How to Find Square Root Without Calculator
This guide shows you how to find a square root by hand, using only paper and pencil. You’ll learn the prime factorization method for perfect squares and the long division method for any number, getting answers accurate to several decimal places. No calculator, no tables, just basic arithmetic.
What You’ll Need
- Paper
- Pencil or pen
- Basic knowledge of multiplication and division
- Optional: a ruler to keep digits aligned
Step-by-Step
Method 1: Prime Factorization (for perfect squares)
- Write the number as a product of primes. Divide by the smallest prime (2, 3, 5, 7, 11, …) until you get 1. For example, for 144: 144 ÷ 2 = 72, ÷ 2 = 36, ÷ 2 = 18, ÷ 2 = 9, ÷ 3 = 3, ÷ 3 = 1. So 144 = 2^4 × 3^2.
- Pair up identical factors. For 144, you have two pairs of 2s and one pair of 3s: (2×2) × (2×2) × (3×3).
- Take one number from each pair and multiply them. That gives 2 × 2 × 3 = 12. So √144 = 12.
- If any factor is left unpaired, the number is not a perfect square. For example, 18 = 2 × 3^2 has an unpaired 2, so √18 is not a whole number. You can still simplify: √18 = 3√2.
Method 2: Long Division Method (for any number, including decimals)
- Group the digits in pairs from the decimal point. For whole numbers, start from the right. For 529, groups are 5 and 29. For 2, write it as 2.00 00 00, so groups are 2, 00, 00, 00.
- Find the largest digit whose square is ≤ the first group. For 529, first group is 5, so the digit is 2 (2²=4, 3²=9 which is too big). Write 2 above the group.
- Subtract that square from the first group. 5 – 4 = 1.
- Bring down the next pair of digits. For 529, bring down 29 to get 129.
- Double the current answer (the number above) and write it as a divisor with a blank. Current answer is 2, doubled is 4, so write 4_ (a blank for the next digit).
- Find the digit that fills the blank so that (4_digit) × digit ≤ 129. Try 3: 43 × 3 = 129, which fits exactly. Write 3 above the second group.
- Subtract and bring down the next pair. 129 – 129 = 0. Since there are no more pairs, you are done. The answer is 23, so √529 = 23.
- For numbers that are not perfect squares, continue after the decimal point. For √2: first group is 2, digit is 1 (1²=1), remainder 1, bring down 00 to get 100. Double the answer (1) to get 2_, find digit: 24 × 4 = 96, 25 × 5 = 125 too big, so digit is 4. Answer so far 1.4, remainder 4, bring down 00 to get 400. Double 14 to get 28_, digit 1 (281×1=281), remainder 119, and so on. Continue until you have the desired decimal places.
Common Mistakes
- Forgetting to pair digits correctly. For a number like 225, pairs are 2 and 25, not 22 and 5. Always start from the decimal point.
- Using the wrong divisor in long division. When you double the answer, you must double the entire number above, not just the first digit. For example, if the answer so far is 14, double it to 28, not 8.
- Assuming every number has a rational square root. Most numbers do not. The long division method gives decimals that go on forever without repeating (irrational numbers).
- Mixing up the order of operations in prime factorization. Always divide by primes in increasing order, and do not skip any.
Tips
- Memorize squares up to 20 or 30. This speeds up both methods enormously.
- Estimate first. For √50, note that 7²=49 and 8²=64, so √50 is slightly above 7. This helps you catch errors.
- Use the long division method for non-perfect squares. It works for any positive number, including decimals.
- For fractions, take the square root of numerator and denominator separately. √(4/9) = √4 / √9 = 2/3.
- Check your answer by squaring it. If (answer)² is close to the original number, you are correct.
Troubleshooting
- Your long division answer is too big or too small. Go back to the step where you chose the digit. The digit must be the largest possible so that (divisor with digit) × digit ≤ current number. If it is too big, the product will exceed the number; if too small, you will get a larger remainder than necessary.
- You get a negative remainder. That means your digit was too large. Reduce it by 1 and try again.
- The prime factorization does not give whole pairs. That is normal for non-perfect squares. You can simplify the radical, but you cannot get a whole number.
- You are unsure if your answer is correct. Square your answer using long multiplication. If it does not match the original number, recheck each step.
Frequently Asked Questions
Can I find the square root of a negative number?
No, not with real numbers. The square root of a negative number is an imaginary number (e.g., √(-4) = 2i). This guide covers only positive numbers.
What is the fastest method for perfect squares?
Prime factorization is usually fastest for numbers under 1,000. For larger perfect squares, long division works too, but memorizing squares up to 30 helps.
How many decimal places can I get with the long division method?
As many as you want, but each extra decimal place requires another pair of digits (bring down 00) and more arithmetic. In practice, 4-6 decimal places is reasonable by hand.
Is there a way to approximate square roots without long division?
Yes, you can use the Newton-Raphson method (also called the Babylonian method). Start with a guess, then average the guess and the number divided by the guess. Repeat until the guess stops changing. For example, for √2, guess 1.5, then (1.5 + 2/1.5)/2 = 1.4167, then repeat.
Why does the long division method work?
It is based on the algebraic identity (a+b)² = a² + 2ab + b². Each step finds the next digit by solving for b in that equation.
Can I use this for cube roots?
No, this method is specific to square roots. Cube roots require a different algorithm, often more complex.
Related
Method: the standard formula for this calculation, rounded as each result states. See how the calculators work or report an error.