Magnus… so basically “whoa”
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”
Slide Rule.
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.
en.wikipedia.org/wiki/Pierre_B%C3%A9zier
The founder of the mathematics upon which 2 and 3 D surfaces are defined within computer systems.
npr.org/sections/ed/2014/10 … n-the-moon
Just three points in space relative to an origin.
X over 1.1 over X
For example, 9/1 and 1/9 ?
And what are the decimal representations of those fractions?
Posting the same stuff over and over is not helpful.
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 isn’t a number. But at least it’s well defined.
Feel free to research it.
Greetings from way back.
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.
They don’t seem to call it imaginary unit. They call it imaginary number and they say it’s a complex number.
See here:
en.wikipedia.org/wiki/Imaginary_number
Beside that:
(i = 0 + 1 \times i)
You said that (\infty) is not a number because (\infty \div 2) cannot be expressed more simply.
More generally, your point seems to be that (a) is not a number unless (a \div 2) can be expressed more simply.
I mentioned complex numbers for two reasons:
because complex numbers are widely recognized as numbers (and because (i) is widely recognized as a complex number)
because (i \div 2) cannot be expressed more simply
This refutes your point.
My point is that if infinity is a number then infinity/2 ought to be fairly easy to calculate.
So why isn’t it? Why do you need to say that it’s irreducible or why do you need to create a new symbol?
Because it’s not a number.
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.
Examining this example shows some possible interpretations:
Both 5 or 2 are numbers in the natural number system. So what about the result of the division?
Either the result is undefined because 5/2 does not exist in the natural number system.
Or the result of 5/2 is 2 or 3 depending on how division of natural numbers is defined.
Now consider infinity within the real number system.
Assuming that infinity is a number, there appears to be no reason why the result of infinity/2 should not exist in the real number system.
Therefore, my assumption that infinity is a number must be wrong. I see no other explanation.
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.