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. |
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