How do you calculate the square root of a number?

Question

I know what square roots are but I'm wondering how to calculate them other than pressing the square root key of a calculator.

Answer 1

If you want good approximations, the following works quickly and has been used to find approximate square roots by hand:

Suppose you want to find the square root of 5. You know it is between 2 and 3, so take x=2 as your first approximation.
Now, if you have an approximation x, get a better approximation by taking the average of x and 5/x:
new approx=(x+5/x)/2=(x^2 +5)/(2x).
With x=2, we get 9/4=2.25. The next approximation will then be
((9/4)^2 +5 )/(2*9/4)=161/72=2.236111..
Then next approximation will then be
[(161/72)^2 +5]/[2*161/72]=
51841/23184=2.2360679779...

Since sqrt(5)=
2.23606797749...
this is a very good approximation. One more time through the averaging will give at least 16 decimal place accuracy. To find the square root of any other number, replace 5 by the number you want to find the square root of.

There is also a proceedure similar to long division which will give the square root one decimal place at a time,but it is a bit hard to describe in this forum. it is also not nearly as fast as the averaging technique.

Answer 2

If you only want an approximation, try this (but it won't work for zero):

Finding the Square Root of Some Positive Number X.

1. Start with a guess. (Try a positive guess, like 1, or X itself.) Call it G.
2. Divide X by G to get X/G. The better your guess, the closer G and X/G will be together.
3. Calculate the difference G - X/G, and take the negative if the result is negative, getting a positive number. (Thus, if G is .25 and X/G is 8, then G - X/G = -7.75, so you take 7.75 instead.) Call this difference, D.
4. If D is not as small as you want (say, less than .001), then add: G + X/G and divide the result by two to get your new guess; call it A for average. The idea is that if G was too small, X/G will be too large, so A should be closer than either.
5. Return to step 2, with A in place of G, and keep going, until you achieve your desired precision.

Example: What is the square root of 2 [sqrt(2)]?

1. I guess sqrt(2)=1.
2. 2/1=2
3. 1 - 2/1 = 1-2 = -1. I will use positive 1. Too large.
4. (1 + 2)/2 = 1.5, which is my new "guess".
5. Return to step 2:

2. 2/1.5 ~= 1.333
3. 1.5 - 2/1.5 ~= .1667 Still too large.
4. (1.5 + 1.333)/2 ~= 1.416
5. Return to step 2:

2. 2/1.416 ~= 1.412
3. Difference is now 0.004902 Better, but still too large.
4. Average (and guess) is now 1.414
5. Back to #2.

2. 2/1.4142... ~= 1.4142...
3. Difference is now 4.2478E-06, or about 4 millionths. Perfect!

The algorithm ends with sqrt(2) ~= 1.4142...

Questions?

Answer 3

You calculate it by doing x*x=x^2

Like, the square root of 16 is 4, because 4*4=16
If you're talking about numbers that are irrational when a square root is taken, such as 3, the square root would be rt 3, because
(rt 3)*(rt 3) = 3

Answer 4

We learned how to take square roots long hand many, many years ago. It is a process that really can't be described by words, it takes a few equations that I don't care to generate.

You can also do it on a slide rule, if you know what that is.

You can also do it with logarithms, if you know what they are.

I'm sure you can find it on the web.

Answer 5

Sometimes it's just trail and error to figure out the answer.

What you need to bear in mind is, you need to find a number that when multiplied by itself, will give the required answer.

Answer 6

memorized it.
or press # times # on a calculator

Answer 7

You have to figure out what number times itself gives you the number your looking for. Unfortunatly, most numbers are huge decimals