Find Square Root by Hand without Calculator

Learn how to find the square root of a number by hand approximated to at least two decimal places. In this video we approximate the square root of 38 out to two decimal places without a calculator.

#squareroots
#byhand
#withoutcalculator

[DarkAngelEU]

My high school teacher insisted there was no way you could do square roots by hand. I insisted there was a way and she challenged me to come up with a method. I never figured it out, but I knew there HAD to be a way. How else could people do them without calculators for all those centuries?

You've shown the way! :D

[Devlinflaherty]

Thanks for being the one video that actually explains this process instead of just a “trick” to do rough estimates.

[HHHGeorge]

I was once shown this by a math teacher when I was 13 and have been trying to remember it ever since. Thank you for this video.

[achrace.profrichardachara]

A true teacher!
You don't presume we know but take the trouble to take us through all (not some) of the logical necessary steps to the answer.
Thank you, sir!

[methodstomathness]

This is how I had learnt to find square roots some 45 years ago in Indian schools. The format and layout were slightly different but the flow was the same. Calculators weren't readily available then. Thanks for this demo.

[vanlepthien]

Fortunately, my 3rd grade math teacher taught us this. It has turned out useful on many occasions.

[fifthavenue]

What can I really say to explain how grateful I am for your clear explanation? Understanding mathematics does so much for self-confidence not to mention giving you the ability to breakdown important concepts and it is so often so difficult to make the process of learning clear and not something that makes you doubt your abilities. Thank-you very much.

[geitekop]

Note, you don't have to actually 'double' the number to get the next number in sequence: the pattern shown is this: let's say we have 12_ at the start. You find it to be the number 1. Hence, 121. The next number will be 121+(number you found=1)_. Thus (121+1)_ = 122_. We found in the video that the missing number was 6. So the next number in the sequence will be 1226+(number you found, thus 6)_ = 1332_, etc.

[harveybc]

For reference, I turn 72 this month. Back when I was in the 5th or 6th grade there was a TV comedy called "My Favorite Martian". In one episode the martian, "Uncle Martin", encountered a 13 year old, I think?, kid who was a genius and he could calculate square root. I asked my father if he could and how to do it. He showed me this very same technique. He didn't use the <= but the technique was the same. Never forgot it.

What is remarkable was that my father, who grew up during the depression, had dropped out of school in the 9th grade. Amazing how much they use to teach kids back then. I graduated college just as scientific pocket calculators were being introduced and was never taught this in school. (Remember slide rules? I was one of those kids that always carried one.)

This video brings back pleasant memories. Thanks.

[kathleengarvey]

Thank you! I am 67 and had never had that explained to me ! ( Girls used to be told " you can't do this. ") You are very clear, knowledgeable and good at explaining math. Thank you.

[notanyaleiff]

This neat method is based on the binomial coefficients. For square root, it's derived from a^2+2ab+a^2 as such: a^2 + b(2a+b). A similar but more complex method gives cube root: a^3 + b(3a^2+3ab+b^2). For fourth root, it's a^4 + b(4a^3+6a^2b+4ab^2+b^3). Fifth root is a^5 + b(5a^4+10a^3*b+10a^2*b^2+5ab^3+b^4), and so forth. You would bring down groups of three 0s, four 0s, and five 0s, respectively.

It's doable by hand for cube root, but rapidly grows to involve gigantic multiplicands when used for higher roots. Nonetheless, it works.

[Rinnyman]

I believe you'd have to always calculate an extra digit to determine approximation, for example, if you just calculated the first decimal, you'd get 6.1, however, in the video you can see that this would approximate to 6.2, so calculating that one extra digit every time should increase the consistency of success in this method.

[humester]

If you have to give an answer to 2 decimal places you must go to three because, if the third decimal place is 5 or greater, you would need to increase the second decimal place by one. And, if you go with the convention that even digits before 5 do not get increased, you would have to go even further.

[kaczka]

Simple, but not precise method is optionally:
1. Estimate the root as well as possible - Lets take the same example (y = 38) and first estimation a = 6
2. Compute the next value as b =(y + a^2)/(2a) e.g. b = (38 + 36)/12 = what is quite good result (delta<0.1%)
Remark:
Using that method twice (i.e. with start value of 6.166) the result is much more exact
Warning:
The better first estimation, the better (more exact) is result. In case of e.g. SQR(45) it is better to take 7 instead of 6 because 45 is next to 49 not to 36. Don't hesitate to use inbetween values as for example 6.5, if the number is more or less in the same distance from both quadrats (e.g. 42)

If the first estimation is made with a relative accuracy of Delta, the result will have an accuracy of (Delta^2)/(2*(1+Delta)) e.g. by estimation at about 20% the result will be better than 1.7% but if You could get better start (say 2%) the result will be practically accurate with 0.02% error.

[EvilSandwich]

You should also do a video on how to do logarithms by hand. Its an equally interesting process... and gave me a ton of respect for what Napier and Briggs had to go through to make their first log tables. Yikes!

[kaiji]

FORMULA:

X = the square root answer
Y = the square closest to the number
Z = the square root number
Difference = the Difference you get after Subtracting Z from Y




√Z = X .

Z - Y = Difference

Difference / X * 2 + X = X

Example:

√26 = 5 .

26 - 25 = 1

1 / 5 * 2 = 1 / 10 = 0.1 + 5 = 5.1

√26 = 5.1

[JesusAlbertoPinto]

It is still better just using a Taylor expansion centered at the closest squared number to your desired value:

√x ≈ √xo + (x-xo)/(2√xo)

Use xo = 36 since this the closest number to x=38 with exact square root:

√38 ≈ √36 + (38-36)/(2√36)
√38 ≈ 6 + 2/12
√38 ≈ 6 + 1/6
√38 ≈ 37/6 = 6, 16…

With this method you will always get some excess in the aproximation, this is: √38<37/6, but it’s still a great approximation.

You can expand the series further to reduce the error but, for an approximation there is no need.

[gort]

AWESOME! Thank you. Its just what I have been looking for to do Fire Dept Hydraulic water calculations for Gallons per minute coming out of the fire hose. Without a calculator. Formula is: (GPM=29.7 x d squared x square root of nozzle pressure) (d squared is the nozzle tip size. ie 1 ", 2", etc) (nozzle pressure can be 80psi or 50psi, etc)

[johnmetcalf]

A very good explanation - thank you.
I would add that it would be better to show the next decimal place, too. I was taught that, if the answer was to 2 decimal places, you had to work out the third dp. If that figure is 5 - 9, you have to round up.

[brentsoper]

Thanks for this. My teacher showed me how to use this technique when I was 10 years old to extend me and I had no trouble following it as it was much more interesting than the horrendous long divisions required of us. I am grateful to find the technique so well presented. Yes I had a great teacher!