You no longer have a field- infinity will not have a multiplicative inverse. It doesn't matter if you agree with the definition of the real numbers that keeps the reals as a field. It's the definition that's been accepted universally and the one that people mean when they say "the real numbers". Look at the extendend reals (either version). They have thier place in mathematics. Some things get nicer (they are compact) some things get worse (no longer a field), but "the real numbers" will always mean the ones satisfying the complete field axioms.
How is that relevent to 360/0 being "not defined"? I was pointing out that the definition is a very good one. So what if it doesn't fit into a group? That doesn't matter most of the time.
If you're talking about limits of real numbers you have no choice but to talk about the reals. QQ and SL both had statements involving 360/0, this is undefined. I see no limit in their posts. "He" is either QQ or SL. I don't see any difference between a "concievable real number" and a plain old "real number" then, or how it has any bearing on anything at all here. Then there's nothing to argue about here unless we want to delve into what we each mean by "mathematically compelling". pi is very 'natural', use something other than radians for sin(x) and it's derivative will no longer be cos(x) but a multiple of it (for example).
It is completely relevant. Look at the definition of the reals, see that division by 0 is not defined. I don't know what else you have in mind for "not defined", but it doesn't get any clearer than this.
N/0 is defined as ∞ where ∞ is not a REAL NUMBER Why the fuck do you keep bringing up REAL NUMBERS? See, that is why pi was chosen, because it is mathematically easy. Before trig, 360 degrees for a circle was mathematically easy.
What else are we talking about? What is 0? What is N? Are these real numbers? No? What then? Or do you just want to talk about these symbols without having any idea at all as to what they are or what rules they obey? Where do you see N/0 defined as infinity by the way? This would be normal in projectively extended reals (i.e. their 1 point compactification), but I'm not sure this is the context you're refering to, or if you even have a context? This is getting into what I would consider 'mathematically compelling". There are many other reasons why sin and cos are 'natural' functions that come up without even considering any geometrical intrepretation of them eg. their relation to e^x, solutions to a simple differential equation f''=-f, their power series, etc. All the dandy analytical structure we are used to is apparent before you even think about what a radian or even pi is. When you want to have a geometrical meaning, the units of measure for an angle become 'obvious' as this structure is somehow 'natural' and something we'd like to preserve.
You are hopelessly lost. N can be any number you wish it to be. In fact, unless you are talking about integers, N is most likely going to be an approximation. Likewise, 0 is an approximation unless you are talking about integers whereby 0 is directly between 1 and -1. Once you understand this context, that the numbers we are dealing with are approximations, then you must accept that N/0 is infinity. Now why are the numbers approximations? N was a degree measurement, correct? Any measurement you are going to make is going to have a +/- value which signifies the degree of accuracy. We could say N = 360.04 +/- .01 which means N was measured to be between 360.03 and 360.05. Furthermore, when we divide N, we need not divide by an integer. So, likewise, the number we are dividing by has a degree of accuracy as well. Both N and 0 exist within a range determined by our measuring capabilities and it just so happens that 0 may be measured in the range of 0 and .01 for instance. That means .000000000000000000000000001 is included in this range as well as even smaller numbers. It is in that context that N/0 is defined as infinity. Got it? It's not that hard. Even my TI-89 will tell me that N/0 is infinity.
What the hell are you talking about? In any case 360 and 0 are both integers, so let me say "division by zero is not defined in the integers" so we don't have to get into what on earth you think approximations have to do with anything relevant to this discussion (or how on earth you are trying to reason with them). The definitive source- "my calculator said so". Now for the appeal to authority by me, "Maple told me N/0 is "Error, division by zero"". (tongue firmly in cheek)
Well, in that case - you are not working in the real world and you are only talking about the ideal mathematical world. The only thing measured in integers are quantities of objects. Such as: I have 5 apples. I don't have approximately 5 apples, or 4.9 apples, I have -5- apples. Measurements of degrees, i.e. 360 degree measurement is not an integer. If it is measured, it is an approximation no matter how you slice it. Go shove your head back into your theoretical mathematics book. We have no use for you in the real (as real as this forum is) world. Yes, computer languages will warn you when you divide by zero. Here is what MATLAB gives (MATLAB is used by a lot of real world people because of it's vast variety of toolboxes, I believe Maple is more of a stictly mathematical tool): >> 360/0 Warning: Divide by zero. (Type "warning off MATLAB:divideByZero" to suppress this warning.) ans = Inf >> 0/0 Warning: Divide by zero. (Type "warning off MATLAB:divideByZero" to suppress this warning.) ans = NaN See, I get the warning in both instances. But for 360/0 the ANSWER is Infinity and for 0/0 the ANSWER is Not A Number (undefined). I suppose by your logic, I am not to trust my calculator when it says 2+2=4 until I can prove it.
0 is most definitely not an approximately tiny number. End of story. (Note, just in case you try to argue the point: What is 360/0 if as an approximately tiny negative number? Whooops.)
Your inconsistency is astonishing. In another thread you attempted to write a program to "prove" that a right triangle with integer sides couldn't have the other angles as integers (measured in degrees). If you believed that measurements of degrees were not integers (not to mention the sides themselves), than what the hell were you trying to do? I guess this isn't a "Physics & Math" forum, but a "What Aer declares is the real world" forum? Sorry, I must have missed a memo. Calculators or programs are usually designed for ease of use, they are not designed to be a source of mathematical definitions. Maybe you think the constant "e" is defined to be "that thing you get when you enter exp(1) in your calculator", but it's not. Why do you think Matlab warned you about 360/0? I'm willing to make a bet that it's to make the user aware that Matlab is going to replace the symbol "360/0" with "inf", since it's not a universal thing to do but can make a program more robust. The user should be aware that if their program causes a division by zero, this is what matlab will do rather than crash altogether, and they should be aware of potentially "spooky" output if this happens.
You are a mathematician as well. I do not expect you to understand. Approximations happen in reality. Exact numbers happen in theory. Get your head out of your ass. I did not ask for the theoretical story. Negative numbers mean a specific thing. If I know the quantity I am measuring must be positive and I measure it as -.00000000001, then I am going to assume it is 0, not -0. 0 does not have a sign, I would expect a mathematician to know this. Whoops.
I've never been inconsistent here. What is wrong with this? That is strictly theory. We wouldn't have any integer 0 sides or angles either. Notice how my sides started at 1 and not 0. Likewise, the angles would have to start at 1 - THAT DOESN'T MEAN YOU CANNOT HAVE AN ANGLE OF 0.1 OR LESS! This is comparing apples to oranges. I proved that in theory, you cannot have integer sides/angles where angles are measured in degrees with a right triangle. The angles will always be non-integers. Are you trying to tell me that an angle must either be 2 degrees or 3 degrees and cannot be 2.345 degrees? That is retarded. Well, since you cannot figure out what is the real world, I must take on the responsibility of informing you Please Register or Log in to view the hidden image! I suppose when my calculator tells me that 0!=1, that is not a definition either. Calculators are programmed with definitions built into them. Geez.
You are so out of your league, it's not even funny. Do you really think -0 and 0 are different numbers? (Hint: They aren't.) Anyway, you're taking the limit of 360/x when x goes to 0 from the right (i.e. from the positive side of 0) lim x -> 0+ 360/x = ∞ and using that as the definition of 360/0 because 360/0 is undefined, if 360 is a real number, and / is standard division. You have not justified this, and I even showed why it's a bad idea, since lim x -> 0- 360/x = -∞ Your bullshit comment about 360/0 being infinity was wrong. You can define a new binary operator on something like the reals extended with infinity in which "Aer's new division operator" is defined when the second operand is zero as being infinity, but that's not the same thing as 360/0 being infinity. You. Are. Wrong.
I get it. The only time theory is allowed is when you post it. Are you the apple or the orange? There was a reason I put "prove" in quotes, you only checked up to a finite bound, and ignored any possible internal rounding 'problems' with matlab (not strictly problems, but just how it deals with arithmetic). I have no idea where this came from, but it doesn't seem to be in response to anything I wrote. I was simply questioning why you would bother to "prove" that there couldn't be a triangle with whole sides and integer angles if you already believed that the angles couldn't be integers since they were approximations. I mean, why the extra effort? Oh, right. You're allowed to post theory, but no one else is. When have I ever made any claims that anything I said was "the real world"? Please point them out to me. They are, but they sometimes make sacrifices for robustness. Do you think that Maple isn't programmed with any definitions?
What part of my response did you not understand? I suspect it was this part where I clearly said 0 does not have a sign. Exactly how does that mean that -0 and 0 are different? I told you that there was no difference! You are out of your league when it comes to reading comprehension..
I know it is a limit, that is what I've said all along! We are working with limits when talking about approximate numbers. N/0 is defined as infinity unless you are talking strictly about integers. It is as simple as that.
You are confused again. Theory needs to be applicable to what you are applying it to. Saying that N and 0 must be defined as integers is where your argument breaks down because they need not be integers and furthermore may only be approximations. Get real.
Yes, I agree it wasn't a formal proof, but any more than 10,000 would have taken more than the couple of seconds of computation my computer could handle. I could have written a more efficient program to skip similiar triangle calculations but a quick and dirty check up to 10,000 was all I cared to do.
