ms.sue, how would you solve the square-root of two manually, without a calculator?
I don't know. But this site explains how to do square root manually.
http://www.homeschoolmath.net/teaching/square-root-algorithm.php
Newton's method.
Estimate the square root.
Divide the estimate into the original number. Then look at the result. If it is higher than the estimate, go half way of the difference an repeat.
Example: sqrt 25
estimate:4
25/4=6.25
half way between 4 and 6.25 is 5.13
25/5.13=4.87
halfway is 5.
25/5=5 Bingo.
another example: sqrt 136
estimate 12;
136/2=11.3
half way...11.7
136/11.7 = 11.6
half way: 11.65
136/11.65=11.67
half way: 11.66
136/11.66=11.66 Bingo.
To manually find the square root of a number, such as the square root of two (√2), you can use a method called the Babylonian method or the Heron's method. Here are the steps:
1. Start by making an initial guess for the square root.
- In this case, you can start with an initial guess of 1.
2. Use the guess to find a better approximation.
- Divide the number you want to find the square root of (2 in this case) by the guess (1).
- Take the average of the guess and the result from the division, and use it as a new guess.
(1 + 2/1) / 2 = 1.5 (new guess)
3. Repeat this process to refine the approximation.
- Repeat step 2 multiple times, using the previous guess to calculate a new guess.
- Each time, divide the number you want to find the square root of by the previous guess,
- And take the average of the previous guess and the result from the division to get the new guess.
For example:
Guess 1: 1
Guess 2: (1 + 2/1) / 2 = 1.5
Guess 3: (1.5 + 2/1.5) / 2 = 1.4167
Guess 4: (1.4167 + 2/1.4167) / 2 = 1.4142
Continue this process until you reach the desired level of accuracy.
4. Keep refining the approximation until you reach your desired level of accuracy.
- The more iterations you make, the more accurate your approximation will be.
- Generally, you can continue refining the guess until the desired number of decimal places or precision is achieved.
Using these steps, you can approximate the square root of two manually without a calculator.