Math Is Fun Forum

Sven

Member

Registered: 2014-05-02

Posts: 18

Square root algorithm

Why do we take 2 digits initially in the algorithm also why do we take 3 digits for cube roots etc?
God bless

Last edited by Sven (2014-06-06 15:22:01)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,140

Re: Square root algorithm

hi Sven,

Back in the dark ages when I was a lad, we didn't have calculators and computers.  You could either use a slide rule or look things up in a book of tables.  I had a book that had logarithms, sine, cosine and tangent tables, and square roots.  For square roots there were two pages, with different numbers.  You had to know roughly what the root would be to use the correct page.  I'll illustrate with an example.

Suppose you wanted  √2.

On one page it said 1414 and on the other 4472.  It is obvious which answer is correct.  So what is the other answer for?

Well √20 ≈ 4.472

The table only showed roots for decimals in the range  1 to 10.  But you could easily adapt the answers for numbers outside that range.

√200 ≈ 14.14
√2000 ≈ 44.72
√20000 ≈ 141.4

As you can see you had to switch between pages when you went up by a multiple of ten; and put in your own decimal points.

Why was two pages enough?

Well √200 = √2 x √100 = √2 x 10
and  √2000 = √20 x √100 = √20 x 10

The algorithm (which you'll have to teach me as I don't know it) relies on this property of roots.  For cube roots you'd need 3 digits because cube root(1000) = 10.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Square root algorithm

Hi Bob

I'm guessing he's referring to this one

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“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 24,974

Website

Re: Square root algorithm

I agree with #3, but NewtonRaphson is easier to do.

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'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

Sven

Member

Registered: 2014-05-02

Posts: 18

Re: Square root algorithm

I am referring to this algorithm http://www.basic-mathematics.com/square-root-algorithm.html
Also why do we take twice of that number?

Last edited by Sven (2014-06-09 17:45:08)

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 24,974

Website

Re: Square root algorithm

Hi; look at post 3

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'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,140

Re: Square root algorithm

hi Sven,

Thanks for the link.  The example I saw there was for √2685 .  This is how I think it works.

Grouping the digits in pairs means you can use what I said before.  √2600 = √26  x  √100, so that's why you start by looking for √26.  We know that is just over √25 = 5; so start with a 5 in the algorithm.

But remember, we are really saying √2685 = 50 + n, where  is a small number.

So next we try to improve on 50 by trying to find the next digit

................................................................

So, by writing down 10 and looking for z so that (10z) times z comes to just below 185 , we are improving the answer by finding the units digit.

101 x 1 = 101
102 x 2 = 204.  This is too big so z=1 does it.

So we know the root is 51 + n, where n is now less than 1.

So we can put 51 in the answer space, put in the decimal point and subtract 101 from 185 = 84.

So we're using

Now we want a digit to put in the tenths position.  Is it 51.0, 51.1, 51.2, 51.3 etc

So we're looking for the single digit z, so that (102z) times z makes 8400.

By trial we find that z = 8  ie. 1028 x 8 = 8224

So add a .8 to the answer ... 51.8  Subtract 8224 from 8400 = 176 and carry on as before to find the hundredths digit.

Hope that helps

Bob

Last edited by Bob (2014-06-09 20:34:26)

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Sven

Member

Registered: 2014-05-02

Posts: 18

Re: Square root algorithm

Thank you all for the answers & God bless you all
I don't want to sound very pestering but I still have a slight difficulty in understanding this:
I don't understand why do we take twice of the number? & also why do we take 10z x z where we also write z next to 5? I am not able understand that, sorry for the trouble but thanks again & God bless

Bob

Administrator

Registered: 2010-06-20

Posts: 10,140

Re: Square root algorithm

I don't understand why do we take twice of the number?

The middle term of this:

why do we take 10z x z where we also write z next to 5

We are trying to find the right one from these choices:

101 x 1
102 x 2
103 x 3
........

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob