Answer to Pi Day question
Published Thursday, March 16th, 2006 at 10:15 pm
Ok, now for the answer to the Pi Day question. The answer is:
Neither. Both values are the same.
How? Well I gave the hint that 3.14159… is exactly equal to π. The dot-dot-dot (…) means a continuation of the numbers in the sequence of Pi. This is not that hard for people to understand or believe. If one were to write out the digits of Pi you would be writing a number approaching Pi. The more digits you write the closer you are to Pi but never exactly there. In order for the decimal number to be equal to Pi you need to write all infinite digits of Pi. The dot-dot-dot is just a short hand for all the infinite digits of Pi.
Now what about 0.9999…? This is much harder for people to grasp even though it is based on the same principle (it was difficult for me at first as well). If you were to write 0.9999 followed by continuing digits of 9 you would be writing a number approaching 1 but never really reaching it. 0.9999… however is an infinite number of nines. Can one write infinite nines? No, but what what would infinite nines mean? Well, infinite nines is exactly 1 just like 3.14159… is eactly Pi, not a number approaching it.
So if 0.9999… is exactly equal to 1 then 0.9999… / 3.14159… = 1/π = π-1. Both values are equal.
Ok, don’t belive that 0.9999… is equal to one? Try this:
Let x = 0.9999... Then 10x = 9.9999... And 10x - x = 9.9999... - 0.9999... Reduces to 9x = 9 With the solution x = 1
Therefore 0.9999… is exactly equal to 1. Don’t belive me? Ask Dr. Math.