Decimal notation. Whole numbers over one, and fractions of 1 over a whole number.
We all know 1/2 of 1 is one half, 1/2 of 1/2 is 1/4, and 1/2 of 1/4 is 1/8… 1/2 is larger than 1/4 which is larger than 1/8, 1/8 is larger than 1/16 which is larger that 1/32, which is larger than 1/64, 1/64 is larger than 1/128. 1.5 is larger than 3/4 but it is half of 1.5.
1/9 is larger than 1/99 which is larger that 1/999.
[attachment=0]999999.png[/attachment]
I may live in my own little world but decimal notation says there are no 1’s in .9 recurring, some of you may be forgetting that 0.9 recurring is another way of saying 10/9 recurring. But I could be wrong…
Is 1.0 = 10/9 recurring?
I propose that any of the digits 0 - 9 in a base-ten system repeated infinitely isn’t a number. But I would venture the thought that any digit recurring over itself recurring is 1.
I would also agree with Magnus that 1/3 is not equal to .3 recurring. It is unresolved in a recurrence, at best, it is as close as a base-10 system allows you to get without being equal. When a recurrence shows up it is telling you it is not equal.
It either equals null or zero, take your pick. At this point, you’re arguing for the sake of arguing, you stopped using mathematical arguments. Your argument is psychosis, no bearing on reality. You want so desperately for there to be orders of infinity, that you have ceased rational discussion.
and yes, N and “2N” are equal in value, what so many people have tried to explain to you, is that you CANNOT use operators on infinity!!
At what point does the sequence terminate? Sure N and 2N have different values, but ONLY in termination, infinity does not terminate though, and so many people in this thread keep trying to explain that to you.
This invites a kind of meta-discussion – a discussion about discussion.
The following isn’t specifically directed at you, so you can safely ignore it, but if you want, you’re welcome to take it in and bake a response for it.
You can interact with people in a large number of ways – indeed, an infinite number of ways, a pretty large infinite number of ways (: – and each one of these ways have certain consequences; and each set of consequences can be compared to every other in order to determine the most preferrable one.
If, for example, you want to be respectful, I believe you absolutely must abide by the rule that says “Make sure that the other person wants to hear what you have to say”. Just in case, I will repeat, you don’t have to be respectful to other people, you act as you see fit, but if you want to be respectful, I believe that’s the way to go.
If you think it’s fruitless to have a discussion with me, that’s fine, and you’re absolutely free to act in accordance with that belief, by say, not trying to explain stuff to me. But by telling me things I don’t want to hear – e.g. that you think that I’m arguing for the sake of arguing, that I stopped making valid arguments, etc – you are being disrespectful. The question is: do you really want to be disrespectful?
And then there’s the general question of the extent to which it is useful to talk to people when they don’t want to listen to you or when they don’t want to listen to what you have to say.
If you think you can educate people that way, make them more intelligent, more capable in life, I don’t think that’s the way to go. Forcing people to do things against their will can certainly make them do things you want them to do but at the cost of becoming confused.
Let that be the end of this meta-discussion.
Let us return to the subject.
I don’t know what it means that N and 2N are equal in value. N and 2N are symbols representing the set of natural numbers and the set of even natural numbers, respectively. You are surely not saying that N and 2N are equal sets i.e. that they have the same elements?
You don’t even understand this discussion. That’s not on you, but on me.
Like I say “everyone is a genius, if you don’t understand something, it either wasn’t explained simply enough or there’s nothing to understand in the first place”
I played with Adobe Illustrator 88. You could crash the program by folding a segment of a line parallel to itself. The stroke cast was infinite. Sent the computer into a recurring loop, attempting to calculate where the parallel lines of the stroke width would converge. Never, and that caused some problems for a graphics program. So how many radial degrees of one will a computer program resolve too… a ridiculously more number of decimal points then were required to fly a man to the moon. But back then they had just 8 bits to work with.
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.