How do I work out square roots without a calculator?
http://www.homeschoolmath.net/teaching/square-root-algorithm.php
Working out square roots without a calculator can be done through a process called the "long division method" or by using an estimation method known as the "guess and check method."
1. Long Division Method:
- Start by separating the integer part from the decimal part of the number you want to find the square root of.
- Write the number in pairs, starting from the decimal point and moving left. If there are any remaining digits, put them in the rightmost pair, placing a 0 before them if necessary.
- Look for the largest digit that, when squared, is less than or equal to the first pair of digits (if the first pair is a single digit, consider it as a pair).
- Write that digit as the first digit of the square root.
- Subtract the square of this digit from the first pair and bring down the next pair from the original number.
- Double the first digit of the square root and put a blank on the right side as a placeholder.
- Find the largest number that, when multiplied by the resulting number, gives a product less than or equal to the two-digit number formed by the difference you just calculated.
- Write that number above the placeholder as the next digit of the square root.
- Multiply this entire divisor by the newly added digit and place the product below the dividend.
- Subtract the product from the dividend and bring down the next pair of digits from the original number.
- Repeat the process until you have brought down all the pairs of digits, and continue adding digits to the square root as necessary.
2. Guess and Check Method:
- Start by estimating the square root of the given number to the nearest whole number.
- Square this estimation and compare it to the original number.
- If the squared estimation is too low, increase it by a small increment.
- If the squared estimation is too high, decrease it by a small decrement.
- Iteratively refine the estimation until you reach an acceptable level of accuracy.
Both methods require practice and can be time-consuming for larger numbers. It's important to note that these methods provide approximate values, as finding the exact square root without a calculator can be challenging for complex numbers.