If you can formulate specific questions about what you don’t understand, we can help.
Heh. Let’s just say I’m a veteran of these things. 
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:
[ul]
[li]1/3 expressed in decimal notation is 0.333…, where the three dots at the end mean ‘extended to infinity’.[/li][li]1/3+1/3+1/3 == 0.333… + 0.333… + 0.333… == 0.999… This was accepted just fine.[/li][li]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…[/li][/ul]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.)
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.
:rolleyes:
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.
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.
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.)
I’d call all of you bestial, sadistic necrophiliacs, but that would be beating a dead horse…
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.
Speak for yourself, Hu-man.
Er, anything divided by 1 is itself, in any base. 2/1 in binary is 2 (although you’d write it “10”).
Of course, 1.11111… is equivalent to 2 (“10”) in binary, just as 9.9999… is equivalent to 10 in decimal. :smack: Never mind.
But 1.111111… in binary equals 2. Just like .999999… equals one.
Never mind me either . . .