Walk through Kiselev's classical digit-by-digit algorithm — the same procedure taught in Russian classrooms since 1888.
Choose a difficulty
Step 1 — Group the digits
Difficulty: Easy
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Time: 0s
√=
Round complete
√4082 = 63 r 113
Nicely done.
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The method
Kiselev's classical algorithm
A. P. Kiselev, Algebra, Part I, §111. The recipe in seven steps.
Step by step
Group the digits in pairs from the right; the leftmost group may have one or two digits.
Take the integer square root of the leftmost group — this is the first digit of the answer.
Subtract its square; bring down the next pair as the new remainder.
Double the root found so far; this is your trial divisor.
Divide the tens of the remainder by the trial divisor — that gives a candidate digit.
Append the candidate to the trial divisor and multiply by the candidate. If the product exceeds the remainder, decrease the candidate by one and retry.
Subtract; bring down the next pair; repeat from step 4 until every pair is used.
Worked example: √4082
′
√4 0 8 2 = 63
3 6
───
′
1 2 3 │ 4 8 2
3 │ 3 6 9
─────
1 1 3
63² + 113 = 3969 + 113 = 4082 ✓
Watch out for
If a trial digit comes out as 10 or 11, that is impossible — jump straight to testing 9.
If the trial divisor is bigger than the tens of the remainder, the next root digit is 0; bring down the following pair and continue.
For non-perfect squares, the procedure yields the largest integer whose square fits, plus a remainder.