
The Lost Art of Manually Calculating Square Roots - sohkamyung
https://medium.com/i-math/how-to-find-square-roots-by-hand-f3f7cadf94bb
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schoen
This reminds me of the story of Feynman beating the abacist

[http://www.ee.ryerson.ca/~elf/abacus/feynman.html](http://www.ee.ryerson.ca/~elf/abacus/feynman.html)

However, the correct Portuguese for "cube roots" is "raízes cúbicas", not
"raios cubicos"(—edited with thanks to JetSpiegel for the correction!). I've
thought that Feynman may have been better at speaking Portuguese than at
spelling it, or that most of _Surely You 're Joking_ may have been transcribed
from tapes by Ralph Leighton, who probably didn't speak Portuguese.)

Edit: Also, I was partly inspired to learn Portuguese by Richard Feynman. I
didn't imagine that that would lead to complaining about his spelling. :-)

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JetSpiegel
It's actually "raízes cúbicas", "raiz" is female. I'm impressed to see
"multiplicação" correctly written in an English language page, usually
everyone ignores tildes and ç, but they are important.

~~~
schoen
Oh gosh, thanks for the correction. I searched for what I thought it was and
found lots of native speakers using the form that matched my intuition—but
that didn't make it correct. :-)

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SeanLuke
This seems like a really hard way to calculate square roots. Why not just use
bracketing?

    
    
        1. Pick a number HI whose square is obviously higher than N.
        2. Pick a number LO whose square is obviously lower than N.
        3. Choose a number MED which is roughly (HI+LO)/2.
        4. If MED * MED > N, replace HI with MED, else replace LO with MED.
        5. Go to 3.
        6. MED converges to your answer.
    

Doesn't converge quite as fast maybe, but a million times easier to grok. I
got to where the author got, ~87.8, in 8 iterations even with terrible initial
HI and LO guesses (100 and 50 respectively).

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dozzie
> Doesn't converge quite as fast maybe, but a million times easier to grok.

What didn't you understand about _manually_ calculating square roots? Because
your method requires tons of troublesome multiplication of numbers that get
longer with each step.

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BoiledCabbage
> What didn't you understand about manually calculating square roots?

Why would you be both insulting and wrong in the same post?

Or did you think that multiplication somehow isn't _manual_?

Next time work on phrasing your comment a bit better. It makes the world a
better place.

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mpweiher
What's lost about it? When I first came to the US aged 12 I was confronted
with an entrance exam for the school we had chosen.

The math part had this weird American division symbol I had never seen before,
and the closest thing I _had_ seen was square root. Which I found a bit
ambitious, but I went ahead and did the square roots.

They accepted me despite the fact that I couldn't do division, but put me in
the "remedial math" class. Got out of it when I did the in-class "calculator
exercises" (weird concept, that) in my head, faster than anyone with the
calculator.

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sohkamyung
There are many methods for calculating square roots [1]. The article explain
the Digit-by-Digit with Decimal (base 10) method, I believe, and explains it
quite well.

[1] [
[https://en.wikipedia.org/wiki/Methods_of_computing_square_ro...](https://en.wikipedia.org/wiki/Methods_of_computing_square_roots)
]

~~~
ballenf
The babylonian method is, imo, easier to remember and perform than the "lost
art" presented here:

Start with a ball park estimate of the root of x, then take an average of
(estimate + x / the estimate). That's your new estimate. Rinse and repeat
until convergence to an acceptable degree.

The biggest drawback if doing this by hand is the division problem can be a
pain.

I see the attraction and "purity" of the proposed lost art.

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nicolapede
Fascinating, I didn't know the babylonian method -- it sounds equivalent to
Newton's.

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mark-r
I had always thought of this as Newton's Method too, but that's because I was
simply misremembering the fullness of Newton's Method [1]. I'm grateful for
the education.

[https://en.wikipedia.org/wiki/Newton%27s_method](https://en.wikipedia.org/wiki/Newton%27s_method)

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mml
fwiw, I learned to compute square roots in school in the 80s (I think it was
the estimation method, which I was extremely bad at).

Calculators of any kind were forbidden at school, which sucked, if you had one
of those cool Casio watch calculators. Shorts were also forbidden, because
suffering is good for you, apparently.

I've never had to do it since, of course. Though I do now wear shorts on
occasion.

~~~
Osmium
> fwiw, I learned to compute square roots in school in the 80s

FWIW, I did too in the 90s/00s, though I confess I don't remember the method
now. I think it was Newton-Rhaphson.

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scottLobster
Interesting. I was always taught to simply reduce the square roots as much as
possible (ie sqrt(8) = 2*sqrt(2)) and leave the square roots, as it's often
more precise than an arbitrary number of decimals.

~~~
BrandoElFollito
Thanks for that, it's an interesting approach, never heard of it before. I
will have a closer look at the precision.

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nils-m-holm
Nice! But even Newton's method works fine on a sheet of paper!

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Turing_Machine
Even rough interpolation converges pretty fast for plain vanilla square roots,
especially if you don't need an answer accurate to more than a few decimals.
Newton-Raphson or more sophisticated methods can be a big win for more
complicated functions, of course.

Estimate sqrt(17):

Clearly it has to be between 4 and 5.

Try 4.5. Result: 20.25 Try 4.2. Result: 17.64 Try 4.1. Result: 16.81 Try 4.15.
Result: 17.01. Probably good enough for most practical purposes.

~~~
Retric
An engineer friend memorized the first 4 digits of all square roots from 1 to
100 which you can quickly turn into a great approximation. AKA sqrt(1730) is
between 4.123 * 10 and 4.242 * 10. Even better 30 is low so it's between 41.23
and (4.242+4.123) / 2 * 10 = 41.825.

In this case the value is sanity checking a calculator / program.

~~~
tpeo
Why would he do that, though? I do think it's neat, but I think most people
would just memorize an algorithm instead.

Were any memory tricks involved (e.g. pegs, palaces) or did he just go through
a table of square roots every morning?

~~~
Retric
I assume because he regularly worked with formulas involving squares and
square roots. This is a guy who did long division in his head on long car
rides to stay awake. He also memorized the square and cube of every number
from 1 to 100.

His justification for starting was trying to keep track of the order of
magnitude of everything he calculated to catch fat finger mistakes. And a
sanity check. However, you need a few digits on the front of a number before
you can really safely keep track of exponents that way.

~~~
jacobolus
Memorizing a log table is a heck of a lot better bang for the buck than
memorizing square roots, cube roots, etc.

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pasta
It's strange that it never occured to me that the square root also can be used
to calculate the sides of a square.

That's because in Dutch we just call it 'root' (wortel).

I really wish teachers used more graphic forms in math because it helps to
understand what kind of problems it can solve.

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BrandoElFollito
Isn't _wortel_ a short form ? (I do not speak Dutch).

I am asking because in France we have the official term _racine carrée_
(square root), abbreviated to _racine_ (root) when the "square" is implied.

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OJFord
In English, too.

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kronos29296
Been a long time since I saw this. Reminds of School math. Nowadays sqrt is
just a calculator button press away and so I have all but forgotten how to do
this. This refreshed my memory.

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masswerk
Those who interested in how to go from there to computer algorithm may have a
look at [http://www.masswerk.at/spacewar/inside/insidespacewar-
pt6-gr...](http://www.masswerk.at/spacewar/inside/insidespacewar-
pt6-gravity.html#square_roots)

This is part of an excursus trying to explain, how basic math was implemented
on the DEC PDP-1, namely in Spacewar!, the first digital video game (1961/62).
The excursus starts at [http://www.masswerk.at/spacewar/inside/insidespacewar-
pt6-gr...](http://www.masswerk.at/spacewar/inside/insidespacewar-
pt6-gravity.html#excursus) (there's a tab on the bottom right for an
instruction list of the PDP-1, mind that the PDP-1 uses 1's-complement for
negative numbers).

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latkin
I must be missing something silly. What is the purpose of the original
rectangle?

This describes how to find the side length of the square, who's area is
smaller than the rectangle​.

~~~
Mithorium
I was wondering that too, the best I could come up with is that the unused
part of the rectangle represents the remainder we haven't "pulled down" yet,
so in the first iteration, 0.17

I don't think its important to the calculation though

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Fire-Dragon-DoL
I actually studied how to resolve square roots by hand in school. It looked
like solving a division with some different rules. I honestly don't remember
the process but can definitely get my math homeworks and find them (yes I
still have them!). I'm 28 years old, so it's not that far come on :P

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lukaa
In my college professor offered passing grade immediately to someone who could
manually calculate root.Nobody knew,although everyone learned it in high
school.It's amazing how much people today depend on computer.In some sense all
that automation makes us stupid.

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cblock811
Just because we don't do something anymore that doesn't make it a "lost art".

~~~
simplicio
Definition of lost art

    
    
        :  something usually requiring some skill that not many people do any more 
    

ex: Writing letters has become something of a lost art.

"[https://www.merriam-webster.com/dictionary/lost%20art"](https://www.merriam-
webster.com/dictionary/lost%20art")

~~~
cblock811
I cant help but chuckle at the exact definition being thrown at me. I guess
that by definition this is a lost art. I still think it's a bit over the top
though.

~~~
simplicio
I hear what your saying, taken individually you'd think a "lost art" would
literally be an art no one remembers how to do. But like a lot of compound
words and phrases in English, its actual meaning as used is different. In my
experience, its pretty much always used in the sense of the article.

(SNL had a whole bit on this back in the 90's: "a chickpea is neither a chick
nor a pea, discuss")

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chirau
How is this a 'lost art'? I had to do this in college. Didn't we all?

~~~
steaknsteak
I graduated college a year ago and I'm not sure if I've ever in my life been
asked to manually compute a square root (except for a perfect square integer).

~~~
yazan94
Same here. And I distinctly remember asking a math teacher in middle school if
a number was a perfect square and we manually ended up checking and
incrementing numbers until we decided that the number was not. It makes me
think that this was possibly not such a common task to assign in school?

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reacweb
My mother taught it me when I was 12. Normally it is presented like a
division. At 17, I have coded it in turbo pascal with arrays of digits. If I
remember well, I had found a way to apply the same method for cubic roots.

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quantum_state
just to add to the conversation ... once one learns algebra and expresses a
number as polynomial of 10, one can just derive the usual manual algorithm for
any integer root ... though it gets tedious very quickly with hisgher roots
... in addition to methods using Taylor expansion and such ... maybe this is a
reason why there is no need to teach it ... apart from the practical reason of
a calculator can do it much quicker ...

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Terr_
Huh, I thought this was going to be about Taylor Series.

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minademian
the most un-Medium article I've read on Medium in a long time. great
execution!

