How can icalculate sequare root of any number ?

Question

For example 113 not by calculator.

Answer 1

Trial and error
Log Tables

Answer 2

Step 1: Split the number into groups of two digits starting for the right handside

like 113 => 01 13

Step 2: now find the nearest square number for the first two digit number i.e for 01 its 1 (so first digit of square root is 1)

Step 3: Now mutiply the digit found previously by 2 (here 1 by 2 gives 2) and find the nearest number x such that 2x * x is closer to the reminder previously... here you are left with 13 previously so the nearest number is 0 (0 is the next digit.. so far two digits found out 10)

Step 4: now since all the actual digits exhausted, append two zeroes at the end to make 1300 and find nearest x such that the root found so far (10) * 2 = 20x * x is nearer to 1300 ==> 206 * 6 is 1236... and go on ..

so comprehending...
1: 113 = 01 13
2: Square root:1 rest 13
3: Square root:10 rest 13
4: Square root:10.6 rest 64
5 Square root: 10.63... and goes on

for simple one say 144,
1: 144 = 01 44
__12___________
1---| 01 44
-----| 01
-----| --- 44
22--| --- 44
|_______
| 00 00

Answer 3

You can do it by the long division algorithm as follows

Step 1

Mark off in pairs from the units column to the right and in the decimals to the left (I have used commas (,) as markers)

1,13.00,00,00,00

Step 2

Seek the nearest smaller or equal square root of the first pair
___1_____________
1 ) 1,13.00,00,00,00

Step 3

Multiply the 1 on the top by the 1 in the divisor and establish the remainder with the next pair

___1_____________
1 ) 1,13.00,00,00,00
......1___
..........13

Step 4

Double the number at the top and find the number that will then be placed after the doubled numbber and in the quotient to make a muliple less than the 13 remainder (in this case it will be 0 as the number we are playing with is 2_ X _ <= 13 and the only solution is 20 X 0 = 00 (<= 13)

___1_0___________
1 ) 1,13.00,00,00,00
......1___
....20)13

Now keep repeating steps 3 - 4

__1_0_._6_3_0__1__
1 ) 1,13.00,00,00,00
......1___
....20)13
...........0
..206)1300
..........1236
.....2123)6400
................6369
.......21260)3100
....................0000
......212601)310000
.....................212601
........................97399
etc

Thus sqrt (113) = 10.6301............


PS By calculator sqrt (113) = 10.63014581........

Answer 4

It helps if you already know the squares of some integers, for example:

10^2 = 100
11^2 = 121

So, since 113 is between 100 and 121, you know that it's square root must be between 10 and 11

My 6th grade teacher actually taught me how to do this by hand (that was a million years ago before the days of calculators) and I wish to God I knew how to do it today, but I don't.

Sorry, this is as far as I can help you on this one!

Answer 5

Group the digits pairwise from the decimal point.
113 = 1 13.00 00 00
Set it up like long division
Find a square <= to the first digit or pair. this becomes your first divisor.
...1
1)1 13.00, multiply
..-1........., subtract, multiply quotient by 2_ and use
2_|..13.00, find trial digit (0 here)add to 20 and multiply

....1..0
20|..13.00
.0|...00, bring down
.20_|13.00, new trial digit (<1300/200)

1 0.6
206|13.00 00
....6|12.36
212_|...64 00, (6400/2120) = 3

...1 0.6 3
2123|6400
.... 3|6369
2126_| 31, (31/21260 = 0)

... 1 0.6 3 0
21260| 3100
...... 0| 00
21260_|310000, (310000/212600 = 1

... 1 0.6 3 0 1, calculator value = 10.6301458
212601|310000
........ 1|212601
...212602_|7399

...and so on
...never derived the method, just memorized it...It has to do with "what pair of rectangles with side = previous estimate added to the estimate squared is still less than F?"

Hope this is readable.

Answer 6

Take a guess and then est.square root of the difference and add to next decimal then do for a few more itterations. So lets take 113, guess 10, 10x10=100 then guess the square root of the difference 113-100=13 divide by 2= 6.5, add to 10 as decimal to get 10.65 x 10.65 = 113.5 so now the diff again /2 = .025 so 10.65-.025= 10.625x10.625=112.9 and keep itteration and itteration till you get it.

Answer 7

I so much want to know this too. I used to know how to do this without calculator when I was in high school but I haven't done this for so long that I can't remember any more. Anyone? please.
And i want to know this in details. Otherwise I can do the trial and error, too. But what is the mathematical process of calculating square root of any number without using a calculator or guessing?

Answer 8

you find an estimate. Better the estimate faster it is

say for N you estimate x say 10
now calculate 1/2(x+N/x) = 1/2(10+113/10) = 1/2(10+11.3) = 10.65
now 10.65 is closer.

once more 1/2(10.65+113/10.65) = 1/2(10.65+ 10.61)
= 10.63
good enough

Answer 9

Browse this site it can help you understand square root