How would you find the square root of a number?

Answer 1

There is a "long division" style method which gives the square roots of all numbers:

1. write the number out in pairs of digits starting from the decimal point;
2. There may be a single digit to the left of the decimal point; this is fine;
3. If there is a single digit at the end after the decimal point put a zero after it;
4. Multiple double zeros can be added to the end of the number AFTER the decimal point to allow for the required degree of accuracy;
5. Now do a kind of long division:
6. Start at the left of the number like a normal division but work with digit pairs (the first pair may be a single digit)
7. Put a decimal point in the quotient (square root result) over the decimal point in the dividend (number)
8. Take the quotient so far (ignoring the decimal point) and multiply by 20 to create a divisor.
9. Now calculate a single digit to add to the divisor so that when multiplied by this digit the result is as large as possible not greater than the partial sum so far.
10. Write this digit over the next pair of digits
11. Multiply the divisor with this digit added by this digit, write down below the partial sum and subtract
12. Bring down the next two digits
13. repeat from step 8 until there are no more pairs of digits to bring down, or the required degree of accuracy has been reached
14. The quotient is the required square root
15. Note that if the final subtraction was not zero, then the last digit of the quotient should be used to round the quotient to one less digit.

Examples:

√87025

____________2__9__5_____

_________--------------_____

______2_|__8_70_25_____←0 × 20 = 0; 0 + 2 = 2 since (0 + 3) × 3 = 9 > 8

________|__4_____________←2 × 2

________|__--_____________

____49_|__4_70_________←bring down 70; 2 × 20 = 40; 40 + 9 since no higher digit

________|__4_41_________←49 × 9

________|__------_________

___585_|____29_25_____← bring down 25; 29 × 20 = 580; 580 + 5 (as equals required valued)

________|____29_25_____← 585 × 5

________|____--------_____

__________________0_____← zero partial result; no more digits, quotient = square root

→ √87025 = 295

√12.96

____________3_⋅__6________

_________------------_______

______3_|_12_⋅_96_______←0 × 20 = 0; 0 + 3 = 3 since (0 + 4) × 4 = 16 > 12

________|__9_____________←3 × 3

________|__--____________

____66_|__3____96______←bring down 96; 3 × 20 = 60; 60 + 6 (since gives required value)

________|__3____96______←66 × 6

________|__----------______

__________________0______← zero partial result; no more digits, quotient = square root

→ √12.96 = 3.6

√167.5 to 2 dp (nearest hundredth) so calculate to 3 dp (thousandth's digit) to round:

____________1__2_⋅__9__4___0

__________-----------------------

_______1_|_1_67_⋅_50_00_00_← note single 1 at left, but .5 becomes .50 and two extra 00 for 3 dp

_________|__1_________________←1 × 1

_________|__--_________________

_____22_|__0_67______________←bring down 67; 20 × 1 + 2 = 22

_________|____44______________←2 × 22 since 3 × 23 = 69 > 67

_________|__------______________

____249_|____23___50________←bring down 50

_________|____22___41________← 249 × 9

_________|____----------________

___2584_|_____1___09_00____← bring down 00

_________|_____1___03_36____←2584 × 4 (2585 × 5 = 12925 > 10900)

_________|____---------------____

_25882_|___________5_64_00_← bring down 00

_________|___________5_17_64_←25882 × 2

_________|___________----------_

_________|_____________46_36_←as want to 2 dp and now got 3 dp can stop.

→ √167.5 ≈ 12.94 to 2 dp.

The finding of the units digit of the divisor at each stage is tricky and requires some trial and error. When calculated in binary, this algorithm is very efficient as the units digit can only be a 0 or a 1, and multiplying by 20 becomes a shift left; this makes it an efficient algorithm for digital computers to use.

Answer 2

One way to estimate the square root of a number is by iteration. This entails making a guess at the answer and then improving on it. Repeating the procedure should lead to a better estimate at each stage. One such is the Newton-Raphson method.

If you want to find the square root of kkk, define f(x) = x^2 – kkk. Then finding the square root of kkk is equivalent to solving f(x) = 0.

Let f’(x) = 2x. This is the derivative of f(x) but you do not need to know that to use the N-R method.

Start with x0 as the first guess. Then let xn+1 = xn - f(xn)/f’(xn) for n = 0, 1, 2, … Provided you made a reasonable choice for the starting point, the iteration will very quickly converge to the true answer. It works even if your first guess is not so good:
Suppose you want the square root of 7 and you you start with x0 = 5 (a pretty poor choice since 5^2 is 25, which is nowhere near 7).
Even so, x3 = 2.2362512515, which is less than 0.01% from the true value. Finally, remember that the negative value is also a square root.

Answer 3

If you have anything more than a basic calculator, it should have a square root key. If it has a key for finding a number raised to a power use x^0.5 to find the square root of x.

Otherwise, one way to estimate the square root of a number is by iteration. This entails making a guess at the answer and then improving on it. Repeating the procedure should lead to a better estimate at each stage. One such is the Newton-Raphson method.

If you want to find the square root of k, define f(x) = x^2 – k.

Then finding the square root of k is equivalent to solving f(x) = 0.

Let f’(x) = 2x. This is the derivative of f(x) but you do not need to know that to use the N-R method.

Start with x0 as the first guess.

Then let xn+1 = xn - f(xn)/f’(xn) for n = 0, 1, 2, …

Provided you made a reasonable choice for the starting point, the iteration will very quickly converge to the true answer.

It works even if your first guess is not so good:

Suppose you want the square root of 7 and you you start with x0 = 5 (a pretty poor choice since 5^2 is 25, which is nowhere near 7).

Even so, x3 = 2.2362512515, which is less than 0.01% from the true value. Finally, remember that the negative value is also a square root.

Answer 4

To find the square root of x, use x^0.5