Bruce Douglas wrote: > so... > > you're saying that 7.99999 (repeating) is equal to 8.0.... > > i say prove it.. as i recall the numbers might be for all practical > purposes the same, they are in fact vastly different... > > so, prove your assertion...
Claim: 7.99999999... (repeating) is EXACTLY equal to 8. Period. Remember converting repeating decimals to fractions back in 5th grade? Convert 0.333333... to a fraction. Let x = 0.333333... 10x = 3.33333333... 10x = 3.3333333... - x = -0.3333333... ============================= 9x = 3 x = 3/9 x = 1/3 You did that, right? Okay, now try it with 7.99999999... Let x = 7.99999999... 10x = 79.999999999... 10x = 79.9999999999... - x = -7.99999999999... ================================= 9x = 72 x = 72/9 x = 8 QED You can do this with *ANY* repeating decimal to get a fractional respresentation EXACTLY equal to that repeating decimal. x = 0.66666666... 10x = 6.66666666 - x = -.66666666 =================== 9x = 6 x = 6/9 x = 2/3 All rules of mathematics prove these are EXACTLY equal. Not "close." EQUAL. TRY IT! 0.222222222 0.444444444 0.1212121212121212 (You have to use 100x on this one) You get the fraction which is EQUAL to the repeating decimal. Now go for: 0.9999 1.99999 2.999999 0.5999999 0.1234999999 If it ends in ???x999999.... you *WILL* find out that there is a corresponding ???(x-1)0000000000... to match it. I *KNOW* it seems counter-intuitive, but there it is. PS I've often wondered why Cantor's Diagonal Theorom never bothered to take this into account -- Sure, you could always just pick a non-9 and non-0 digit instead of just the 'next' digit, but... A contrived counter-example to Cantor's Diagonal Theorom is trivial to create and "dis-prove" it as it is written in most textbooks. Oh well. Somebody Else's Problem. This is sooooooo off-topic! But it does sort of point out that even in the mathematics you've used every single day of your life, things are not always what they seem. It's just even more not what they seem inside a computer using PHP. :-) PPS Don't take my word for it. There are only a few zillion mathematicians and/or 5th-graders who can provide an abundance of examples. :-) :-) :-) -- Like Music? http://l-i-e.com/artists.htm -- PHP General Mailing List (http://www.php.net/) To unsubscribe, visit: http://www.php.net/unsub.php
