RE: 34.123*89 = 3036.95 (3036.947)

*To*: mathgroup at smc.vnet.net*Subject*: [mg47919] RE: [mg47905] 34.123*89 = 3036.95 (3036.947)*From*: "DrBob" <drbob at bigfoot.com>*Date*: Fri, 30 Apr 2004 19:26:57 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

No, it's actually the following binary representation: x = 34.123*89 RealDigits[x, 2] Precision[x] 3036.9469999999997 {{1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1}, 12} MachinePrecision How that looks in decimal depends on how many digits you display, how the conversion to decimal is done, whether it's rounded up or down, et cetera. But the product IS that string of binary bits, in this case 0.10111(and so forth) times 2^12. Most rational numbers cannot be represented exactly in binary. They can be ONLY if the denominator of the rational fraction, in reduced form, is a power of 2. But you can get as close as you want to, by specifying inputs to more decimal places. You can control display with NumberForm. For instance, x = 34.12300000000000000000000000000*89 RealDigits[x, 2] Precision[x] NumberForm[x, 20] 3036.947`30.533047206146456 {{1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0}, 12} 30.533047206146456 3036.9470000000000000 I specified more than $MachinePrecision digits in the input; otherwise it defaults to a machine precision number, like your original input. But this is a tempest in a teacup. Unless you happen to know the first multiplier is EXACTLY 34123/1000 because it's derived from an exact formula, I doubt that you really know it even to machine precision, and that's what Mathematica is assuming when you input 34.123. If you know the number exactly, you can enter it as a rational number, and then there's no approximation error at all. If you don't know the number that exactly, just relax. The visible "error" you think you're seeing is smaller than your actual uncertainty. In this case, the original answer in your post is "off" by less than 10^-10. The relative error is less than 10^-14. Does it really matter? error = 3036.946999999970000000 - 34.123*89 error/3036.946999999970000000 -2.955857780762017*^-11 -9.732990996425178*^-15 DrBob www.eclecticdreams.net -----Original Message----- From: Thomas Schulz [mailto:thomas-k-schulz at t-online.de] To: mathgroup at smc.vnet.net Subject: [mg47919] [mg47905] 34.123*89 = 3036.95 (3036.947) I am a NewBe - and i dont understand the Result. Mathematica says that 34.123*89 = 3036.94699999997 but its 3036.947... How can i change this? i use a IBook (G4) with Mac OS X. THX Thomas