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O girl! You with eyes like a young fawn! O mine of cleverness! Auspicious Lilavati! If you know the methods of multiplication by Form, Place, Division, and Parts, then tell me: what is the result if 135 (one hundred thirty-five) is multiplied by 12 (twelve)? Tell me this using all the various methods. Also, tell me what the result is when that same product is divided by 12. // 2 //
Multiplicand original: guṇya: 135
Multiplier original: guṇaka: 12
By performing this, the result is 1620.
Explanation: According to the previously mentioned rule of multiplication, take the 5 at the end of the multiplicand 135 and multiply it by the multiplier 12.
| 135 | Multiplying by twelve gives 60 (sixty). From this sixty, the zero |
|---|---|
| 12 | is placed in the units place below the multiplicand and multiplier, and the remaining 6 (six) |
| --- | is kept in memory. Then, multiply the 3 (three) next to the end by the multiplier; |
| 1620 |
3 times 12 is 36 (thirty-six). To this, add the 6 from the previous sixty, which makes 42 (forty-two). From this, the last digit 2 is placed in the tens place to the left of the previous zero, and the remaining 4 (four) is kept in memory. Then, multiply the third digit, 1 (one), by the multiplier—meaning 12 times 1 is 12. To this twelve, add the 4 from the previous forty-two, which results in 16 (sixteen). These digits are placed to the left of the previously placed digits. Thus, the product is 1620 (one thousand six hundred twenty). This method is commonly used everywhere.
original: aṅkānāṃ vāmato gatiḥ — a fundamental rule in Indian mathematics stating that while we read from left to right, the place value increases as we move toward the left.
| 1 2 3 5 | Multiplication of the last digit. | Numbers are counted from the right side; according to this method, the last digit in the multiplicand is 1 (one). |
|---|---|---|
| 1 5 6 5 | Multiplication of the second digit. | Multiplying it by 12 (twelve) results in 12. The text here describes placing the product 12 starting at the appropriate place value. |
| 1 6 2 0 | Multiplication of the third digit. | Multiplying the second digit, 3 (three), by the multiplier 12 gives 36 (thirty-six). |
| This is the result. | Of this, 6 is placed in the position of the multiplicand 3, and 3 is written below the 2 in the hundreds place and added. |
This makes 1565 The text is demonstrating the step-by-step accumulation of products.. Then, multiplying the third digit, 5 (five), by the multiplier 12 gives 60 (sixty). From this, the zero is written in the place of the multiplicand 5, and the 6 (six) is added to the 6 in the tens place, which equals 12 (twelve). The 2 (two) is written in the tens place, and the remaining 1 (one) is carried to the hundred-