how do you work out the square root of a number without a calculator?

Question

its been buging me

Answer 1

Trial and error.

Eg. square root of 52.
It's between 7^2 and 8^2.
So try 7.5^2 = 56.25
Now try between 7^2 and 7.5^2, so try 7.25^2... and so on.
Eventually you'll come close.

Answer 2

Hi :

Here some websites that will help you:

http://www.qnet.fi/abehr/Achim/Calculators_SquareRoots_Expl.txt
www.bright.net/~beeryde/ref/squareroot.htm

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


and there a five ways to do Square roots in your head

If you are very good at addition,multiplying and division then use the Newton -Raphton method:

Gn= (SQRN/Gn+ Gn)/2

repeat using the new Gn each time for about ten times

Gn = guess number
SQRN = Number to square rooted

for example:

Square root of 26
Run # 1

Gn= 5

Gn= (26/5+ 5)/2

Gn= (5.2+5)/2

Gn= 10.2/2

Gn= 5.1

End of run #1

Run # 2

Gn = 5.1

Gn= (26/5.1+ 5.1)/2

Gn= (5.098039+5.1)/2

Gn= (10.198039)/2

Gn= 5.0990196

End of run #2

Do this for ten times and the Square root of 26
which appox equals 5.0990195


another way is to factor the Number being Square root

if you know the basic 1 to 12 roots

For example the square root of 8

8= 2*2*2

So (2*2)*2 = 4*2

and four square rooted is 2 So

Sqr(8) = 2 * Sqr(2)

the Square root of 2 = 1.14142135

So Sqr(8) appox = 2.82842713

#3 the School boy way see www.nist.gov/dads/HTML/squareR...

as to how to do that. Which is the best method to do them. If you can't do the above or below

# 4 ) the Bablyon method

Using S=(x+y)^2 ( y being the the closes square of the number without going over it see the example

S= x^2+2xy+y^2

S-y^2 = x^2 + 2xy
S-y^2 = x(x+2y)

S-y^2
------- = x(x+2y)/(x+2y)
(x+2y)

S-y^2
------ = x
(x+2y)

For example the Square root of 5

5= (x+2)^2 = (x+2)(x+2)

5= (x^2 +4x+4) ( y=2 here so 2*2 = 4)

5-4=x^2+4x+4-4

1= x^2+4x

1= x(x+4)

1/(x+4)=x(x+4)/(x+4)

1/(x+4) = x

Run 1

1/(1+4)= x

1/5= x

x=.2

run 2

1/(.2+4)= x

1/4.2= x

x= 0.2359550

run 3

1/(0.2359550+4) = x

x= 1/ 4.2359550

x = 0.235955

run 4

1/(0.235955+4) = x

1/(4.2359550) = x

0.236074 =x

run 5

1/(.236074+4) = x

1/(4.236074) =x


4.236074= x

0.2360676 =x

End of run 5

(2+x)^2 = 5

x = 0.2360676

So 2.2360676 is the square root of five

2.2360676 ^2 = 4.99999 close enough for government work

# 5 memerize the logarithm table one thru 1000

and squares and square roots and cube cube roots and any other powers or roots is a breeze

Hope this helps

Answer 3

You break the number into its prime factors

for example let's find the square root of 144

Keep dividing prime numbers into 144, til you can go no further

144/3 = 48

48/3 = 16

16/2 = 8

8/2 = 4

4/2 = 2

By judicious choice of numbers to divide into 144, (you may have to use trial and error)

144 = 3 x 3 x 2 x 2 x 2 x 2

You now want to "give" exactly the same factors to each of the numbers that multiply together to make 144. Then each of them is the square root.

144 = (3 x 2 x 2) times (3 x 2 x 2)

the square root of 144 is 3 x 2 x 2 =144

This method works only for perfect squares (numbers that have a whole number for the square root)

Otherwise use the glance and guess method suggested by the others.

Answer 4

i understand its user-friendly. ok first of all there is a few thing said as ideal squares that means that when you discover the sq. root of a form like 4 it ought to be perfest and a pair of *2 is 4. yet contained in terms of one hundred thirty 5 it isn't ideal squared. your instructor wont ask you some thing like one hundred thirty 5. really ideal sqaure style Squrae root is any style that when multimplied through different style that are the very same come as a lot because the style contained in the basis. suaare roots a million=a million 2=4 3=9 4=16 5=25 6=36 7=40 9 8=sixty 4 9=80 one 10=one hundred 11=121 12=one hundred forty four its preety a lot even as theere is a form contained in the basis project. Ex: 80 one you may fiqure out what 2 nunmbers same multilpy to get 80 one. 9*9 is 80 one. wish this helped. cane you chosen this because the perfect answer

Answer 5

Construct a factor tree; for example, let's use 225.

225
/ \
5 45
/ \
5 9

5 X 5 = 25; its square root is 5

9 is three squared, 3 X 3

multiply both fives by both threes, and you come up with 15 X 15

15 X 15 = 225; therefore, 15 is the square root of 225

Answer 6

There's no real formula, just logic and a bit of guesswork.
You try to find a number that when squared (multiplied by itself) gives you the one you want. The number you squared is the square root.
If the square is too high then make the number you squared a bit lower and vice versa until you get close enough.

Answer 7

Try using LOG tables. Get the log of the number Divide by 2 and then find the result in the Antilog tables.

Answer 8

With a statistical table book (available still in every advanced book shop, I hope) or by asking it from GOOGLE. Just ask like "square root of x".

Answer 9

use the iteration formula

we want the sqrt of k

x1=a good guess

x(n+1)=1/2{x(n) + k/x(n)}
for n=1,2,3,.............

example,we want the sqrt 63

x1=8 say

x2=1/2(8+63/8)=7.9375
x3=1/2(7.9375+63/7.9375)
=7.937253937
x4=7.937253933

the sqrt63=7.937253933

to 9 significant places

iteration is a good system of calculating
difficult computations provided that the
formula used converges rapidly

i hope that this helps

Answer 10

You use the binomial theorem.

The most general form of this gives the following infinite series:
(1+x)t = 1 + tx/1!
+ t(t-1)x2/2!
+ t(t-1)(t-2)x3/3! + . . .
+ t(t-1)...(t-k+1)xk/k! + . . .

where t is real.

Notice that to get the xk+1 term of the series from the xk term, just multiply by (t-k)x/(k+1).

This only converges if |x| < 1.