Vedic Maths Sutra 12 : Śeṣāṇyaṅkena Carameṇa
Meaning & Essence
This Sūtra means “The remainders by the last digit”.
It is mainly used for converting proper fractions into recurring decimals. We first generate the sequence of successive remainders when dividing by the denominator, then multiply each remainder by the last digit of the denominator and keep only the last digit of that product. Those last digits form the recurring decimal.
Mathematical Principle
For a fraction \( \dfrac{1}{d} \) (or more generally \( \dfrac{n}{d} \)):
- Perform successive division of the remainders by \( d \) to obtain the remainder sequence
- Multiply each remainder by the last digit of \( d \)
- Take only the units digit of each product
- These units digits form the recurring decimal expansion
Worked Examples
- Successive remainders when dividing by 7: 1 → 3 → 2 → 6 → 4 → 5 → 1 …
- Last digit of the divisor is 7
- Multiply each remainder by 7 and keep only the last digit:
- 3×7 = 21 → 1
- 2×7 = 14 → 4
- 6×7 = 42 → 2
- 4×7 = 28 → 8
- 5×7 = 35 → 5
- 1×7 = 7 → 7
- Therefore \( \dfrac{1}{7} = 0.\overline{142857} \)
- Successive remainders: 1 → 10 → 9 → 12 → 3 → 4 → 1 …
- Last digit of 13 is 3
- 10×3 = 30 → 0
- 9×3 = 27 → 7
- 12×3 = 36 → 6
- 3×3 = 9 → 9
- 4×3 = 12 → 2
- 1×3 = 3 → 3
- Therefore \( \dfrac{1}{13} = 0.\overline{076923} \)
Enter a numerator and denominator (proper fraction). The demo generates the remainder sequence and applies the “remainders by the last digit” method.
When to Use This Sūtra
• Converting proper fractions into recurring decimals
• Especially useful when the denominator ends with 1, 3, 7 or 9
• Mental calculation of well-known recurring expansions such as 1/7, 1/13, 1/17, etc.
• Understanding the relationship between remainders and decimal digits