Trick 9 found roots of perfect squares. This method finds the square root of ANY number, digit by digit, to as many decimal places as you want — like straight division (trick 19) with the duplex.
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The trick
Dwandwa Yoga — “the duplex (with straight division)”
Pair the digits from the right; the biggest square in the first pair gives the first digit. Divisor = twice that digit.
Each step: bring the remainder to the next digit, take away the duplex of the root digits found after the first one, divide by the divisor.
Take one less if the next step would become negative. When the digits run out, bring down zeros — those are decimals.
Use it for: Square roots to 1 decimal place without a calculator (answers here are 4-digit numbers).
Bring 3 to 3 → 33. 33 ÷ 8 = 4 would make the next step negative, so 3 rem 9.
Bring 9 to 0 → 90, take the duplex of 3 (9) → 81. 81 ÷ 8 = 9 rem 9.
Answer: 43.9.
Check: 43.9² = 1927.21 and 44² = 1936, so √1930 is between 43.9 and 44.0 ✓
✏️ Practice
Work it out in your head or on paper, type the answer and press Check. Stuck? Press “Show me how”.
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🏁 Trick 30 test
20 sums. Check each answer, then go to the next one. Get 16 or more right to pass.
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