I am still waiting for Silhouette to define “undefined”.
His argument so far has been that the word “undefined” cannot be defined. This implies the word is meaningless given that only meaningless words cannot be defined. But why would someone use meaningless words when discussing mathematics?
I have absolutely no idea what it means for a number of anything (not only 3s) to be undefined.
My best guess is that what he’s saying is that the number of 3s is unspecified.
But the number of 3s in (0.333\dots) is certainly not unspecified. The ellipsis tells us that the number of 3s is infinite. That’s not the same as unspecified. Unspecified means we didn’t specify (i.e. say) how many 3s there are.
Even if the number of 3s in (0.333\dotso) is unspecified, which it isn’t, how does it follow that (0.333\dotso) equals (\frac{1}{3})? If the number of 3s is unspecified, what follows is that (0.333\dotso) is indeterminate because it can be any of the following: (0.3), (0.33), (0.333) and so on. It means that (0.333\dotso) is a CATEGORY of numbers, not a SINGLE number.
On the other hand, phyllo made an argument that (\infty) is not a number because (\infty \div 2) cannot be expressed more simply. I responded by saying that (i \div 2), which is widely accepted as a complex number, cannot be expressed more simply either. His response was to conveniently ignore this point.
i is a perfectly good number. It is in fact a complex number, a perfectly legitimate member of the mathematical universe. If you like, you can think of it as just another notation for the point (0,1) in the Euclidean 2-plane you hopefully studied in high school. If you believe in the Euclidean plane from high school it’s easy to believe in the complex numbers.
Throughout history from ancient times to the present, mathematicians have continually expanded their notion of what is a “number,” which actually has no universal definition in math. Rather, a number is just “Whatever a consensus of contemporary working mathematicians think a number is.” In other words “number” is a historically contingent and evolving notion. People didn’t used to accept negative numbers, or zero, or fractions, or irrational numbers. Now we do.
The complex numbers were first glimpsed in the 17th century (with much distrust) and only became full-fledged, widely accepted numbers in the 19th. Today they’re part of the mathematician’s toolkit. In application they’re an essential part of modern electrical engineering and quantum physics. They’re not really that mysterious. The number (i) is just a formalism that lets you calculate with ideas such as phase shift and periodicity. Maxwell discovered that the electric and magnetic fields are perpendicular to each other; and this is expressed in the modern formalism using complex numbers. My point being that complex numbers are part of nature.
The number (i) is a perfectly good number; as is (\frac{i}{2}), which is just another name for the point ((0, \frac{1}{2})) in the high school Euclidean plane. As a complex number, (i) represents a quarter turn counterclockwise in the plane. Do it twice in a row and you’re facing the opposite of where you started, and you’re at the point (-1,0). So (i^2 = -1).
Feel free to continue to try and bastardise my arguments.
That “a thing which is undefined cannot be defined” is a tautology. Obviously. Take your very first logic class some time soon, it will help you think - for the very first time.
Again I try in vain to remind you of the difference between a legitimate definition, which is complete and exhaustive, and how laymen like you think of “definition”, which is “some vague description that you can sort of associate with some idea of a concept that you’re trying to dance around”.
Be precise. Be specific. Be free from contradiction and be in the slightest bit aware of counter-arguments before you insist upon them as unequivocally certain - as though your very identity depended upon it.
Then maybe we can have something that remotely resembles a constructive conversation that isn’t a waste of everyone’s time…
I know you have absolutely no idea.
This is the problem.
You actually think (0.\dot3) can be “any of the following: (0.3, 0.33, 0.333) and so on”?
(0.\dot3) is completely different, nevermind a “category” of these numbers, or whatever nonsense you’re trying to push now…
I’m tired of dealing this, can’t you tell? Why do you insist on calling upon me to correct your ridiculous assertations? Do you enjoy humiliation? Because that’s not my thing to indulge that kind of shit.
To define a word means to verbally or non-verbally describe its meaning.
That’s what I am asking you to do. I am sking you to verbally describe the meaning of the word “undefined”. The goal is for me to understand your point.
If you’re saying that the word “undefined” cannot be defined, this implies the word “undefined” has no meaning. In other words, it implies it’s a meaningless word.
Only meaningless words cannot be defined. This is because they have no meaning. You cannot verbally describe the meaning of a word if the word has no meaning.
Are you saying the word “undefined” has no meaning? If so, why are you using it? Why are you using meaningless words in order to discuss mathematics?
That’s not what I think. Read my post once again.
You have yet to explain what it is.
I’m merely pointing out that you have no arguments and that you are far more interested in psychological mind-games than you are in logical argumentation.
Yet, pretty much every single mathematician says that (i) is a complex NUMBER. But sure, if you’re operating with a different definition of the word “number”, you might be correct. The question is: how is that relevant?
I feel the need to remind everyone that there is no place for statements such as “Feel free to research it” within this thread. This is supposed to be a logical debate.
i is the imaginary unit. It’s a concept. It’s not a number. A complex number has the form a+b*i. The i identifies the imaginary part of a complex number.
You switch to whatever number system you feel like whenever you feel like it.
Complex numbers have nothing to do with infinity or the question of 1=0.9… or 1<>0.9…
Therefore I do not want to discuss it in this thread. It’s off-topic.
You can research for yourself if I’m right or wrong about the ‘imaginary unit’ but I don’t care if you do or you don’t.
And my point is that your conclusion does not logically follow.
If (a \div 2) cannot be expressed more simply, it does not logically follow that (a) is not a number. It can simply mean that we did not invent a simpler way of representing the number represented by the expression (a \div 2).
This is evident in the case of (i). (i \div 2) canot be expressed more simply, and yet, no mathematician will take this to mean that (i) is not a number.
Indeed, back when only natural numbers existed, there was no way to express (5 \div 2) more simply.
Let’s stick to the standard definition of division – the one that we apply to integers and reals the same way.
According to such a concept of division, the result of (5 \div 2) is neither (2) nor (3). In fact, the result is not a natural number.
Let us now suppose that our mathematical language is restricted to natural numbers and basic arithmetic operators (+, -, * and /).
This means that (5 \div 2) cannot be expressed more simply. There are other ways to express it, sure, but there are no simpler ways (i.e. there are no equivalent expressions that use fewer tokens.)
Does that mean the result is “undefined”?
Depends on what you mean by “undefined”.
Does that mean the result is not a number?
Not really. It simply means we have no way to express the resulting number more succinctly.
Does that mean that (5) is not a number?
Most definitely not!
Remember that your argument is that (a) is not a number if (a \div 2) cannot be expressed more simply.
The symbols (\pm \infty ) are defined rigorously as extended real numbers. They’re used as convenient shorthands in real analysis and measure theory.
But even as extended real numbers, division of those symbols by other real numbers is not defined. Which just means that it’s not defined. It has no standard definition and there’s no sensible way of creating one.
The specific rules for the defined arithmetic properties of (\pm \infty ) are here:
I ignored it because you’re repeating a point I addressed earlier.
(\infty) is not a real number. Neither is (\infty \div 2). Nonetheless, they are both numbers.
(Albeit, as wtf noted, the latter expression has no recognized meaning within the official language of mathematics, which means nothing with regard to this topic.)
You appear to be saying one of the following:
if (a) is not a real number then (a) is not a number
if (a \div 2) is not a real number then (a) is not a number
Neither of those is a logically valid argument.
I said it earlier: the set of real numbers is not the set of all numbers. There are numbers that are not real numbers e.g. complex numbers and hyperreal numbers.
Thanks, but I’m afraid I’m getting off the roller coaster. Magnus wrote, “(Albeit, as wtf noted, the latter expression has no recognized meaning within the official language of mathematics, which means nothing with regard to this topic.)”, my emphasis. If math isn’t what the thread’s about, I’m at a loss to contribute.
I did write extensively in this thread several years ago explaining why .999… = 1, but clearly to no avail. Long answer short, though, it’s because .999… = 1 is a theorem that can be proved (by a computer if one likes) in standard set theory.
There’s no other reason. Once you define the symbols as they are defined in standard math, the conclusion follows. If you define them differently, you can say that .999… = 47 or anything else you like. There is no moral or absolute truth to the matter, it’s strictly an exercise in defining the notation and then showing that the theorem follows. Just like 1 + 1 = 2. If you give the symbols different meanings, you get a different truth value. With the standard meanings to the symbols, the statement is true.
But none of this is of much interest, I gather. It’s “not what the thread’s about.” Perhaps I never understood what this thread is about.
If you have been following this thread, then you know that he has rejected the operations on infinity which are described in that wiki article:
[attachment=0]extreal.JPG[/attachment]
I was trying to show the contradictions within his own ideas about infinity.
None of this bears on the original topic. The infinity of the extended reals has nothing, repeat nothing, to do with the infinite cardinals and ordinals of set theory, or the meaning of positional notation in decimals. It’s a red herring and a distraction to the question of .999… = 1.
The notation .999… is a shorthand for the infinite series 9/10 + 9/100 + 9/1000 + … which sums to 1 as shown in freshman calculus. There truly isn’t any more to it than that, though if desired one could drill this directly down to the axioms of set theory.
But this point has been made repeatedly by myself and others in this thread to no avail. I confess to not understanding the objections, It’s like asking if the knight in chess “really” moves that way. The question is a category error. Chess is a formal game and within the rules of the game, the knight moves that way. There is no real-world referent. Likewise with a notation such as .999… By the rules of the game, it evaluates to 1. It’s a geometric series. If you make some other rules, you can get a different result. Efforts to imbue this with some kind of philosophical objections are likewise category errors. Rules of formal games aren’t right or wrong. They’re only interesting and useful, or not. The rules of math turn out to be interesting and useful so we teach them.