Understanding Long Division for Square Roots
Long division can be used to find the decimal value of square roots. Here’s a step-by-step guide:
Step 1: Setup
- Write the number whose square root you want to find, for example, 20.
- Create pairs of digits from the decimal point. For 20, you can write it as 20.00.
Step 2: Find the Largest Square
- Identify the largest square number less than or equal to the leftmost pair (in this case, 20).
- The largest square is 16 (4 x 4), so write 4 above the square root symbol.
Step 3: Subtraction
- Subtract 16 from 20, leaving a remainder of 4.
- Bring down the next pair of digits (00), making it 400.
Step 4: Double the Quotient
- Double the number above the square root (4 x 2 = 8).
- This will be the beginning of your new divisor.
Step 5: Guess the Next Digit
- Determine a digit (let’s say x) to complete the divisor (8x) such that (8x)x is less than or equal to 400.
- In this case, 87 x 7 = 609 (too high) and 86 x 6 = 516 (too high), but 85 x 5 = 425 is also too high, so try 84 x 4 = 336 (valid).
Step 6: Update and Repeat
- Write 5 above the root (making it 4.5), subtract 336 from 400, leaving 64.
- Bring down another pair of 00 for 6400.
- Double 45 (90) and repeat the guessing process.
Step 7: Continue for Precision
- Keep repeating the steps for additional decimal places until you achieve the desired level of accuracy.
This method allows you to derive any square root to as many decimal places as needed!