To find the square root of a number without a calculator, you can use several manual methods depending on the number and desired precision:
Common Methods to Find Square Roots by Hand
1. Factoring by Squares
- Find factors of the number that are perfect squares.
- Separate the square root into the product of the square roots of those factors.
- Calculate the square roots of the perfect square factors and multiply them.
Example:
225=25×9=25×9=5×3=15\sqrt{225}=\sqrt{25\times 9}=\sqrt{25}\times \sqrt{9}=5\times 3=15225=25×9=25×9=5×3=15
2. Long Division Method
- Group the digits of the number into pairs starting from the decimal point.
- Find the largest square less than or equal to the first group and subtract it.
- Bring down the next pair and double the current root estimate to form a divisor.
- Find the next digit by trial so that the product of the divisor and this digit is less than or equal to the current remainder.
- Repeat the process for more digits or decimal places.
This method works for perfect squares and non-perfect squares alike and can give precise decimal approximations
3. Estimation and Approximation
- Find the nearest perfect squares above and below the number.
- Use linear approximation:
a2+b≈a+b2a\sqrt{a^2+b}\approx a+\frac{b}{2a}a2+b≈a+2ab
where a2a^2a2 is the nearest perfect square less than the number and bbb is
the difference.
Example:
11≈3+22×3=3+26=3.3333\sqrt{11}\approx 3+\frac{2}{2\times
3}=3+\frac{2}{6}=3.333311≈3+2×32=3+62=3.3333
. 4. Prime Factorization
- Factor the number into primes.
- Pair the primes and take one from each pair outside the root.
- Multiply the numbers outside the root to get the square root
Summary
Method| Best For| Accuracy| Complexity
---|---|---|---
Factoring by Squares| Perfect squares| Exact| Easy
Long Division Method| Any number| Exact or decimal approx.| Moderate, requires
practice
Estimation/Approximation| Quick approximations| Approximate| Easy
Prime Factorization| Perfect squares| Exact| Moderate
For numbers that are not perfect squares, the long division method is a reliable way to get an accurate square root by hand, while estimation methods provide quick approximations
. Additionally, there are mental math tricks and ancient formulas that can help approximate square roots quickly with some practice
