The Long Division Method Works for Any Square Root

You can find the square root of any number using long division, the same way you learned to divide by hand in school. The method takes patience but no special tools — just paper, pencil, and a few steps you repeat until you reach the precision you need. Most people can learn it in under an hour and use it to solve square roots that would otherwise require a calculator.

The long division method for square roots is sometimes called the digit-by-digit method because you find one digit of the answer at a time, working from left to right. It works whether you are finding the square root of 16, 256, or 7,392. The steps are identical; only the number of repetitions changes.

Key Takeaways

  • The long division method finds square roots one digit at a time by pairing digits from right to left and testing which number, when multiplied by itself, fits into each pair.
  • You start by grouping the digits of your number into pairs from the decimal point outward, then find the largest single digit whose square does not exceed the first pair.
  • Each step involves subtracting, bringing down the next pair of digits, and testing the next digit of your answer by trial and error.
  • The method produces an exact answer for perfect squares and a decimal approximation for non-perfect squares, accurate to as many decimal places as you have patience to calculate.

Step 1: Pair Your Digits from Right to Left

Write your number and draw a vertical line to its left, as if you were about to do long division. Then mark the digits in pairs, starting from the decimal point and moving right and left. For example, the number 7,392 becomes 73|92 (two pairs). The number 256 becomes 2|56 (two pairs, with the first pair having only one digit). The number 15,625 becomes 1|56|25 (three pairs).

If your number has a decimal part, pair the decimal digits in the same way, moving away from the decimal point. The number 7.3456 becomes 7.|34|56 — one digit before the decimal, then pairs after. This pairing tells you how many digits your answer will have before the decimal point.

Step 2: Find the First Digit of Your Answer

Look at the leftmost pair (or single digit if the first pair has only one). Find the largest single digit (1 through 9) whose square does not exceed this pair. Write that digit above the pair as the first digit of your answer.

For example, if your leftmost pair is 73, test: 8 × 8 = 64 (fits), and 9 × 9 = 81 (too large). So 8 is your first digit. Write 8 above the 73. If your leftmost pair is 2, test: 1 × 1 = 1 (fits), and 2 × 2 = 4 (too large). So 1 is your first digit.

Step 3: Subtract and Bring Down the Next Pair

Subtract the square of the digit you just found from the leftmost pair. Write the result below. Then bring down the next pair of digits next to this result. This creates a new working number.

Continuing the 7,392 example: 73 − 64 = 9. Bring down 92 to make 992. Now you are working with 992. For the 256 example: 2 − 1 = 1. Bring down 56 to make 156. Now you are working with 156.

Step 4: Find the Next Digit by Trial and Error

This is the core step, and you repeat it for each remaining pair. Take the answer digits you have found so far, double them, and write that number to the left of your working number. Then test single digits (0 through 9) in a blank space next to this doubled number. For each test digit, multiply the doubled number with the test digit appended by the test digit itself. The largest test digit whose result does not exceed your working number is the next digit of your answer.

For 7,392: your answer so far is 8. Double it to get 16. Your working number is 992. Test: 160 × 0 = 0, 161 × 1 = 161, 162 × 2 = 324, 163 × 3 = 489, 164 × 4 = 656, 165 × 5 = 825, 166 × 6 = 996 (too large). So 5 works. Write 5 above the next pair. Your answer is now 85. Subtract: 992 − 825 = 167.

Step 5: Repeat Until You Reach Your Target Precision

Bring down the next pair (or next pair of decimal digits if you are past the decimal point). Double your answer so far and repeat the trial-and-error test. Continue this cycle for as many digits as you need. For a perfect square, the working number will eventually reach zero. For a non-perfect square, you can keep going into decimal places as long as you want, adding pairs of zeros after the decimal point.

The square root of 7,392 is approximately 85.98 using this method. The square root of 256 is exactly 16 — the working number reaches zero after two cycles. The more pairs you process, the more decimal places you obtain, and the more precise your answer becomes.

A Worked Example: Finding the Square Root of 256

Start by pairing: 2|56. The leftmost pair is 2. The largest digit whose square does not exceed 2 is 1 (1 × 1 = 1). Write 1 above the 2. Subtract: 2 − 1 = 1. Bring down 56 to make 156.

Double your answer (1) to get 2. Test digits: 20 × 0 = 0, 21 × 1 = 21, 22 × 2 = 44, 23 × 3 = 69, 24 × 4 = 96, 25 × 5 = 125, 26 × 6 = 156. The digit 6 works exactly. Write 6 above the 56. Your answer is 16. Subtract: 156 − 156 = 0. The working number is zero, so you are done. The square root of 256 is 16.

Frequently Asked Questions

What if the first pair is a single digit?

Treat it the same way. Find the largest digit whose square does not exceed it, write that digit as your first answer digit, subtract, and bring down the next pair. A single-digit first pair straightforward means your answer will have fewer digits before the decimal point.

How many decimal places should I calculate?

Stop whenever your answer is precise enough for your purpose. For most everyday uses, two or three decimal places is sufficient. For a perfect square, the working number reaches zero and you stop naturally. For non-perfect squares, you can keep going indefinitely by adding pairs of zeros after the decimal point.

Can I use this method for very large numbers?

Yes. The method works for any number, no matter how large. Larger numbers straightforward require more repetitions of the same steps. The process is mechanical and does not change, only the number of cycles increases.

What if I make an arithmetic mistake partway through?

The error will show up in your working number — it will not match what you expect. Go back and check your subtraction and your trial-and-error test. Recalculate the step where the mismatch appeared. The method is forgiving because each step depends only on the previous working number, not on the final answer.