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#1
Old 01-15-2006, 08:24 PM
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What is the deal with dividing 10 by 3?

why is it, if you divide 10 by 3, you gett 3.333333 or something, but then if you multply that times 3, you only get 9.9999999. Where did the missing numbers go? Do mathematicians have a name for this?
#2
Old 01-15-2006, 08:28 PM
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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.
#3
Old 01-15-2006, 08:36 PM
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(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.
#4
Old 01-15-2006, 08:55 PM
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Quote:
Originally Posted by StewieGriffin420
why is it, if you divide 10 by 3, you gett 3.333333 or something, but then if you multply that times 3, you only get 9.9999999. Where did the missing numbers go? Do mathematicians have a name for this?
It is call calculator math. Your calculator only display 8 digits. The real answer to 10/3 is 3.3 followed by and infinate number of 3's. When you multiply this by 3 you get 10 again.

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
#5
Old 01-15-2006, 11:15 PM
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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.
#6
Old 01-15-2006, 11:39 PM
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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 floating-point 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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#7
Old 01-15-2006, 11:50 PM
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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 right-most digits of both factors.

Well, you can't do that. An infinite number like 0.333333... does not have a right-most digit. If there were a right-most digit, it wouldn't be infinite, you dig?

Therefore, grade-school math cannot solve this problem and we turn to calculus. Or to Unca Cecil.
#8
Old 01-16-2006, 12:00 AM
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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.
#9
Old 01-16-2006, 02:47 AM
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Quote:
Originally Posted by Derleth
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.
Trying to head off the inevitable argument?
#10
Old 01-16-2006, 07:17 AM
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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.
#11
Old 01-16-2006, 08:33 AM
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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.
#12
Old 01-16-2006, 08:48 AM
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Quote:
Originally Posted by Philster
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.
Reality and mathematics only approximate each other. In the case of your 10 metre rope, if you join it together, you'll need to use knots of splices. Either way it will be less that 10 metres long. Another example: take a piece of timber 10 metres long, cut it into three, then rejoin it -- it will be less than 10 metres long by the thickness of two saw cuts.

With an infinite long decimal fraction like 3.333333333333333..., if it represents length in metres, past about the 20th figure you're into unmeasurable sub-atomic 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.
#13
Old 01-16-2006, 11:31 AM
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Has no one cited the Master yet: An infinite question: Why doesn't .999~ = 1?, answering the hotly-debated and deservedly dead thread Why doesn't .9999~ = 1?.
#14
Old 01-16-2006, 11:33 AM
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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!
#15
Old 01-16-2006, 02:19 PM
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Quote:
Originally Posted by Philster
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.
If the tolerance for error is that low, then yes, you should. That's the difference between a math problem and a real-world calculation. Enough people don't get that that it's worth mentioning.
#16
Old 01-16-2006, 04:32 PM
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Quote:
Originally Posted by Philster
....Also still a machine is the brain.
Thank you, Yoda


I really want a pizza now.
#17
Old 01-16-2006, 06:04 PM
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Quote:
Originally Posted by Giles
. . .In base 2 arithmetic. . .
This is the crux of the issue. It's not about math as some abstract thing vs. reality, it's not about reality vs. machines, it's about the base 10 system. The number 1/3 can be represented exactly as 0.1 in base 3. The number 1/7 can be represented exactly as 0.1 in base 7.

It's pretty much an accident of the fact that we were born with 10 fingers.
#18
Old 01-16-2006, 06:08 PM
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Quote:
Originally Posted by Derleth
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.
I reject your reality and substitute it with my own!
#19
Old 01-16-2006, 07:19 PM
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Quote:
Originally Posted by Tevildo
(a) This is going to be a long thread.
You know it's going to be a long thread when the Straight Dopers who know the answer are making the thread longer by arguing with each other about the best way to answer it.

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.
#20
Old 01-16-2006, 07:32 PM
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Quote:
Originally Posted by Fish
You know it's going to be a long thread when the Straight Dopers who know the answer are making the thread longer by arguing with each other about the best way to answer it.

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.
Well I don't understand the answers, however not because the answers given are no good; I am just dumb as a rock when it comes to math.
#21
Old 01-16-2006, 07:49 PM
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If you can formulate specific questions about what you don't understand, we can help.
#22
Old 01-16-2006, 07:52 PM
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Quote:
Originally Posted by Excalibre
Trying to head off the inevitable argument?
Heh. Let's just say I'm a veteran of these things.

Quote:
Originally Posted by Fish
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.
Not only that, but we've come across people who simply refused to understand that how you write a number doesn't have any impact on what number it is.

The disputes ran like this:
  • 1/3 expressed in decimal notation is 0.333..., where the three dots at the end mean 'extended to infinity'.
  • 1/3+1/3+1/3 == 0.333... + 0.333... + 0.333... == 0.999... This was accepted just fine.
  • The problems came when we attempted to take them the final step by stating 0.999... == 1.000... They couldn't, or wouldn't, accept that, and kept insisting that there was some value 'between' 0.999... and 1.000..., such that 1.000... - 0.999... couldn't equal 0.000...
This got frustrating for those of us who know a bit of math. I think the misunderstanding hinged on the notion of infinity: A mathematician knows it as not a process but an entity, whereas the querulous querants kept insisting that somewhere, somehow we'd 'run out of' nines.

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.)
#23
Old 01-16-2006, 08:42 PM
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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.

#24
Old 01-16-2006, 09:18 PM
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Look, it's an artifact of the decimal representation system and symbolic representation in base 10.

1/3 is a repeating, non-terminating 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.
#25
Old 01-16-2006, 09:33 PM
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Quote:
Originally Posted by ultrafilter
If you can formulate specific questions about what you don't understand, we can help.
Beautifullly put, and it could almost be a tagline instead of "It's taking longer than we thought". Although I like that one better.

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.
#26
Old 01-16-2006, 10:04 PM
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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.)
#27
Old 01-16-2006, 10:08 PM
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Quote:
Originally Posted by Derleth
Philster: You're missing the point.
I'd call all of you bestial, sadistic necrophiliacs, but that would be beating a dead horse...

#28
Old 01-18-2006, 04:07 PM
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Quote:
Originally Posted by Fish
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.
Just coming off teaching calc 1, I'm feeling cynical. The reason it takes such supreme effort is that (a) new askers can't be bothered to read old threads, so we have to rehash the entire discussion each time, and (b) they refuse to accept valid arguments. I'll stop short of contradicting your assertion about our assumptions.
#29
Old 01-18-2006, 05:10 PM
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Originally Posted by CookingWithGas
It's pretty much an accident of the fact that we were born with 10 fingers.
Speak for yourself, Hu-man.
#30
Old 01-18-2006, 06:04 PM
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Quote:
Originally Posted by Giles
In base 2 arithmetic, 2/1 = 1.111111111...
Er, anything divided by 1 is itself, in any base. 2/1 in binary is 2 (although you'd write it "10").
#31
Old 01-18-2006, 06:07 PM
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Of course, 1.11111... is equivalent to 2 ("10") in binary, just as 9.9999... is equivalent to 10 in decimal. Never mind.
#32
Old 01-18-2006, 06:08 PM
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Originally Posted by Mr2001
Er, anything divided by 1 is itself, in any base. 2/1 in binary is 2 (although you'd write it "10").
But 1.111111... in binary equals 2. Just like .999999... equals one.
#33
Old 01-18-2006, 06:09 PM
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Originally Posted by Excalibre
But 1.111111... in binary equals 2. Just like .999999... equals one.
Never mind me either . . .
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