.
5. Multiply the deviations:
– Multiply -2 × -3 = 6.
6. Subtract the adjustment:
– Subtract 6 from 9500: 9500 – 6 = 9494.
Result: 98 × 97 = 9494.
Why this works: This method leverages the base (100) to simplify the multiplication process:
1. The differences from the base split the problem into manageable steps.
2. The deviations create a correction factor that’s mathematically straightforward.
3. The sutra applies in the subtraction step, where one number is reduced by ‘one more’ than its difference.
Verification with Traditional Multiplication
To confirm:
98 × 97 = 98 ⋅ (100 − 3) = (98 ⋅ 100) − (98 ⋅ 3) = 9800 − 294 = 9494
The result matches perfectly, validating the method.
Key Benefits of Ekadhikena Purvena
1. Intuitive Logic: The sutra provides an easy-to-follow framework that avoids tedious calculations.
2. Universal Application: Works well for squaring numbers ending in 5, approximating reciprocals, and simplifying multiplications close to powers of 10.
3. Quick and Efficient: Especially useful for mental math and competitive exams.
This example highlights the elegance and efficiency of Ekadhikena Purvena in simplifying complex calculations!
Happy Learning!