So if you say an essential thing to understanding the topic is irrelevant, you can imagine it’s not an issue?
I’ve been getting this impression from you for a long time now. You throw around the accusation of irrelevance as soon as a problem comes up. It’s either intentional out of bad faith or unintentional from incompetence - I think a mixture: except I don’t think you are explicitly trying to argue in bad faith, I think it’s just the usual psychological protections that people usually fall foul to that are getting in your way of changing your mind. If you weren’t merely trying to bolster your own (lack of) argument, and instead you tried to understand see the sense in the arguments of others - take a leaf out of Mowk’s book for this one - then we could get somewhere, but I think the inability also plays a part against you here too.
You like to say these psychological analyses of you are also “irrelevant”, but they really aren’t - for the same reasons as above ^
The problem is, if this is truly enough, why did you feel the need to talk about radians and unusual numeral systems (such as the base-10 system with 16 digits)? I believe it’s because . . . it’s actually not enough. Hopefully, you will agree the quoted is a redundant statement. It’s a statement that has to be proven, so you can’t use it as a premise.
Thanks for noting.
Alright, that’s your position, and I will confirm (I believe once again, it’s not the first time) that I understand it.
The problematic part is your claim that dividing (10) by (3) is the same exact process as dividing (9) by (3). That’s where we disagree. I don’t think it’s the same exact process.
That’s a good question. On the other hand, I do not really think it’s important. Certainly, it’s not necessary to answer it. But I’ll do my best anyways.
But . . . I already explained what’s wrong with the long division proof, didn’t I? So I’ll be repeating myself in all likelihood. Be warned.
When you divide (9) by (3), you get to a point when every subsequent digit is (0). When you divide (1) by (3), no such point is ever reached, simply because there is no such point. And this is a signficant difference even though it may not appear so at first.
In plain terms, in the case of (1 \div 3), the long division process is never finished. (And this is quite simply because (\frac{1}{3}) has no decimal representation.)
The question I need an answer to is: why do you disagree with this? This will help me proceed with more accuracy. Less redundancy, fewer assumptions about what you think and so on.
As it is, I can only guess . . . shoot blindly.
The never-ending process of long division, such as the one in the case of (1 \div 3), tells us that there is no number of digits used in that particular numeral system (such as (3)s) that can satisfy the condition. This is a subtle but very important point.
The long division tells us that the number of (3)s necessary to satisfy the condition is greater than any number we can think of. The problem is . . . there is no such number.
Suppose that such a number exists. Being greater than every number, it means it’s greater than itself. But a number cannot be greater than itself. Hence, there is no such number.
That tells us that there is no number of (3)s that can satisfy the condition. It does NOT tell us that the number of (3)s necessary to satisfy the condition is infinite since infinity has a different meaning: that of a number greater than every integer (not every number in general.)
You keep using the word “undefined” in your own particular way while refusing to define it when other people tell you they have trouble processing it.
Note that if a word is meaningless there is generally no place for it in mathematics (such words can only be useful in the art of deception and perhaps cryptography.)
If you mean that the word “infinity” is undefined in the sense that its meaning has not been verbally described (at least not properly) then I disagree. The meaning of the word “infinite” is perfectly captured by the sentence “greater than every integer”.
It is really hard to illustrate base-10 decimal system mapping as it is a logarithmic scale not a linear one.
The distance between each number isn’t the same distance apart on a line. As an example the distance between 0 and 9 are separated by 9 units, the distance between 9 and 99 is 90 units, the distance between 99 and 999 is 900 units, the distance between 999 and 9999 is 9000 units and so on. To the right of the decimal point it is logarithmic as well. The distance along the line between 1 and 0 is divided into 10 segments, the line is divided into 100 segments then a thousand segments, then 10,000 segments, then 100,000 segments and on and on and on. Well actually the line has all these divisions all at once because the distance between 1 and 0 can be divided into an infinite number of parts. So if we plot the 9/10ths, the next digit is 9 represents 9/100ths, the next 9/1000ths, and the next 9/10,000ths and so on. As you plot the points along the line the segment distance between iterations of 10 gets smaller and smaller the closer to 0 you get. But as a result of having 9/over what ever number of powers of 10 you are working with, you get closer and closer to zero but never get there.
This is why I believe that .9 recurring is actually a better description for zero then it is for one but it doesn’t work well for zero either. While it is remarkably close to zero it will never get to zero because of the infinitely recurring 9. It is really difficult to plot the point along a line that is divided infinitely.
Look up decimal notations if you don’t believe me.
Yes the numbers you have represented are what I thought too, but how are you mapping them on a line between 1 and 0?
1.0 9/10 9/100 9/1000 9/10,000 9/100,000 and zero is over here.
By this time you are very close to zero already. So how does it work that 1.0 = 0.9 recurring? When by the time you’ve gotten to the 9/100,000th digit in the number you’re already close to zero not 1?
Hey Sil,
Yeah I had really shitty math teachers growing up. Really. Plot .9 on a line representing 1.0 and 0, then plot .999999 on the same line.
I did OK in algebra did really well in geometry. Calculus I barely passed. Didn’t take Trig.
Never took a computer class in my life, but I managed to realize plenty. Enough to be the “go to” guy in a tech college with over 1500 employees. My understanding of graphics applications is intense. I’ve saved companies I’ve consulted for hundreds of thousands of dollars. So yeah, I really appreciate your condensing attitude. But I don’t let it go to my head.
I don’t think it’s relevant and I’m not the only one. JSS didn’t think it’s relevant either. Phyllo probably knows this.
Now, I may be wrong, but if I am, I’ll have to realize the relevance of these concepts, and if others want to show me their relevance, they must make an adequate effort.
This is very much in bad taste. Complaining about others not behaving the way you want them to behave (even if it’s better for them to behave in such a way) is a negative self-portrait (:
Suppose you’re right and I’m wrong. Either you want to help me or you don’t. If you do, you have to make an effort to do so, and most importantly, you must not be surprised if you fail. You make a decision to try to help, you own what follows. If you don’t – and NOBODY and I really mean NOBODY – is holding a gun to your head, then you simply don’t and you mind your own business.
Otherwise, redundancy ensues in the form of unnecessarily many unnecessarily long posts that say nothing of value.
. . . unless, of course, you want to psychoanalyze others, which can be of value, but has nothing to do with the subject.
Well, if you want to stick to the subject, they are irrelevant.
I don’t spend my time psychoanalyzing Ecmandu, Carleas, wtf, Uccisore, Gib and others who are in disagreement with me, not because I am stupid and don’t realize this is a useful thing to do if you want to explore the subject, but quite simply because I am on-topic and don’t see much value in psychoanalyzing random people on the Internet.
To my understanding you are repeating the 9’s and not representing them as single digits. 9 10ths the next digit 9 is in the 100th column, the next digit 9 is in the 1000th column.
In whole numbers the number 5,324 has just 1, 5 in the thousands column, just 1, 3 in the hundreds column, a 2, in the tens column and a 4, in the ones column.
To the right of the decimal point, .9999 has just 1, 9 in the 10ths column, just 1, 9 in the 100ths column, and just 1, 9 in the thousandths column.
Nope I don’t think what you have written is getting anywhere.
9 divided by 10 = .9
99 divided by 100 = .9
999 divided by 1000 = .9
All of your representations are going no where, stuck to the fist digit. You aren’t plotting different numbers. It isn’t getting any closer to one or zero.
I could make a snide remark here but I’ll just keep that to myself. That’ll learn me.
9-10th + 9-100ths + 9-1000ths + … now plot the numbers on a line between 1 and 0.
The standard notion of parity, as indicated by Wikipedia, applies only to integers – and infinity is not an integer. Of course, you can extend the concept to other kinds of numbers, but in such a case, you’d have to define what parity means.
Yes, you can use it to count digits and elements in a series and though necessary it’s not a sufficient condition to classify something as an integer.
From Wikipedia, once again:
No mention of infinity. Hyperreals aren’t considered integers either.
What would it mean for an infinite quantity to be even or odd?
Why is that even important?
And you’re asking exactly what?
“The value of an expression” generally means “An equivalent expression that is of one’s interest”. “The value of 4 + 4 is 8” means that “8” is equivalent to “4 + 4” and that “8” is the kind of expression that one needs at that point in time (which is usually a single number.)
So I suppose you’re looking for an equivalent expression . . . but what kind of equivalent expression? There are many expressions that are equivalent to (\frac{\infty}{2}) e.g. (\frac{\infty}{4} + \frac{\infty}{4}).
Given a number ‘n’ we have ways to calculate n/2 which is also a number. If infinity was a number we ought to be able to calculate infinity/2 and it ought to be a number.
You only gave expressions with pending operations. You gave no resulting number.
I don’t know why I would do any sort of mapping when we don’t even agree to the value of 99/100. If you think that it’s .9 and I think think that it’s .99 then we have a major problem that needs to be resolved before moving on.
You have a strange definition of the word “inconsistent”.
You didn’t answer my question. You didn’t state what you’re looking for. If you’re looking for a real number that is equivalent to (\frac{\infty}{2}), well, there isn’t one. But if something isn’t a real number, it does not mean it’s not a number. Real numbers are a subset of numbers, they aren’t the set of all numbers. There are other numbers . . . such as complex numbers.
What’s the value of the following expression?
( -1 + 3i )
What’s the resulting number of that operation?
It’s actually a pair of numbers ((-1, 3)) that has no equivalent real number. And yet, noone minds calling it a number.
Complex numbers have a real part and an imaginary part. That number is already reduced to it’s simplest form. Now, if you asked me to calculate half of that number, I would be able to do so.
Think of “0.9” as filling up a jug of water 90% of the way.
Then “0.09” more filling it up to a total of 99% of the way.
Then “0.009” more fills it up to 99.9% of the way.
Stop right there are the jug is not 100% full. it’s just close.
Very “approximate”.
But wait, now fill it 0.0009 more. It’s getting closer right?
Keep doing that and stopping somewhere along way the way, and it’s still only approximate.
It’s only if you do it infinitely that there’s no gap left. Stop at any point along that way and it’s only “close”, but it’s the infinity of the steps that completes the tendency to no other value than 100%.
Obviously all feeble attempts to try to define a consistent value otherwise are contradictory - any quantity you propose can be divided further and will never be sufficient.
Those who stare at the clouds will conclude that you will never ever get to 100% and yet they will never ever be able to prove any consistent quantity that “tops up” the water to 100% full. But they always have those clouds!
Mathematicians respect convergence (and divergence). They understand that its not “irrelevant” that the tendency unequivocally goes to 100% and no other value.
There is zero margin for error in where the tendency goes, and there is no equivocation over whether it “stops” getting there.
The very fact that it never stops is exactly what infinity means (endless) - and it’s only through this property that we can exactly get there - anything short of that and we cannot.
Anybody who thinks we can’t get there is confusing infinity with finity. Plain and simple. Case closed.
So can we got onto making this thread into something productive yet?
And the fact that there is no real number that is equivalent to (\frac{\infty}{2}) is supposed to be a proof that (\frac{\infty}{2}) is not a number?
I think not.
As I’ve said, there is no real number that is equivalent to (-1 + 3i), and yet, noone minds calling it a number.
Very curious. (-1 + 3i) has more complexity than both (\infty) and (\frac{\infty}{2}), and yet, you have no problem with it (:
You can divide (-1 + 3i) by (2), of course, but the result will be a relatively complex mathematical expression (not a single natural or real number.) For some strange reason, you have no problem with this.
You can divide (\infty) by (2) to get (\frac{\infty}{2}).
But you don’t accept this . . .
You can divide (\frac{\infty}{2}) by (2) to get (\frac{\infty}{4}).
But you don’t accept this . . .
Yes, (-1 + 3i) is the simplest possible expression of the idea, but so is (\frac{\infty}{2}). There is nothing simpler than it. Unless, of course, you’re willing to invent a new symbol, call it (hi), that stands for “half-infinity”. In such a case, (\frac{\infty}{2} = hi).