Then please, teach me. Where is my error is this proof? I can’t find it and I need your help.
- x = 0.333…
- 10x = 3.333…
- 3 + x = 3.333…
- 10x = 3 + x
- 9x = 3
- x = 3/9
- x = 1/3
- 0.333… = 1/3
Moderator Warning
7777777, I instructed you previously to stick to mathematics and factual information, and not base your posts on your personal opinions. This is an official warning for ignoring moderator instructions.
Stick strictly to mathematics in your posts.
Colibri
General Questions Moderator
I’m not saying it’s “more right”. I’m saying it represents something real, and it’s the type of math that most people think about when you propose math problems to them.
Perhaps I should have said that such “innate” types of math are maths that are mathematical models of physical/real things. That’s the technical term, but I hadn’t thought of describing it that way. Most people think of math in terms of that, because that is by far the most useful and most used math, and because most math taught, even in college, is math like that. Whereas the other stuff is useful, but it doesn’t directly represent any physical properties/things; the fact that those “weird” maths is useful is sort of… “roundabout”, perhaps?
The vast majority of the time when people are presented with math problems, it’s a math problem that uses mathematical models that are meant to model the physical/real world, so that’s what they assume the kind of math is every time you present a question/discussion.
And I find it interesting to think of even grade-school math as forms of more and more complex mathematical models. And again, I think this ties into some innate ability to learn math, which I would guess is pre-programmed to focus on such directly observable things.:
Integers/counting is a model of the basic human concept of abstraction/generalization. It represents the “theory” that you can have “kinds of things”. You can have 5 rocks, you can have 6 gourds, etc. It’s the most basic representation of human abstract thought. You can’t PROVE that there really are kinds of things, that everything isn’t just some other random object, it’s entirely conceptual, but there it is, we humans do it all the time, and so we came up with counting at first.
Then from there you have addition/subtraction arithmetic, which could be described as being more complex, involving the modeling of grouping things together.
Then higher up you have fractions/rational numbers, representing things that are continuous.
And so on.
Just popping in because this reminds me of the famous quote:
But let me also add: Some of that “weird” math is there because someone just thought it fun to play with, but the vast majority was developed to solve some problem that arose from things developed earlier. Most of math developed not necessarily to apply to the real world, but to apply to other parts of math. It may no longer be intuitive (but neither is modern physics), its direction of development arguably wasn’t always necessary (we had a brief thread on that some years ago), but it’s ultimately grounded in something intuitive.
Which is not to say that I think there’s necessarily a facility of the brain that developed specifically to count things. That’s an interesting, and possibly empirical, question. It could just be that the development of general reasoning faculties entails the ability to count (which may be part of what you’re saying).
That’s sort of what I’m saying. I’m also saying that all these pre-programmed abilities tend to be focused around the real-world stuff, because it was the ability to understand that kind of math that had the most evolutionary advantage.
There are many omni-cultural trends, ,one of which is the development of math around large-scale building and economic counting. This can’t be a coincidence. I mean, I remember seeing this show about tribesmen in Papua New Guinea, and they’re showing how they arrange a certain pandan leaves (I think they might have been pandan) into a shiplap pattern to make shingles, just like everyone else in the rest of the world also came up with, but with hard shingles. From a very young age kids love legos. There’s even that personality brekadown where one type is the “artisan”
I’m just saying there’s something there. Again, like I mentioned, the more formal studies on geometry understanding in people who never learned any math.
Counting is not an “omni-cultural trend.” There are societies in which people don’t count beyond some small number like three or twenty. There are apparently society in which people don’t count at all. It’s just not necessary for them, so it was never developed. There is no reason to think that counting is a basic human ability that’s part of every brain. It’s just another application of the human mental capacity to use (mental) tools. Counting is no more than another human invention that’s useful in some societies, and the society develops it (or borrows it from another society) when it’s necessary for them.
I am going to offer at least a partial disagreement here.
I don’t think there would be any cultural groups at all that did not possess the concept of one to one correspondence – even if those societies lack some counting numbers. There is a distinction between concept and notation here.
Also I recall reading (admittedly some time ago) Richard Skemp The Psychology of Learning Mathematics where he pointed to evidence that there were regions within the brain dealing with mathematical concepts (space, number, comparison) that were analogous to the known language centres of the brain.
I think the anthropological observation that some societies operate without a need to count is incidental to the discussion.
Now back to the question of whether 0.9999…=1, I think it is clear that some higher order mathematical thinking is required to resolve that one. Empirical evidence before us right now would suggest so.
Codswallop! WHy is anyone pushing this claim?
This is the same thing as people trying to claim marriage is solely a cultural invention, and pointing to a few tribes here and there who have some weird form of marriage
Of course bushmen here and there can’t count. But most of humanity, if not the lion’s share, eventually all independently developed relatively centralized nation states supported by agriculture, usually with significant technologies. The bigger the society and the larger structures they built, the more math they came up with INDEPENDENTLY. Numerous societies across the globe all figured out all the basic stuff like the area of triangles, pythagorean’s theorem, etc.
Are you really going to tell me this is just some random-ass coincidence?
Certain things just ARE cultural universals, and often they’re based on a mix of biology and the practicalities involved with human biology.
Would you quit saying, “Codswollop”? Do you think that impresses people with your immense learning? Look, you don’t have degrees in linguistics or linguistic anthropology, right? You don’t know about the research into this matter. There are societies and languages without numbers beyond twenty or three or apparently even numbers at all:
If a society doesn’t need numbers beyond a certain point, they don’t develop them. There’s no proof that there is a mechanism in human brains just for counting. There’s no proof that there is a mechanism in human brains just for language. They very well might be just what any human society develops using a mental ability for developing tools. The fact that the numbers they develop resemble each other proves only that if numbers are going to be a useful tool, they have to work in a certain way. Also, it appear that most of the time societies didn’t develop their system of counting numbers independently. They borrowed them from other societies that had them already.
7777777, do you intend to address my most recent statements? Or, you know, just address the statements about R being closed with regards to addition. This is a basic quality of R. So is infinity an element of R, or is Z not an element of R?
j_sum1, The Psychology of Learning Mathematics was first published in 1971. We’ve learned a lot since then. I believe that the notion that there are centers of the brain for particular skills like mathematics or language is no longer considered to be clearly true.
I said that Z cannot be equal to infinity because adding 1 to infinity does not increase
its value, whereas adding 1 to Z increases its value.
You must see what you yourself say: “add+1 to get 2,3,4,5,etc. The thing about this, though, is that at no point does this end”.
You say that in this way you will get to infinity, and you get to infinity with numbers which are real numbers. Number 1 is a real number, and add infinite amount of 1s equals infinity: 1+1+1+1+…=∞. Why should Z not be a real number as well
if we can get to infinity by adding 1 to it?
Or are you saying that adding infinite amount of real numbers is different than
adding a finite amount of real numbers? How exactly real numbers transform into
a symbol ∞ no-one knows as I have said.
You write also that " There is not a single real number Z for which Z+1 does not equal another real number". Apparently this rule does not apply if you are adding an infinite
amount of real numbers because their sum is infinity and not a real number.
It seems that there are exceptions to the rules when dealing with infinity.
Can we be sure if any rules work with infinity?
If so, why should not there be Z, my equation deals with infinity, do the rules work
now?
The main problem with Z is, as I said already, it does not seem to have a value.
If you write down a largest number, for example 100000, then you can add 1 can
arrive at 100001 and so on. You did not tell a solution to this problem.
I said that the wrong answer is that there is no Z. You can never write down all the
decimals of π, it does not mean there is no π.
Z = ∞ is a wrong answer as I have said many times. The main property of Z is that
its value increases if you add 1 to it, whereas adding 1 to ∞ does not increase its
value because ∞ = ∞ + 1. Of course , if you assume that Z = ∞ then x = 0
as you said, but it is false. The infinitesimal is not equal to 0.
If Z + 1 = ∞
What does ∞ - 1 equal??
7777777, I’m still waiting for an answer to my question.
If two numbers are not equal, then you can find another number that is between them. Do you agree with that?
If you do, what number is between 1 and 0.9999…?
Yes, I agree. The answer is:
The infinitesimal x
Z = -log(x)
x is a variable whose value cannot be written down for the same reason that all the decimals of π cannot be written down. Nevertheless π exists, as a symbol it represents all the digits of π.
To answer that question, one need first reach ∞
You cannot subtract 1 from a number if the number is inaccessible.
Tell me how do you reach ∞?
It is illegal to divide by 0. So that 1/0 = ∞ is not a correct answer.
1+1+1+1+1+1+…= 1/(1-1) because 1+1+1+1+…is a geometric series
whose sum s = 1/(1-r), now r=1 so that s=1/(1-1)=1/0
Nope. Doesn’t work.
What is x/x in this case? What is 10*x? What is x^x?
In the case of pi, these answers are all straightforward. Not so this entirely new, made up number you’ve created. You are implicitly defining a new number and assuming arithmetic works out ok. Without proof.
By the way, you can actually define a system where such a number exists (how many times have I’ve said this now?) and many people have done so but that system certainly isn’t the standard real number system.
I rather thought you assert that you add one to Z. Do you still assert this? If you do why are you asking about reaching ∞ ?
OK, so you accept that the value of the geometric series s = Σar[sup]n[/sup] is a/(1 - r). Now then, let’s see what this means for the sum 0.9 + 0.09 + 0.009 + 0.0009 + …
Plainly, in this case, a = 0.9 and r = 0.1. So, can you tell me what the value of s = a/(1 - r) then is?
Wait, I thought you said that the “infinitesimal x” was the difference between 1 and 0.9999…
Now you’re saying it’s a number that’s super-duper close to one, but not exactly? Those are completely different.
So again, if 1 and 0.9999… are different, please name just one of the infinitely many numbers that must lie between them.