How do you find out the square root of a decimal?
ex. 2.31
You could use a calculator.
I need to do it without a calculator
This used to be taught to 6th graders back in the 1910's. With the advent of calculators and computers, the method got lost from the system. Some home schooling programmes still do it.
1. By modern calculator
Assuming you have a calculator, press 2.31 followed by the √ key.
2. By simple calculator
If your calculator does not have a √ key, you can do it by trial and error. Try to square 1.5 (=2.25), too small. Try 1.6² (=2.56), too big. So the square root is somewhere in between Keep trying and halving the gap, for example, try 1.55 in this case.
3. By simple calculator using Newton's method.
From a given estimate x, calculate a better estimate using
x1=x + (2.31-x²)/(2x)
Keep repeating the same procedure until the desired accuracy is achieved.
Example:
x=1.5
x1=1.5+(2.31-1.5²)/(2*1.5)=1.52
x2=1.52+(2.31-1.52²)/(2*1.52)=1.519868
which is already accurate to 6 figures after the decimal point.
4. Manual method.
It is rather difficult to describe the method without having an easy way to align numbers in the response. So I will refer you to a web-site that describes the method:
http://www2.qnet.fi/abehr/Achim/Calculators_SquareRoots.html
To find the square root of a decimal, such as 2.31, you can follow these steps:
Step 1: Estimate
Start by estimating the square root value. In this case, you can estimate that the square root of 2.31 is approximately between 1 and 2, since its value is close to 2.
Step 2: Guess and Check
Next, you can make a reasonable guess within the estimated range and then check if the square of that guess is close to the given number. Let's try a guess of 1.5:
1.5 * 1.5 = 2.25
Since 2.25 is less than 2.31, you'll need a slightly bigger guess.
Step 3: Refine the Guess
You can then refine your guess to a more accurate value. Let's try a guess of 1.6:
1.6 * 1.6 = 2.56
Since 2.56 is greater than 2.31, you'll need a slightly smaller guess.
Step 4: Continue Refining
Keep refining your guess until you find a value that is sufficiently close to the given decimal. Let's try a guess of 1.55:
1.55 * 1.55 = 2.4025
Since 2.4025 is closer to 2.31, but still bigger, you'll need a smaller guess.
Step 5: Final Answer
Continue refining your guess until you find a value that, when rounded to the desired decimal place, matches the given decimal. Based on the previous iteration, let's try a guess of 1.54:
1.54 * 1.54 = 2.3716
Since 2.3716 is still bigger than 2.31, you'll need a slightly smaller guess.
Finally, based on this estimate and guess-and-check method, the square root of 2.31 is approximately 1.52 when rounded to two decimal places.