Is there a technique to work out the square root of a given number?

Question

Without using a calculator. I think I was taught it at school, but that was a long time ago

Answer 1

There is, but it's easier to show you than to tell you.

Look here --> http://jwilson.coe.uga.edu/EMT668/EMAT4680.folders/Nowlen/squareroot.html

Answer 2

To find the square root of 63:

1. Pick a number, eg. 5.
2. Square it to get 25.
3. Do 63-25 to get 38.
4. Divide it by 10 to get 3.8.
5. Add 3.8 to 5 to get 8.8.
6. Repeat this with 8.8.

Continue doing this until the answer is accurate enough for you. You will have to do long hand multiplication. Check your final answer.

Having looped through this nineteen times I got 7.937254... which my calculator agrees with.

Initially the numbers vary a lot around the true answer but then they become much more accurate. I hope it does not take you too long!

Added Stuff
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I slept on this and found that it did not always work. You will need to experiment with step 4 so that the answer from step 3 gets nearer to zero every time. Try dividing by 100 (or even more sometimes). I think that for numbers up to about 900 that you need to find the square root of, you should make the answer to step 4 less than 10. If the number is greater than 1000 then experiment!

Answer 3

If you're working with whole numbers 16, 25, 49, 121 then your best bet is to "guess" till get close ... e.g. if you are finding suare root of 35 ... 5 squared is 25 so must be more, then 7 squared is 49 .. must be less ... 6 squared .. wahoo!!! answer!

If not then: Newton's Iteration is probably the easiest but it still does require the use of algebra.

Newton's iteration is simply an application of Newton's method for solving the equation x*x -n = 0.

or x*x = n or x = square root of n.

To use the iteration you basically use the following:

x (sub k+1) = 1/2 * (x (sub k) + (n/x (subk) )

or

x = 1/2 * ( x + n / x) and keep plugging in numbers for x till you get closer to the answer.

I'm providing a link to the page that explains it below cause Yahoo Answers isn't condusive to answering math equations.

Also there's another link to other Square Root Algorithms.

Answer 4

if you are only going up to 25*25

2*2 =4 3*3= 9 the difference is 5.if i keep adding 2
so 9+7= 16 4*4 the difference is 7 so +2 again
so16+9= 25 5*5 the difference is 9 so +2 again
so25+11=36 6*6 the difference is 11 so +2 again
so say number is 240 You would think of say 15 i give up sorry i have headache

Answer 5

You should be aware that this is not a very useful algorithm. Your calculator, for example, uses an entirely different procedure for finding square roots. Actually no procedure with "find" the square root of 13, since the root of 13 is irrational, that is, it is not an integer and cannot be written as a common fraction of the form a/b. The square root of an integer is either an integer, as in the case of 9 or 16, or is irrational and the decimal part is infinite and non-repeating.

Answer 6

I've spent many an hour trying to make my own algorithm to calculate the square root of a given number but to no avail.

Nice one DoctaB01, you get the best answer vote for me.

Answer 7

Yeah. You know the multiplication table by heart, hopefully. So, if you are doing the square root of 144, your answer is 12, because 12X12=144. The square root of 25 is 5, because 5X5=25.

Answer 8

Finding the square root of a number is the inverse operation of squaring that number. Remember, the square of a number is that number times itself.

Examples of the formula are here

Good luck

Answer 9

yup...

do u remember hw to take ot L.C.M(lowest common multiple)

for eg ur number is 144

lets see the factors..

2 |144
2 | 72
2 | 36
2 | 18
3 | 9
3 | 3
| 1

then make the pairs of 2 of the multiples

2
2
2
2
3
3

pairs 2,2,3

multiply them 2x2x3= 12

so the square root of 144 is 12

i hope u get it!!!

Answer 10

new x=.5(x+b/x) where b is the square root you want and x is a random number for the first try. This is the equation used in all computers and calculators.
b=25 x=21
.5(21+25/21)=11.1
.5(11.1+25/11.1)=6.676
.5(6.676+25/6.676)=5.21
.5(5.21+25/5.21)=5.004

Answer 11

I know of 8 ways to get a square root of a number

1) Use the x^y key ( type the number to square rooted, press the x^y , type .5,and press the " = " key or enter

2) Type the number, use the inv key, press x^2 , and press the " =" key or enter

3) Type in your number press the "log" key then divided by 2 press the " = " or "Enter" and use the inv and press the "log" key


4) Type in your number press the "ln" key then divide by 2 press the " =" key or "Enter" then select the inv spot and press the "ln" Key then press the " =" key or enter { this is the same thing as #3 but using a different log base }

5) use the follow method:

gn= (N/gn)+ gn)/2

repeat for about five to ten time ( example # 2 below) this is faster than example # 1 below:

6) Use the School boy method the web link for it is:

www.nist.gov/dads/HTML/squareRoot.

7) Use The Handbook of Mathematical Formulas and Tables and find the Square and Square root table for the square

8) use this method ; if you got a lot of time to kill and your bored:
( See example # 1 below )

(for example the square root of 26)

start with a start square root in the first digit


1. start with the closest square without going over in this case 5
2. add .1
3 store in memory, then multiply by 5, add result to the square number
4) square it or times that number by itself
5 ) get the answer
6) If it too low then add number in memory to squared numbered then # 10
7) if it too high for the number square rooted then # 11
8) replace your square number with new square number
9) multiply number in memory by .1
10) have you gone to as digits you want to go to? then Stop
11) goto # 4

11) subtract number by memory
12 goto # 4

Example # 1 :

square root of 26 :

Step # 1 :

5

Step #2 :

.1 in memory

Step # 3:



5*.1 = .5

Step # 4:

5+.5 = 5.5

Step # 5

5.5* 5.5 = 30.25

Answer too high

5.5 - .1 = 5.4

doing Step #5 again

5.4^2 = 29.16

Answer still too high

5.4-.01= 5.3

doing step # 5 again

5.3 *5.3 = 28.09

Answer still too high

5.3-.1= 5.2

doing step 5 again

5.2*5.2= 27.04

answer still too high

5.2-.1= 5.1

5.1*5.1= 26.01

answer close. check to make sure it it not lower

5.1-.1 = 5.0

doing step 5 again

5.0*5.0= 25.

answer too low

gone to as many digits you want to go?

no

recall .1*.1 = .01

.01* 5 = .05

5.05 *5.05= 25.5025

answer too low

5.05+.01= 5.06

5.06*5.06= 25.6036

answer too low

5.07*5.07= 25.7049

answer too low

5.07+.01= 5.08

5.08* 5.08= 25.8064

answer too low

5.09*5.09= 26.01

answer close

Have we gone to as many digits as you want to go?

no

.01*.1= .001
.001 * 5 = .005

.005 + 5.095

5.095 * 5.095 = 25.959025

answer too low

5.095+ .001= 5.096


5.096*5.096= 25.969216

answer too low

5.096 + .001= 5.097

5.097*5.097= 25.979409

answer too low

5.097+ .001= 5.098

5.098*5.098= 25.989604

answer too low

5.098+ .001= 5.099

5.099*5.099= 25.999801

answer close

have you gone to as many digit you want to go?

no

.001*.1= .0001
.0001*5= .0005
5.099+.0005= 5.0995

5.0995*5.0995= 26.00490025

answer too high

5.0995-.0001= 5.0994

5.0994*50994= 26.00388036

answer too high

5.0994-.0001= 5.0993

50993*5.0993= 26.00286049

answer too high

5.0993-.0001= 5.0992

5.0992*5.0992= 26.00184064

answer too high

50992- .0001 = 5.0991
5.0991*5.0991= 26.00082081

answer too high

5.0991-.0001= 5.099
5.099* 5.099= 25.999801

answer too low

have we gone to as many digits you want to go to?
yes

5.099 is the apoximate square root accurate to 4 decimal places

I said that this is a time waster but it will give you accurate results

Example #2

the square root of 37

gn=( (N/gn)+Gn)/2

gn = guess root or square root
N= number to square root ( in this case 37)


start of run # 1

start with gn = 6

((37/6) + 6)/2= gn

(6.16666...)+6)/2= gn

12.6666.../2= gn

6.083333333..= gn

End of run #1


Start of run # 2 :

gn = 6.083333333

((37/6.083333333)+6.083333333)/2 =gn

(6.082191780...+ 6.083333333...)/2 = gn

12.165525114/2 = gn

6.082762557= gn

End of run #2

Start of run # 3:

gn = 6.082762557

((37/6.082762557) + 6.082762557)/2= gn

( 6.082762503+6.082762557 )/2 = gn

12.165525060/2 = gn

6.082762530= gn

End of run # 3


Start of run # 4:

gn = 6.082762530

((37/6.082762530)+ 6.082762530)/2= gn

(6.082762530+ 6.082762530)/2 = gn

12.16552506/2 = gn

6.082762530 =gn

End of run # 4

The Square root of 37 is appoximately equal to 6.082762530 accurate to nine decimal places