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When you divide 10 by 3, you get 10/3, or 3 + 1/3. Multiply that by 3 and you get 10.
Representing these operations in decimal notation requires infinite precision, which no real calculator has. So the "missing numbers" are due entirely to rounding error. 




(a) This is going to be a long thread.
(b) One answer that I've found fairly convincing to most people: With every extra "9" on the end of the recurring decimal, the value gets closer to 1. 0.9 is 0.1 away from 1. 0.99 is 0.01 away from 1. 0.9999999 is 0.0000001 away from 1. So, with an _infinite_ number of "9"'s, the difference between the decimal and 1 becomes infinitely small  that is, zero. 1  0 = 1. 




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The other name for it is a machine error. A really good calculator or a program like MS Excel handles this math with no problems at all. Jim 






Stewie, I don't know if this will help. We've had threads here in the past discussing the fact that 9.999999... (out to infinity) is exactly equal to 10. Actually, we were saying 0.99999... = 1, but that's pretty much the same thing.
Anyway, your calculator calculates with only a certain number of digits, so there is a fraction of the last digit that can be in error. If you accumulate these errors, they can cause the very last digit to be off by a few. That's why more digits means more accuracy. 




As has been said, it's possible to handle this kind of thing transparently such that rounding never has to occur. One common method is to never convert ratios into floatingpoint values except when you display them on screen: The user sees 0.333333 (out to some predefined precision) but the program knows it's really 1/3 (with two integers representing the number and some clever functions that can do the math correctly).
This only works if the number is rational; that is, if it can be written as the ratio of two finite integers. Two very important numbers that are irrational (that is, not rational) are π and e. Additionally, I think the majority of square roots are irrational. To represent those numbers you have to make do with some form of compromise: Either you find the closest possible ratio within the limits of the size of integers you want to work with, or you fall back to floating point representation and find the closest possible floating point value. Errors creep in regardless, especially if you have to do serious math with irrational numbers. The field of mathematics involved in minimizing those errors is called 'numerical analysis'.
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"Ridicule is the only weapon that can be used against unintelligible propositions. Ideas must be distinct before reason can act upon them." If you don't stop to analyze the snot spray, you are missing that which is best in life.  Miller I'm not sure why this is, but I actually find this idea grosser than cannibalism.  Excalibre, after reading one of my surefire millionseller business plans. 




When people talk about multiplying 0.333333... by 3, it might help to review the rules of multiplication that you learned in grade school.
1. Start by multiplying each of the two rightmost digits of both factors. Well, you can't do that. An infinite number like 0.333333... does not have a rightmost digit. If there were a rightmost digit, it wouldn't be infinite, you dig? Therefore, gradeschool math cannot solve this problem and we turn to calculus. Or to Unca Cecil. 




And, finally, not because I think it will change anything but because it needs to be said: Mathematics isn't amenable to argument. It is amenable to proof, and proof is absolute. You cannot argue against a valid proof, and you cannot argue in favor of an invalid one. 'Acceptance', 'belief', 'persistence', and 'persuasiveness' have nothing to do with anything whatsoever.





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OK, there is a reason. It's because 10 is 3^2 + 1.
In base 2 arithmetic, 2/1 = 1.111111111... In base 5 arithmetic, 5/2 = 2.222222222... In base 17 arithmetic, 17/4 = 4.4444444444... In base 26 arithmetic, 26/5 = 5.5555555555... etc. 




You mathguys. Jeesh.
Take a pizza pie that weighs 10 pounds (big pizza!) Divide it into three equal sections. You have: 1/3 of a ten pound pizza plus 1/3 plus 1/3 = one full pizza (3/3) = 10 pounds. Your calculator can't do a pizza pie demo; it tries to communicate the concept of dividing stuff up using these things call 'numerals' or 'numbers' that we humans use. If you weighed the three sections, the digital scale would have the same problem, but you know you are holding 1/3 of ten pounds when you hold a section. So, nothing is lost. It is just that getting a machine to represent 1/3 is a pain in the arse.  Or think of a length of rope, 10 meters long. Divide it by thirds and you have 3 sections, each 1/3 of 10 meters. Put them together and you still have a total of 10 meters. Mathematically, it jives. Getting a machine to represent this (pencil/paper/human hand) makes it seem tricky. Also still a machine is the brain. 




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With an infinite long decimal fraction like 3.333333333333333..., if it represents length in metres, past about the 20th figure you're into unmeasurable subatomic distances that have no real meaning. And to an engineer, 9.999999 metres = 10 metres, because an engineer building 10 metre object does not worry about micrometres. 




Has no one cited the Master yet: An infinite question: Why doesn't .999~ = 1?, answering the hotlydebated and deservedly dead thread Why doesn't .9999~ = 1?.





Giles, I really was tempted to put in disclaimers about material lost to cutting, but didn't think anyone would actually go there.
We're not talking about literally cutting a friggin pizza and the amount of crust lost to cutting. Come on now. Ooh...wait, I should calculate the kerf of the knife when I use giant pizza pies as an example in a math problem. STOP! 






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It's pretty much an accident of the fact that we were born with 10 fingers. 




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Apologies to the OP, who hasn't returned yet for the answer: we get this question a lot. All of this effort to express the answer isn't because we assume you're stupid, but because (in the past) it has frequently taken supreme, inhuman effort to find a way to express the answer so that it can be understood. I'm not a mod, but I would like to personally vote that we let the discussion die out here unless the OP returns to ask for clarification. 






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The disputes ran like this:
Their arguments may have been irrelevant to the truth of the matter*, but they sure took up space in a thread. *(Mathematics does indeed deal in truth, as opposed to facts. Truth in mathematics flows from an axiomatic system, so what is true in one axiomatic system is untrue in another. The axiomatic system of interest here is the real numbers.) 




I'm sorry but this is ridiculous, Dopers.
If you have a pizza and divide it into thirds and you reattach the thirds, you have the whole pizza back, which helps someone who is confused get around the math. Argue all you want, but for the sake of explaining math to someone who is confused, the example is basic, and a good starting point to demonstrate how numbers are merely trying to represent concepts, yet they are not precise enough at all levels. My example implies (and I really didn't believe I'd be arguing about this).....anyway, it implies that there is nothing lost, because I did not indicate that something less than a third was being rejoined. Yeah, I magically took any lost cheese/crust/grease/misc and reattached them for the example. 




Look, it's an artifact of the decimal representation system and symbolic representation in base 10.
1/3 is a repeating, nonterminating number: 1/3 = 0.333333333333...the 3s never stop...333.... In base ten series representation, 1/3= 3x10^(1) + 3x10^(2) + 3x10^(3) + ... 1/3= .3+ .03 + .003 + .0003 + ... The series is an infinite representation of the real number (1/3). Forget about practical examples about slicing pizzas our cutting fabric...the idea requires indefinitely infinitessimal subdivisions. 






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I love these kinds of threads. I don't understand half of what goes on either, but they are great threads, and I do learn a little each time. 




Philster: You're missing the point. The point is that mathematics isn't a physical process and numbers aren't physical entities. Numbers are concepts completely independent of how they are written. 16 is equal to 0x10 is equal to 020 assuming that C's notation is taken as standard and the first is decimal, the second is hexadecimal, and the third is octal. By the same token, 0.999... is the same number as 1.000..
That interlude about the pizza and ropes and knives was an illustration of the fact that mathematics is perfectly precise in precisely the way the real world is not. (Believe it or not, we've had long discussions on this topic because a poster refused to countenance the notion that one number can be written two ways, and insisted that 1 can only equal 1.) 




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