can you find the square root of n using only +, -, :, x ?

logics

Is it possible to find √n using only the times table and the 4 operations?

Office_Shredder

Using only finitely many such operations, you cannot. For example taking the square root of two, after any such calculations you will be stuck with a rational number when you wanted an irrational one. You can calculate very good approximations though

logics

It is possible:

if n = x² , digits up to 6, : you can find (in most cases) the exact result using no tools,
[likewise, you can find in all cases : [itex]\sqrt[3]{x³} [/itex] (9-digit n), and [itex] \sqrt[5]{x^5}[/itex] (15-digit n)]
if n = x² , digits > 6, : you can still guess three digits and get x with a few operations,

with a pocket calculator you can get any root in a few seconds. Can you believe that?

Dickfore

You can find the square root of a number a as the limit point of the following iterative procedure:
[tex]
x_{n + 1} = \frac{1}{2} \, \left( x_n+ \frac{a}{x_n} \right)
[/tex]
However, as this is a limiting process, you can only get an exact result in the infinite limit [itex]n \rightarrow \infty[/itex]. The convergence is fast though, and you get a good approximate result in just several iterations. Actually, you will exhaust the number of digits on your calculator pretty fast.

logics

That is the Babylonian method, according to wiki. Is there a general algorithm that does not use powers of the same order (k) of n = x^k ?

Cam you find[itex]\sqrt{403225}[/itex] using only logics, knowing that n= x²?

TheDestroyer

Why didn't you guys mention Taylor series? It's very easy to calculate the square-root with a series:

(1+x)^n = 1 + n x + n(n-1) x^2/2! + n(n-1)(n-2) x^3/3! + ...

Substitute n with 1/2 and there you got the square root. Keep adding terms till you get the precision that satisfied you.

That's actually how computers calculate square-roots.

http://en.wikipedia.org/wiki/Taylor_series

Hurkyl

It depends on what you mean by "find".

For example, for many purposes, an algorithm to produce arbitrarily good approximations (e.g. the method in this post) counts as "finding" the number.

Hurkyl

Reference? I find this claim rather unlikely....

Office_Shredder

Are you sure about that? Taylor series are typically not a very fast way of calculating the function they are approximating. And your Taylor series only converges for |x|<1 which is pretty limited (the wikipedia article on 'calculating square roots' mentions how to adapt for this but also says the method is fairly slow compared to the others mentioned)

logics, if you just want the square root of something you know is an integer you can just use a binary search to find the square root without doing anything but multiplication a lot of times. For example to find the square root of 403225:

500² = 250000 so we need a bigger number
750² = 562500 so we need a smaller number. We try one in between these two:

625² = 390625 so we need a number between 625 and 750
687² = 471969

We need a number between 687 and 625
650² = 422500

We need a number between 625 and 650:
637² = 405769
We need a number between 625 and 635
etc, we find the square root is 635.



This technique can also be used to find the square root of any number to whatever precision you want, for example if you want the square root of 403226, we know that it is larger than 635 and smaller than 636. So we try 635.5, and if the square is too large, 635.25, and if the square is too large 635.125, and eventually when the number squared is smaller than 403226 we have a bound for the square root (and at that point we can keep dividing the interval in half in order to get more precision)

Hurkyl

Is it cheating to use the operation "<"?

Dickfore

What operation is that? What is the output of such an operation?

Office_Shredder

It's a binary operation, which takes two inputs a<b and returns 1 if a is smaller than b, and 0 otherwise. Hurkyl was noting my tremendous reliance on it in my example despite it not being one of the permissible operations in the original post. I can't figure out a workaround either

Dickfore

your example is the midpoint method for solving the equation [itex]x^2 - a = 0[/itex]. The Babylonian method is nothing but the Newton-Raphson method for solving the same equation.

logics

When I say can you find x =[itex]\sqrt{635²}[/itex] using only logics I mean using only your mind, knowing by heart the times table, using no calculator, no pencil and paper.

Mentallic

I guess...

[itex]635^2=403,225[/itex] just for reference.

So we're trying to find the answer of [itex]\sqrt{403,225}[/itex]. We know that since [itex]100^2<403,225<1000^2[/itex] then the answer will be a 3-digit number that we will denote as ABC.

To find the hundreds value, A, just realize that the first 2 digits of 403,225 are 40, and that is between 6² and 7² so A=6.

For C, the last 2 digits are 25, so I'd guess that it would be 5.

To find B - arguably the trickiest of all - I guess I'd flex my brain muscles by not pulling out a pen and paper and expanding (600+10B+5)² in my mind, and scrapping all the values except the relevant ones that would be the biggest influence on the ten-thousand's value in the answer.

So [tex](600+10B+5)^2=(600+(10B+5))^2=600^2+1,200(10B+5)+(10B+5)^2[/tex]

Clearly the influence would be that middle term [itex]1,200(10B+5)[/itex]

So now let's do some simplifications. [itex]1,200(10B+5)>12,000B[/itex] and [itex]403,225-360,000 \approx 40,000[/itex] and then [tex]3<\frac{40,000}{12,000}<4[/tex] and so I would make the logical guess that B=3.

Thus, ABC = 635

logics

Well done, Mentallic, if you can manage all that in your mind!
But you must find something simpler if you want to solve the challenge in the twin-thread about the cube root... (try to find x = [itex]\sqrt[3]{723³}[/itex]).
If you do not wish to move on to that thread, here is a slightly tougher [but still easy] challenge: find x = [itex]\sqrt{277729}[/itex]

Mentallic

With a bit of practice, I could answer the square root problems more efficiently, as long as they're either 4 or 6 digits, which is pretty narrowed down and a useless skill to have if you ask me.

And obviously I'd have little luck extending it to the cube roots.