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Using Binomial Expansion to Evaluate [2 + sqrt(3)]^50

Date: 11/29/2006 at 10:45:55
From: Matt
Subject: Why is (2 + root3)^50 so close to an integer?

When you work out (2 + root3)^50, why is it so close to an integer?
I have worked out (2 + root3)^50 on a computer and got the answer 
39571031999226139563162735373.999999999999999999999999999999999974728
...

I have tried expanding it using all the expansions I know (starting 
with the binomial expansion) but have had no luck seeing why it comes
out so close to an integer.  Do you have any ideas?



Date: 11/29/2006 at 11:31:07
From: Doctor Douglas
Subject: Re: Why is (2 + root3)^50 so close to an integer?

Hi Matt.

Are you sure that you're not simply running into roundoff error from 
the finite precision of the computer arithmetic?  Calculations such as 
these on a computer or calculator require considerable care because 
they usually don't keep enough significant digits in memory to be 
resistant against roundoff errors.  This might not matter if you're 
only looking for a result to a few significant figures, but if the 
question is evaluating how close the original number is to an integer, 
one can encounter problems.

Your idea of writing out the original number X using the binomial 
expansion is an excellent one:

   X = [2 + sqrt(3)]^50 =  C(50,0) * 2^50 * sqrt(3)^0
                            + C(51,1) * 2^49 * sqrt(3)^1
                            + C(52,2) * 2^48 * sqrt(3)^2 
                               + ... +
                            + C(50,50) * 2^0 * sqrt(3)^50

Notice how many of these terms (in particular those with 
sqrt(3)^[something even] are integers, and do not affect the 
fractional part of X.  So we can safely ignore them, as far as the 
integer part of X is concerned.  Then, notice that everything else is 
of the form

   D[k] * sqrt(3)^k,                  D[k] is an integer, k is odd,

where k is odd, and if we subsume all of the even powers of 
sqrt(3), we can condense this down to

   D[k] * (n) * sqrt(3)^1             k=2n+1, n=0,...,48.

Thus we see that the only terms that affect the fractional part of 
X are of the form

   E[k] * sqrt(3),                    E[k]=D[k]*n is an integer,

and the only remaining step is to evaluate the sum 

   E = E[1] + E[3] + E[5] + ... + E[49].

This should be a lot easier for your computer, since this is 
guaranteed to be an integer, and you should be protected somewhat from 
roundoff errors.  But note that the numbers in the middle of this sum, 
say E[27], will have 27 or 28 decimal digits or so, and you will still 
have to retain at least this many digits in these integer 
computations.  

The leading digits of E give you information about how close the 
original number X is to an integer.  If the leading digits are close 
to 1/sqrt(3) = .577350269..., then when finally multiplied by 
sqrt(3)^1, the product is near an integer.  If E is zero, then X is an 
integer (ask yourself, can that happen?).

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/ 


Date: 10/22/2011 at 09:46:34
From: Mike
Subject: re:69763.html :  Why is (2 + root3)^50 close to an integer  

I read the cool question from Matt. I wanted a theoretical answer, and
figured something out. Please check it out: 

Let 

   x = 2 + sqrt(3).

Clearly,

   1/x = 2 - sqrt(3).
   
By binomial expansion, we get that ...

   x^n = a + b sqrt(3)

... and ...

   (1/x)^n = a - b sqrt(3),

... where a, b are integers. Therefore, this is also an integer:

   x^n + (1/x)^n

Since lim(1/x)^n = 0, the fractional part of x^n tends to 1.

That's why (2 + root3)^50 is so close to an integer. No need to use
computers!

The most difficult is to conjecture that x^n + (1/x)^n is an integer.

Feel free to use my solution.


Date: 10/24/2011 at 21:08:50
From: Doctor Douglas
Subject: re:69763.html :  Why is (2 + root3)^50 close to an integer  

Hi Mike,

I think that this is a very nice approach. Can you generalize it to other
constants, such as 2 + sqrt(5)?

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 10/25/2011 at 13:03:06
From: Mike
Subject: Thank you (re:69763.html :  Why is (2 + root3)^50 close to an integer)

Dear Doctor Douglas, 

It is a great idea to consider (2 + sqrt(5))^n and generalize.

This, too, is "close to an integer" since 
 
    i)  (2 + sqrt(5))^n + (2 - sqrt(5))^n  is an integer  
 
   ii)  lim (2 - sqrt(5))^n = 0.

The same holds for ...

   a + sqrt(b),
   
... where a, b are integers and

   |a - sqrt(b)| < 1.
  
Mike

Associated Topics:
College Number Theory
High School Number Theory

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