Vedic Math Guru

Sheshanyankena Charamena

शेषाण्यङ्केन चर्मेण

Description

Meaning: "The remainders by the last digit".

Primary Application: Converting Fractions to Recurring Decimals

This sutra provides a method for converting fractions, especially those whose denominators end in 9 (or can be made to end in 9 by multiplying numerator and denominator by a suitable factor), into their exact recurring decimal representations.

How it works (e.g., for 1/19):

  1. The "Last Digit" (Charamena): The denominator is 19. The last digit is 9.
  2. The Multiplier: The sutra implies working with a multiplier derived from the denominator. For denominators ending in 9, like D9 (e.g., 19, 29, 39), the "Ekadhika" of the part before 9 (i.e., D+1) is often used as a key multiplier or related to the process. For 19, the part before 9 is 1. Ekadhika of 1 is 1+1=2. This '2' will be our multiplier.
  3. Process:
    a. Start with the numerator as the first (rightmost) digit of the recurring part of the decimal (or as the first remainder if setting up for division). For 1/19, start with 1.
    b. Multiply this digit by our multiplier (2): 1 x 2 = 2. This '2' is the next digit to the left in our decimal sequence.
    c. Take this new digit (2), multiply by 2: 2 x 2 = 4. '4' is the next digit.
    d. Take 4, multiply by 2: 4 x 2 = 8. '8' is the next digit.
    e. Take 8, multiply by 2: 8 x 2 = 16. Write down 6, carry over 1.
    f. Take 6, multiply by 2 = 12. Add carry-over 1: 12+1 = 13. Write down 3, carry 1.
    g. Take 3, multiply by 2 = 6. Add carry-over 1: 6+1 = 7. '7' is next.
    h. Continue this process: (7x2=14 -> 4 carry 1), (4x2+1=9 -> 9), (9x2=18 -> 8 carry 1), (8x2+1=17 -> 7 carry 1), (7x2+1=15 -> 5 carry 1), (5x2+1=11 -> 1 carry 1), (1x2+1=3 -> 3), (3x2=6 -> 6), (6x2=12 -> 2 carry 1), (2x2+1=5 -> 5), (5x2=10 -> 0 carry 1), (0x2+1=1 -> 1).
  4. Result: The sequence of digits obtained, read from right to left as generated, then reversed for the decimal, is 052631578947368421. The process stops when the starting digit (1) reappears as the result of a step (after considering carries and the full cycle). Since we started with 1/19, the decimal is 0.052631578947368421 (the bar indicates recurring part).

Note: The "remainders by the last digit" part refers to how each step effectively processes a remainder using the last digit (or a related multiplier) to find the next digit of the decimal. The Ekadhika of the digit(s) preceding the 9 in the denominator plays a crucial role as the effective multiplier in this specific application.

Example 1: Convert 1/19 to a decimal using Sheshanyankena Charamena.

Let's convert the fraction 1/19 into its recurring decimal form using "Sheshanyankena Charamena".

Fraction: 1/19

Numerator: 1

Denominator: 19 (ends in 9)

Step 1: Determine the Multiplier

The denominator is 19. The part before the last digit '9' is '1'.

The Ekadhika (one more than) of this part is 1 + 1 = 2.

Our working multiplier is 2.

Step 2: Iterative Multiplication Process (Right to Left)

We start with the numerator (1) as the first digit of our sequence (from the right).

  1. Current digit: 1. (Sequence: ...1)
  2. 1 × 2 = 2. Next digit: 2. (Sequence: ...21)
  3. 2 × 2 = 4. Next digit: 4. (Sequence: ...421)
  4. 4 × 2 = 8. Next digit: 8. (Sequence: ...8421)
  5. 8 × 2 = 16. Write 6, carry 1. Next digit: 6. (Sequence: ...68421)
  6. (6 × 2) + carry 1 = 12 + 1 = 13. Write 3, carry 1. Next digit: 3. (Sequence: ...368421)
  7. (3 × 2) + carry 1 = 6 + 1 = 7. Next digit: 7. (Sequence: ...7368421)
  8. (7 × 2) = 14. Write 4, carry 1. Next digit: 4. (Sequence: ...47368421)
  9. (4 × 2) + carry 1 = 8 + 1 = 9. Next digit: 9. (Sequence: ...947368421)
  10. (9 × 2) = 18. Write 8, carry 1. Next digit: 8. (Sequence: ...8947368421)
  11. (8 × 2) + carry 1 = 16 + 1 = 17. Write 7, carry 1. Next digit: 7. (Sequence: ...78947368421)
  12. (7 × 2) + carry 1 = 14 + 1 = 15. Write 5, carry 1. Next digit: 5. (Sequence: ...578947368421)
  13. (5 × 2) + carry 1 = 10 + 1 = 11. Write 1, carry 1. Next digit: 1. (Sequence: ...1578947368421)
  14. (1 × 2) + carry 1 = 2 + 1 = 3. Next digit: 3. (Sequence: ...31578947368421)
  15. (3 × 2) = 6. Next digit: 6. (Sequence: ...631578947368421)
  16. (6 × 2) = 12. Write 2, carry 1. Next digit: 2. (Sequence: ...2631578947368421)
  17. (2 × 2) + carry 1 = 4 + 1 = 5. Next digit: 5. (Sequence: ...52631578947368421)
  18. (5 × 2) = 10. Write 0, carry 1. Next digit: 0. (Sequence: ...052631578947368421)
  19. The next step would be (0 × 2) + carry 1 = 1. Since '1' was our starting digit (from the numerator), the cycle has completed.

Step 3: Form the Decimal

The recurring sequence of digits, read from left to right (reversing the generated order), is: 052631578947368421

1/19 = 0.052631578947368421

The decimal representation of 1/19 is found! 🔄

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