My Calculator Gives Wrong Answers for Large Values of x ...

Date: 01/20/2010 at 19:22:56
From: Noland
Subject: The number e

Today, my teacher taught us about the number e. She said the number e 
is approximately equal to 2.7182818... and can be described by the 
equation

   (1 + 1/n)^n
   
But when I put the equation (1 + 1/x)^x in my calculator and let 
x = 9999999999999, it equals 2.76057... 

And when I put x = 200000000000000, it equals 7.6207...

And when x gets really big, the calculator says the equation is equal 
to 1.

My teacher says e is an irrational number that never ends -- but
why does my calculator say it equals so many different things?

Date: 01/20/2010 at 21:19:16
From: Doctor Vogler
Subject: Re: The number e

Hi Noland,

Thanks for writing to Dr. Math.

The problem is your calculator.

Your calculator probably only has about 16 digits of precision, so 
when you start to deal with numbers with close to 16 significant 
figures, you will start to get answers which are not very accurate.

For example, suppose that your calculator has 20 digits of precision, 
and you set

   x = 10^30
     = 1000000000000000000000000000000
   
Your calculator stores that in scientific notation as 1.0 * 10^30.  
To evaluate (1 + 1/x)^x, first it computes the term 1/x, which it 
stores in scientific notation as 1.0 * 10^-30.  Next, it adds 1 to 
that number, which is

       1.000000000000000000000000000001

but since your calculator only has 20 digits of precision, it rounds 
this off to the number

       1.0000000000000000000

This is pretty close to 1.000000000000000000000000000001.

However, when you raise this number to the power x, you get a very
different answer:

   1^x = 1, but

   1.000000000000000000000000000001^x is very close to e.

If you used a calculator with more precision, you would find that 
when x = 9999999999999,

  (1 + 1/x)^x = 2.7182818284589093212688645...

which is within 0.00000000000014 of e.

When x = 200000000000000,

  (1 + 1/x)^x = 2.7182818284590384396557163...

which is within 0.0000000000000068 of e.

A closer value for e is

  2.7182818284590452353602874713526624977572470936999595749669676....

If you have any questions about this or need more help, please write 
back and show me what you have been able to do, and I will try to 
offer further suggestions.

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

Date: 01/21/2010 at 15:44:28
From: Noland
Subject: Thank you (The number e)

Thanks, I never really realized that until now.