How to calculate square root by hand

You need a manual method to find a square root, often for a test that bans calculators. Here is how to calculate square root by hand using three methods: estimation, the Babylonian method (also called Newton's method), and the long division method. Each works for perfect squares and irrational numbers like √2. Pick the one that fits the problem and the time you have.

Every positive number has two square roots: a principal (non-negative) root and a negative root. The principal root is what the radical symbol (√) denotes. For a number like 9, √9 = 3, but the equation x² = 9 has two solutions: 3 and -3. For a negative radicand, no real square root exists; you need the imaginary unit i, defined as i = √(-1). The methods below produce the principal root for non-negative radicands.

This is the fastest way to get a rough answer without any written algorithm. You rely on knowing perfect squares: 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 121, 144, and so on.

How to Estimate a Square Root

Find the two perfect squares that bracket the radicand. For √50, the nearest perfect squares are 49 (7²) and 64 (8²). Since 50 is closer to 49 than to 64, the square root is slightly above 7. A reasonable estimate is 7.1. Square 7.1 to check: 7.1² = 50.41, a bit high. Adjust down to 7.07: 7.07² ≈ 49.98. That is accurate enough for many problems.

When to Use This

Use estimation when the radicand is under 200 and you only need one or two decimal places. It fails when the radicand is large or when the test requires an exact digit-by-digit answer. For those cases, use Method 2 or 3.

The Babylonian method, also known as Newton's method, is the algorithm calculators use internally. It converges quadratically: the number of correct digits doubles with each iteration. The formula is x_n+1 = (x_n + a / x_n) / 2, where a is the radicand and x_n is your current guess.

How to Apply It

Pick any positive initial guess. For √10, start with 3 (since 3² = 9). First iteration: (3 + 10/3) / 2 = (3 + 3.3333) / 2 = 3.1667.Second iteration: (3.1667 + 10/3.1667) / 2 = (3.1667 + 3.1579) / 2 = 3.1623.Three iterations give near-calculator precision.

Failure Case

A poor initial guess far from the true root requires more iterations. Starting √10 at 10 gives a first iteration of 5.5, which still works but wastes steps. The method only handles non-negative radicands; for a negative number it diverges. Use the long division method if the problem demands a fixed number of digits with no guesswork.

Historical Note

The YBC 7289 clay tablet, dated 1800-1600 BCE from the Yale Babylonian Collection, shows a sexagesimal approximation of √2 as 1;24,51,10 (1.41421296).

The square root long division method produces one digit per iteration and works for any non-negative radicand, perfect square or not. It requires only paper, pencil, and multiplication skills.

Step-by-Step Procedure

Pair the digits from right to left. For 2025, the pairs are (20)(25). For 2, write it as 2.00 00 00, adding pairs of zeros after the decimal as needed.

Find the largest integer whose square is ≤ the first pair. For 20, that is 4 (4² = 16). Write 4 above the pair. Subtract 16 from 20, leaving remainder 4. Bring down the next pair (25) to get 425.

Double the current result (4), ignoring the decimal, to get 8. Write 8_ as the divisor with a blank space. Find the largest digit (0-9) such that the product of that digit and the new divisor is ≤ the remainder. For 425, try 5: 85 × 5 = 425 exactly. Write 5 above the next pair. The result so far is 45.

For a perfect square you stop when the remainder hits zero. For non-perfect squares, bring down the next pair of zeros and repeat. Double the current result, find the divisor, and pick the next digit.

Worked Example: √2025

Pair (20)(25). First digit: 4 (4² ≤ 20). Remainder: 4. Bring down 25 → 425. Double 4 → 8. Divisor 8_. Digit: 5 (85×5 = 425). Result: 45. Since remainder is zero, √2025 = 45. Check: 45² = 2025.

Worked Example: √2 to Three Decimal Places

Write 2 as 2.00 00 00. Pair (2)(00)(00)(00). First digit: 1 (1² ≤ 2). Remainder: 1. Bring down 00 → 100. Double 1 → 2. Divisor 2_. Digit: 4 (24×4 = 96 ≤ 100). Remainder: 4. Result so far: 1.4.

Bring down next 00 → 400. Double 14 → 28. Divisor 28_. Digit: 1 (281×1 = 281 ≤ 400). Remainder: 119. Result: 1.41.

Bring down next 00 → 11900. Double 141 → 282. Divisor 282_. Digit: 4 (2824×4 = 11296 ≤ 11900). Remainder: 604. Result: 1.414. So √2 ≈ 1.414, accurate to three decimal places.

Common Errors

Pairing from the wrong direction: pair from the decimal point outward, not from the left. Forgetting to double the current result before finding the next digit. Misplacing the decimal point in the answer: it goes directly above the decimal in the radicand. A single multiplication error in the divisor test propagates through all subsequent digits; double-check each product.

If the radicand is a perfect square, prime factorisation gives an exact integer answer with no iteration. This works because a perfect square has every prime factor appearing an even number of times.

How to Do It

Factor the number into primes. For 2025, the prime factors are 3⁴ × 5² (since 2025 ÷ 3 = 675, ÷ 3 = 225, ÷ 3 = 75, ÷ 3 = 25, ÷ 5 = 5, ÷ 5 = 1). Take half of each exponent: 3² × 5¹ = 9 × 5 = 45. That is the square root. This method is faster than long division for numbers under 10,000 that are obviously perfect squares. It does not work for non-perfect squares, because the exponents are odd and leave a radical in the result.

Estimation is for quick checks and rough answers, especially when the radicand is between two small perfect squares. The Babylonian method is best when you need high precision with few iterations, and it is the method calculators use. The long division method is the only one that gives exact digits on demand, which is why many tests require it. Prime factorisation works only for perfect squares, but it is the fastest when it applies.

For a test that bans calculators but allows written work, the long division method is the safest because it does not depend on a good initial guess. For a homework problem that asks for √2 to six decimal places, the Babylonian method needs four iterations; the long division method needs six digit-finding steps. The Babylonian method is faster if you are comfortable with division. The long division method is more mechanical and easier to check for arithmetic errors.

The most common failure is stopping too early. For the Babylonian method, run at least three iterations from a reasonable guess. For the long division method, continue until the remainder is zero or you have the required number of decimal places. Never assume a calculator's decimal is exact; 1.414 is an approximation of √2, not the true value.

Decimal accuracy: The Babylonian method converges to double precision in 5-7 iterations. The long division method produces one digit per step, so 6 steps give 6 decimal places. Estimation rarely exceeds 2 correct decimal places.

Speed of calculation: The Babylonian method reaches high accuracy in 3-4 iterations. The long division method is slower for many digits but does not require division by a decimal number.

Ease of learning: Estimation is fastest to learn. The long division method requires careful procedure and practice. The Babylonian method requires understanding of iteration and initial guess choice.

Applicability to non-perfect squares: All three methods work, but the long division method is tedious for large radicands because each digit requires a multiplication and subtraction. The Babylonian method works regardless of the radicand size.

Handling of negative radicands: None of these methods handle negative radicands in the real numbers. For a negative radicand, the result is an imaginary number using i = √(-1). The SQRT function in Google Sheets and Microsoft Excel returns an error for negative numbers; the POWER function does the same.