how to get the square root of a number manually?

Question

we are currently studying triangles. we are at the pythagorean theorem.

Answer 1

Suppose you had a right angled triangle of sides 3cm and 4cm and were asked to find the hypotenuse, the working could be:-
h² = 3² + 4²
h² = 9 + 16
h² = 25
h = 5 (mentally) because 5 x 5 = 25
However there will not be many that you can do mentally and the usual way is to use a calculator to find the square root of the number on the right hand side.

Answer 2

Well,

I know of a way but I think you may not mind the calculator after you use
it for a while. It is based on an algorithm called Newton's method, but you
don't have to understand that to be able to do the method.

My idea is to first make a good guess, then see how close you are, and then
make a better guess.

We'll do it with 6, which according to my computer is

2.4494897427831780982

We will see how close we can come.

We start by guessing. 2 is a pretty good guess. So we check

2^2 is 4 - too low so we guess 3
3^2 is 9 - too high

So now we need to refine our guess.

The number we are looking for is 6. 4 is 2 lower than six and 9 is
3 higher than six.

So ( here is the important part) to make our next guess we take the total
difference in the results of our first two guesses (9-4 = 5) and then we
find how much the desired answer is greater than the smaller guess (6-4 = 2).

Then we add the quotient to our guess, so our next guess will be 2 and 2/5
or 2.4

2.4 ^2 is 5.76 - too low, so try 2.5 ^2 = 6.25

Now we repeat
6.25 - 5.76 = .49
6 - 5.76 = .24

.24/.49 = .44898 the level of precision here is actually quite small,
so you should always just use the tenths place, which is .4 - so our next
guess would be 2.44

Notice that I did not add .4 to get our next guess; I merely added the digit
four to the end of the number that we are using as our guess.

This method should lead to greater and greater refinement.
As you can see, we already have the first three digits right.

I will do one more example to let you see how this would work for a large
number. Let us consider 700.

My initial guess will be 20
20^2 = 400 - too low so I go to 30
30^2 = 900 - too high, so we go to the method

900 - 400 = 500
700 - 400 = 300

So 300/500 is .6 and our next guess will be 26. Do you see why?

26^2 = 676 - too low, so we try 27
27 ^ 2 = 729

so 729 - 676 = 53
and 700 - 676 = 24

and 24/53 = .4538... but all we need is the tenths place, which is .4

So our next guess is 26.4
26.4^2 = 696.96 - too low, so try 26.5
26.5 ^ 2 = 702.5

702.5 - 696.96 = 5.54
700 - 696.96 = 3.04

3.04/5.54 = .5487

But again we will just take the 5 and add it to the end.
This makes our next guess 26.45.

and 26.45^2 = 699.603.

You can see that we are very close now and that we could get as close as
we wanted to with this method. My computer gives this answer.
26.457513110645905905

so we didn't do too bad, huh?

Hope this helps.

- Doctor Ethan, The Geometry Forum

Answer 3

1. Create a “square root” chart for yourself number square square root 1 1 1.000 2 4 1.414 3 9 1.732 4 16 2.000 5 25 2.236 6 36 2.449 7 49 2.646 8 64 2.828 9 81 3.000 10 100 3.162 11 121 3.317 12 144 3.464 13 169 3.606 14 196 3.742 15 225 3.873 16 256 4.000 17 289 4.123 18 324 4.243 19 361 4.359 20 400 4.472 21 441 4.583 22 484 4.690 23 529 4.796 24 576 4.899 25 625 5.000 26 676 5.099 27 729 5.196 28 784 5.292 29 841 5.385 30 900 5.477 31 961 5.568 32 1,024 5.657 33 1,089 5.745 34 1,156 5.831 35 1,225 5.916 36 1,296 6.000 37 1,369 6.083 38 1,444 6.164 39 1,521 6.245 40 1,600 6.325 41 1,681 6.403 42 1,764 6.481 43 1,849 6.557 44 1,936 6.633 45 2,025 6.708 46 2,116 6.782 47 2,209 6.856 48 2,304 6.928 49 2,401 7.000 50 2,500 7.071 51 2,601 7.141 52 2,704 7.211 53 2,809 7.280 54 2,916 7.348 55 3,025 7.416 56 3,136 7.483 57 3,249 7.550 58 3,364 7.616 59 3,481 7.681 60 3,600 7.746 61 3,721 7.810 62 3,844 7.874 63 3,969 7.937 64 4,096 8.000 65 4,225 8.062 66 4,356 8.124 67 4,489 8.185 68 4,624 8.246 69 4,761 8.307 70 4,900 8.367 71 5,041 8.426 72 5,184 8.485 73 5,329 8.544 74 5,476 8.602 75 5,625 8.660 76 5,776 8.718 77 5,929 8.775 78 6,084 8.832 79 6,241 8.888 80 6,400 8.944 81 6,561 9.000 82 6,724 9.055 83 6,889 9.110 84 7,056 9.165 85 7,225 9.220 86 7,396 9.274 87 7,569 9.327 88 7,744 9.381 89 7,921 9.434 90 8,100 9.487 91 8,281 9.539 92 8,464 9.592 93 8,649 9.644 94 8,836 9.695 95 9,025 9.747 96 9,216 9.798 97 9,409 9.849 98 9,604 9.899 99 9,801 9.950 100 10,000 10.000 1 4 9 16 25 36 49 64 81 100 121 144 169 Notice, the top numbers are whole numbers, the BOTTOM are the “square” of the whole number above it. These are called: PERFECT SQUARES. 2. Identify the whole numbers the “square root” falls between. 3. Think that “.5” falls between the two whole numbers (If the two whole numbers are 8 & 9, then “8.5” is right in the middle) 4. Ask yourself: “Which whole number is the square root CLOSER to?” (If it is closer to the “smaller” number, then the estimate should be LESS THAN the “.5”, if it is closer to the “larger” number, then the estimate should be GREATER THAN the “.5”. Examples Estimate each square root to the nearest tenth: 1. (refer to the chart above) 2. the square root of 10 falls between “9” and “16” (Which is “3” and “4” whole #'s) 3. THINK that “3.5” is right in the middle. 4. Since the square root of 10 is closer to 9 (or whole number 3), your estimate must be LESS THAN “3.5”. Ans. Is about “3.1” or so.

Answer 4

You told that you are studing the triangles problem using Pythagorean theorem so you have little (i.e..small numbers)problems in finding sides or hypotaneous(in exams also you get a small and little bit problems like the sides of a triangles are in manner 3,4,5 etc..) Upto ten you remembers square of a numbers(square of 1=1, square of 2=4........square of 10=100)


Now take a example 324 to find out the square root just follow

-I think u know lcm just follow that procedure


2 324
2162 ( This is lcm finding out )
381
327
39
33
31


-Then you convert all numbers in the type of squres....Just see

by doing lcm we get No. of 2's=2

No. of 3's=3


i.e..324=2*2*3*3*3*3=2*2*9*9

324 =square of 2 * square of 9=square of (2*9)

now square root of 324 = squre root of[ square of (2*9) ]

which implies =18


so square root of 324=18

Answer 5

Hey! Wassssssssup???
pythagoras theorem studying, then ur in 6th or 7th i guess!
now, the only 2 ways i kno to get sq root manually is to--
1. Factorize the no. eg:--
find sq root of 81
3 81
3 27
3 9
3 3
1
nOWU TAKE OUT PAIRS.

81= (3*3)*(3*3)
so, sq root 81 =3*3 =9

2. this a long method u have to divid it but keeping sqres in mind
eg- lets take sqrt 196

First, from units place pair the no.s
1 96, then start ur process.
i don remember it soooooo well nd sinc its looooong. iwont confuse u
stick 2 prime factorization method
cya
hope it wasuseful 2u.
Byeee

Answer 6

Better Guess = (1/2)( Guess + Number/Guess)

the method is iterative.

for example, what is sqrt(21)

well, i know it is between 4 and 5 (because sqrt(16) = 4 and sqrt(25) = 5) so as my first guess i'll guess 4.5

Better Guess = (1/2)(4.5 + 21/4.5)
Better Guess = 4.583333

now use this as the guess:

Better Guess = (1/2)(4.5833 + 21/4.5833)
Better Guess = 4.5825757

Keep going as often as you need for the accuracy you desire. The formula converges quickly, especially if your initial guess is smart.

look up Babylonian Method for more info. Look up Newton's Method for info to find any root (3rd 4th etc)

Answer 7

This link shows how to take the square root of a number by hand.

http://www.nist.gov/dads/HTML/squareRoot.html

Answer 8

without a calculator your teacher will want you to leave it in sqr root form ... she may ask you to reduce such as
sqr(72)=sqr(36)*sqr(2)=6*sqr(2)

Answer 9

you mean without a calculator??