Anybody know any simple (easy to remember) ways to find a square root without using a calculater?

Answer 1

You should know the square numbers:
1 , 4 , 9 , 16 , 25 , 36 , 49 , etc

then if you have a problem like √42 ...
you can estimate that is is between 6 and 7
so you might say 6.5 or 6.6

~~

Answer 2

You can use the Babylonian method, which is a loop you go through as often as you need to get the accuracy you want. Fortunately, the number of significant figures that are correct in the answer are going to double approximately for every trip around the loop, so you don't have to go that many times if you start with a reasonably good guess.

Step 1: Start with an initial guess for your square root.

Step 2: Divide your radicand - the number you want to take the root of - by your initial guess.

Step 3: Add the quotient - the answer from step 2 - to the guess from step 1 and divide the sum by 2. That is, average your guess with the answer. This will be your next guess at the square root.

Step 4: The actual square root will be somewhere between this new guess and the smaller of your old guess and the quotient from step 2. If this is something other than your first time through this loop, the quotient will be smaller and the actual root will lie near the new guess from step 3. If the interval hasn't been narrowed enough yet, go back to step 1, using your new guess.

Let's try this algorithm to find the square root of 2 to two decimal places.

Loop 1, step 1: Let's take a rather bad initial guess of 1 as the square root.

Step 2: 2 / 1 = 2

Step 3: Average our guess 1 and our quotient 2 to get a new guess of 3/2 (1.5):

Step 4: We know that the actual square root is between 1 and 1.5 but is closer to 1.5. Let's go around the loop again.

Loop 2, step 1: Use 3/2 (or 1.5) as our new guess.

Step 2: 2 / (3/2) = 4/3 (1.33333...)

Step 3: Average 3/2 and 4/3 to give 17/12 (1.416666...)

Step 4: We know that 4/3 < √2 < 17/12. In fact, our new guess is already correct within 2 decimal places, but we must go through the loop yet again to find that out, so here goes:

Loop 3, step 1: Our new guess is 17/12

Step 2: 2 / (17/12) = 24/17 (1.411764)

Step 3: Average 24/17 and 17/12 to get 577/408 (1.414215)

Step 4: We now know for sure that 1.411 < √2 < 1.414215, with the result being (much) closer to the upper limit, so we take 1.414215 as our final answer, correct to more than 2, places quite probably much more. In fact the actual value of √2 is a bit less than 1.414214.

In this case, if we wanted to avoid the addition in step 3, we could have verified at step 2 that 1.411 < √2 < 1.417, which is still good to 2 places past the decimal point. However, by performing step 3, we have a much better value than required with relatively little extra work.

Answer 3

Take a wager on the sq. root--something reasonable will do. you may %. some thing unreasonable, too, basically no longer 0: it basically takes longer. call your wager A. Your extensive style, I shall call N. Compute: B=N/A Compute: C=(A+B)/2 Now, enable us to anticipate A>B for the applications of argument. If that's not, basically swap 'em so that's actual... in any case, Compute C' = N/C If C > C' then A = C If C = C' then you definately've have been given your sq. root, good there! If C < C' then B = C Now, recompute C = (A+B)/2 bypass lower back and recompute C' and try this yet lower back. ultimately, you will attain a factor the place C-C' is smaller than your calculator can safeguard; you're surely there. observe that the gap between A & B surprisingly plenty drops through a million/2 each time you run via this cycle... it is somewhat quickly: in 10 repetitions, you're interior of a million area in one thousand; in 20, a million in a million; in 30, a million in one thousand million...

Answer 4

There's an easy method to compute by hand. It's called the Babylonian Method. It's related to a more general technique developed by Newton, but predates his work by hundreds of years!

Start with the number you want to find the square root of. There are three steps:

1. Guess
2. Divide
3. Average.

Just keep repeating steps 2 and 3 to improve your answer.

Here's an example. Let's try and find the square root of 12.

First, start by guessing a square root value. It helps if your guess is a good one but it will work even if it is a terrible guess. We will guess that 2 is the square root of 12.

In step two, we divide 12 by our guess of 2 and we get 6.

In step three, we average 6 and 2: (6+2)/2 = 4

Now we repeat step two with the new guess of 4. So 12/4 = 3

Now average 4 and 3: (4+3)/2 = 3.5

Repeat step two: 12/3.5 = 3.43

Average: (3.5 + 3.43)/2 = 3.465

We could keep going forever, getting a better and better approximation but let's stop here to see how we are doing.

3.465 * 3.465 = 12.006225

That is quite close to 12.

Answer 5

The method is described on the web page below. It is sure to be the same as the one on Roger the Mole's .../homeschoolmath/... web page, but maybe you will find one of them easier to follow than the other.

It is an easy method for me to remember, because I was taught it at school in the early 1950s, and I had to use it so often in worked exercises and examinations - before there was such a thing as a hand-held calculator - that I will never forget it or do it wrongly.

Answer 6

I like the method that looks like long division which is demonstrated under the heading "Finding square roots using an algorithm" at:

http://www.homeschoolmath.net/teaching/square-root-algorithm.php

It's especially useful because you can use it to calculate as many decimal places as you want. Calculators have built-in limits about that.

Answer 7

There are a few.

The most common is:

make a guess (sqrt(10) is about 3)

first approx:

sqrt(10) ~= 1/2(3+10/3) = 6.333.../2 ~= 3.17

if you need it closer, try again:

sqrt(10) ~= 1/2(3.17+10/3.17) = 6.3245.../2 ~= 3.162

and sqrt(10) is = 3.162277.....

Answer 8

I found a very helpful video. I hope it can help you. Here's the link:
http://www.youtube.com/watch?v=rHyaXYtjqvY

Here's another video, that explains the basics. Maybe this will help also:
http://www.youtube.com/watch?v=IqSMNfFtaMk&feature=related

Answer 9

yes.
memorize them.
and eat a lot of iron & vit. A