How would i find the square root of a number without a calculator?

Answer 1

iteration

you have the number.

Guess what the root might be. Square it. Were you too high? guess again with a lower number. Or were you too low? guess again with a higher number.

Then do it again, and again and again, refining it over and over, until you have enough decimal places to satisfy your curiosity.

That's how calculators do it...

Answer 2

the simplest way is using the 'bisection method', this involves finding two initial guesses , the first guess being too low and estimate, the second too high, and then from these two forming a next new estimate by taking the middle point between them, in other words bisecting them.

take for example finding the square root of 12, well our first two initial estimates might be '3' and '4'.

we know that 3 is too low an estimate because 3^2 equals 9 and not 12, also that 4 is too high an estimate as 4^2 equals 16, so we take the number midway between these as our new estimate, that being 3.5

now, (3.5)^2 is equal to 12.25, which is again too high, but it has helped to refine our answer, as now we know that the square root of 12 must lie between 3 and 3.5 (we take the 3 not the 4 because 3^2 is closer to 12 than 4^2 is)

so taking 3 and 3.5 as our new estimates and forming another bisection, that being 3.25, we test this as before, (3.25)^2 = 10.5625

so we now know that the root lies between 3.25 and 3.5 (again taking the 3.5 rather than the 3 because (3.5)^2 is closer to 12 than 3^2 is), so bisecting again gives 3.375, which when squared gives 11.390625

so now we know the root is between 3.375 and 3.5, and bisecting again gives 3.4375

so our new range is 3.4375 and 3.5 - we know the root is in this interval somewhere. bisecting again gives 3.46875

as this estimate is 'closer' to being the root of 12 then we dispense with the 3.5 estimate and go with the 3.46875, so our new interval is now 3.4375 and 3.46875

bisecting again gives 3.453125, this is a better estimate than the 3.4375 so we junk that.

the new interval is now 3.453125 and 3.46875, continuing this way we can further and further refine our answer, the true answer to 2decimal places is 3.46 so you can see that we are getting close already.

this method is the simplest but is rather slow to converge, better methods are 'hero's method' (sometimes called heron's method) or the newton-raphson method.

they are a little more complicated but far faster.

Answer 3

The square root algorithm is very difficult to describe in this forum. I put a link to one site that has an explanation that is not bad. If you put "square root algorithm" in a search engine, you might find a better one.

Another more rudimentary method is called divide and average. You make a guess of the square root's value. Then divide the radicand by your guess. Average your guess with the quotient. Make this your next guess. Any digits that are the same in your guess and the quotient are part of the square root.

Ex. Sqrt (87) ----> Guess 9
87/9 = 9.66666666

(9.66666666 + 9)/2 = 9.33333333

87/9.3333 = 9.3214

(9.3333 + 9.3214)/2 = 9.3274

Since the 9.3 in the guess and the quotient match, that is a pretty good estimate. I would probably go ahead and make it 9.33.

Answer 4

i comprehend its elementary. ok at the start there is something pronounced as ideal squares meaning that as quickly as you detect the sq. root of a variety like 4 it must be perfest and a pair of *2 is 4. yet interior the case of one hundred thirty five this is not ideal squared. your instructor wont ask you something like one hundred thirty five. purely ideal sqaure variety Squrae root is any variety that as quickly as multimplied by making use of different variety that are the coolest same come as much as the variety interior the basis. suaare roots a million=a million 2=4 3=9 4=sixteen 5=25 6=36 7=40 9 8=sixty 4 9=80 one 10=a hundred 11=121 12=one hundred forty four its preety a lot while theere is a variety interior the basis element. 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 of the fact the best answer

Answer 5

you can't find the exact answer you can estimate by thinking of the perfect squares and where the number lays in relation to them.

perfect squares 1, 4, 9, 16, 25, 36, 49, 64, 81, 100

square root of 67 would be somewhere around 8

Answer 6

use the method before calculators, 'Logarithms.
Log (to the base 10) 5.555 = 0.7447 / 2 = 0.3723
anti lg = 2.357
2.357 x 2.357 = 5.555

Divide the log by 3 for cube root or 4 for 4th root etc.
Multiply by 2 to square or 3 to cube etc.
Log tables are found at the back of old maths books

Answer 7

you have an estimate x for sq rt y. Now y=x^2 . So dy/dx =2x.

As an example you wish to find sq rt of 5. You estimate 2. Squaring it gives 4 . dy =1. dy =2xdx In this case x=2 so dy =4dx.

This gives dx =1/4 . So next estimate is 2.25. And so on.

Answer 8

Trial and error.
Say you're after the square root of 200.
You find the two integer squares it sits between, 14 and 15 and try out 14.5 squared. You'll find it's between 14 and 14.5 so you try 14.25 and so on, going to as many decimal points as required.

Answer 9

Three ways here : http://www.homeschoolmath.net/teaching/square-root-algorithm.php

Answer 10

Use papet-based log tables:
Take the number. Log it. half this value. Then anti-log it for you answer.