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#1 2006-12-23 04:29:41

Stanley_Marsh
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Registered: 2006-12-13
Posts: 345

A spooky discovery

I was just wandering around equation and stuff,

After some calculation I get this

√(b^2 - 4c) = |(b^2-2c)/b|

It's so wierd , by it , I can calcultion a square root approximately.
As the number becomes greater , the result become more accurate

when I assume b^2-4c = 99999,  I calculate the sqr99999 =316.22857
which by calculator  it's 316.2261

when I assume b^2-4c=999999 , I get 999.9995 which is the same as the result come out of the calculator

Wierd man,.~

Last edited by Stanley_Marsh (2006-12-23 04:30:15)


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#2 2006-12-23 07:29:55

Ricky
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Registered: 2005-12-04
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Re: A spooky discovery

I guess the real question is *how* did you get that?  Can you post your steps?


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#3 2006-12-23 10:28:49

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

Re: A spooky discovery

Just "powering" the equation:


...

for b,c =/=0:

So if c<<b, then the approximation will be better.
Here's a plot:

Last edited by krassi_holmz (2006-12-23 10:31:13)


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#4 2006-12-23 11:22:38

Toast
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Registered: 2006-10-08
Posts: 1,321

Re: A spooky discovery

Wow, awesome find! Can... uh... you provide a fully worked through example please?

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#5 2006-12-23 12:24:45

mathsyperson
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Registered: 2005-06-22
Posts: 4,900

Re: A spooky discovery

I think I get how this works.

Let's try to use it to find the square root of 1000. The square root of 1000 is an irrational number, and hard to calculate without some kind of computer or calculator. However, 1024 is much easier, because it is just 32².

We can say that √1000 = √(1024 - 24) = √(32² - 4x6)

Using Stanley's equation:
√(b^2 - 4c) = |(b^2-2c)/b|

We can see that in this case b = 32, and c = 6.
Substituting those values in gives an approximate square root of |(32² - 2x6)/32| = (1024 - 12)/32 = 31.625.

The actual square root of 1000 is 31.622..., so the approximation is accurate to 2 decimal places and much easier to work out on paper.

As Krassi says, the answer gets more accurate as b gets larger in relation to c, and so if you wanted to make your approximation better then you could instead work out the square root of 100000 and divide that answer by 10 afterwards.

Or, in general, find the square root of 1000k² and divide the answer by k afterwards. Increasing k should make the approximation more accurate but also make the calculation harder, because you'd be dealing with bigger numbers.

Incidentally, is there any way of tweaking the formula so that the error term, 4c²/b², gets smaller without making the calculation much harder? I have a feeling that this formula has potential to be developed.


Why did the vector cross the road?
It wanted to be normal.

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#6 2006-12-23 12:58:18

Stanley_Marsh
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Registered: 2006-12-13
Posts: 345

Re: A spooky discovery

OMg , Thank you for solving my mystery! I get this from a  x^2+bx+c=0 equation , the determinant of root  b^2-4c . I think I see now .hehe


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#7 2006-12-23 13:02:49

Stanley_Marsh
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Registered: 2006-12-13
Posts: 345

Re: A spooky discovery

And to how I get that , I get to find my notebook , it's been a long time .


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#8 2006-12-23 13:43:23

Toast
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Registered: 2006-10-08
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Re: A spooky discovery

Isn't the determinant b²-4ac?

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#9 2006-12-23 20:22:32

Stanley_Marsh
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Registered: 2006-12-13
Posts: 345

Re: A spooky discovery

Yep , but I assume a to be 1


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#10 2006-12-24 03:49:48

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

Re: A spooky discovery

About Mathsy post - good point.
I suppose the best will be to use the first greater or equal to the number exact square.
And the approximation is really good.
As you can see form the image :

,
which is true for all x,y.
I'll make some tests to see the efficiency of the algoritm.

Last edited by krassi_holmz (2006-12-24 06:05:21)


IPBLE:  Increasing Performance By Lowering Expectations.

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#11 2006-12-24 04:22:45

luca-deltodesco
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Registered: 2006-05-05
Posts: 1,470

Re: A spooky discovery


The Beginning Of All Things To End.
The End Of All Things To Come.

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#12 2006-12-24 05:34:14

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

Re: A spooky discovery

Yes, the same.
Now, i'll uncover my results.
Let



If we use the first greater or equal to the number square indeed, then

and

So: (removing this absolute value for x>y and x>1)

Let us define the error function:

Here are 2 plots of it (fig 1 and 2).
As we see the error is 0 when n is a perfect sqare, and it grows bigger when n=k^2-q for small q.
So in the worst case

But

So we can assume
and we'll get an upper bound for the error function:

And, after some simplification:

And this estimate is very good!
(figures 3 and 4 showing a plot with the error function and the bound).

Last edited by krassi_holmz (2006-12-24 06:01:49)


IPBLE:  Increasing Performance By Lowering Expectations.

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#13 2006-12-24 16:45:16

George,Y
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Registered: 2006-03-12
Posts: 1,379

Re: A spooky discovery

b[sup]2[/sup]-4c≈b[sup]2[/sup]-4c+4(c/b)², when c/b is very small, or b is much larger than c.

That's how he got it.


X'(y-Xβ)=0

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#14 2006-12-24 16:46:35

George,Y
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Registered: 2006-03-12
Posts: 1,379

Re: A spooky discovery

b[sup]2[/sup]-4c+4(c/b)²= (b-2c/b)[sup]2[/sup]


X'(y-Xβ)=0

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#15 2006-12-24 16:48:42

George,Y
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Registered: 2006-03-12
Posts: 1,379

Re: A spooky discovery

But if b is much larger than c, simply b is a quicker result. Nevertheless, b minus 2c/b is more accurate.


X'(y-Xβ)=0

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#16 2006-12-24 23:38:12

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

Re: A spooky discovery

This remids me about something...
Since:


then, for small Δx:


When f(x) = sqrt(x):

So:

I think this is a well-known identity.
We can obtain the spooky discovery from it.

Last edited by krassi_holmz (2006-12-25 21:47:21)


IPBLE:  Increasing Performance By Lowering Expectations.

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#17 2006-12-25 17:06:28

George,Y
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Registered: 2006-03-12
Posts: 1,379

Re: A spooky discovery

f(x+Δx)≈f(x)+f'(x)Δx


X'(y-Xβ)=0

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#18 2006-12-25 21:49:08

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

Re: A spooky discovery

yeap.


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