SOLUTION: HI tutors! I have this problem in my college math book: The decimal expansion of the irrational number pi symbol begins 3.141592653589793 . . . . Use your calculator to deter

Algebra ->  Decimal-numbers -> SOLUTION: HI tutors! I have this problem in my college math book: The decimal expansion of the irrational number pi symbol begins 3.141592653589793 . . . . Use your calculator to deter      Log On


   

Question 645485: HI tutors!
I have this problem in my college math book:
The decimal expansion of the irrational number pi symbol begins 3.141592653589793 . . . . Use your calculator to determine which of the following rational numbers is the best approximation for the irrational number pi symbol:
22/7 , 355/113 , 103,993/33,102 , 2,508,429,787/798,458,000.
Your calculator may tell you that some of these numbers are equal to pi symbol, but that just indicates that the number agrees with pi symbol for as many decimal places as your calculator can handle (usually 10-14). No rational number is exactly equal to pi symbol.
Here is what I get from left to right for these fractions:
3.142857143 , 3.14159292 , 3.141592653 , and 3.141592654.
The book's answer is the last fraction.
Why is the last fraction the right answer?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
The fractions below

22/7, 355/113, 103993/33102, 2508429787/798458000

convert to the following numbers in decimal form

3.14285714285714, 3.14159292035398, 3.1415926530119, 3.141592653589794

----------------------------------------------------------------------
Now compare this with the given value of pi, which is 3.141592653589793...


First Rational Approximation (22/7)

3.141592653589793
3.14285714285714

Note: first number is the given value of pi and second number is the rational approximation

Notice how 22/7 only matches up to 3.14 and then it's off for the rest of the expansion.
------------------------------------------------------------------

Second Rational Approximation (103993/33102)

3.141592653589793
3.14159292035398

Note: first number is the given value of pi and second number is the rational approximation

We got a better approximation this time since it matches 3.141592

So far, the second approximation is better
------------------------------------------------------------------

Third Rational Approximation (355/113)

3.141592653589793
3.1415926530119

Note: first number is the given value of pi and second number is the rational approximation

We got a better approximation this time since it matches 3.141592653

So far, the third approximation is best
------------------------------------------------------------------

Fourth Rational Approximation (2508429787/798458000)

3.141592653589793
3.141592653589794
Note: first number is the given value of pi and second number is the rational approximation

We get the most accurate match this time since it matches 3.14159265358979

The only difference between this rational approximation and the given decimal expansion is that the '3' at the end doesn't match with the 4.
==========================================================================

Answer:

So the fraction 2508429787/798458000 gives the best approximation of pi


--------------------------------------------------------------------------------------------------------------
If you need more help, email me at jim_thompson5910@hotmail.com

Also, please consider visiting my website: http://www.freewebs.com/jimthompson5910/home.html and making a donation. Thank you

Jim
--------------------------------------------------------------------------------------------------------------