have to round off pi — Metaphysician Undercover
to round off at some point, carry it to two decimals, three, whatever — Metaphysician Undercover
if there are issues with similar division problems we simply round things off (like with pi, and some square roots, and other division problems) — Metaphysician Undercover
Call me crazy, but why isn't a base 9 and a base 10 representation of the same number a base 9 and base 10 representation of the same number?If 1 is divisible, its divisibility is different in base nine from what it is in base ten. — Metaphysician Undercover
If fractions bother you then we can use exponents instead. — Michael
1919 in base 10 is equal to 110110 in base 9, so 0.111...0.111... in base 10 is equal to 0.10.1 in base 9. It's divided equally. — Michael
Meanwhile, we all understand that half a dozen is six, and what's meant by a third of the area of the lawn, so that works fine (presumably for you as well). But of course, we don't speak of a ninth of dislike for pizza with pineapple, at least not without some further clarification. — jorndoe
That is, there's something to round off. Seems you've already presupposed what you want to deny. (The division procedure isn't really the problem here.) — jorndoe
Call me crazy, but why isn't a base 9 and a base 10 representation of the same number a base 9 and base 10 representation of the same number? — InPitzotl
I don't think you can represent a solution to a division problem in base nine... — Metaphysician Undercover
Lousy example. The number's representation is no more the number than you are a white M in a pink rectangle.Take a look at the divisibility of "1" in base two, and compare it with the divisibility of "1" in base ten, for a good example — Metaphysician Undercover
It is the decimal equivalence which you are claiming that is what bothers me. That is the issue of the thread. You said 1/10 in base nine can be represented as .1 in base nine, and this is equivalent to .111... in base ten. I don't think you can represent a solution to a division problem in base nine, as a decimal, (.1), because decimals are proper to base ten, and that would be to conflate base nine and base ten representations. — Metaphysician Undercover
Lousy example. The number's representation is no more the number than you are a white M in a pink rectangle. — InPitzotl
You keep telling people to take a look at binary. Okay. 1/9 = 0.(000111)2. And? That's just another name for 1/9. Do you have a real point or a confused one? — InPitzotl
But nomenclature not withstanding, my point stands. In both base 10 and base 2, 110=0.1110=0.1, and 0.12=0.5100.12=0.510. — Michael
Sorry, but we haven't resolved that there's an actual problem here (not to me, or to anyone else here that I've seen).Sorry, but we haven't resolved the question — Metaphysician Undercover
If turtles are animals, why do they lay eggs? Since when does being called a prime have anything to do with being a number? The very fact that you even asked this question and actually think it's relevant shows that something is majorly wrong with your "problems".If you think it represents a number, then why is this number not a prime number? — Metaphysician Undercover
The point is that 1/9 is not a name for anything. — Metaphysician Undercover
the question of whether "1" is the representation of a number or not. — Metaphysician Undercover
We already know what numbers are and what expressions mean. The only problem here is you, and we don't even have to deal with that problem. But you've diverted 11 pages on this thread so far on your ego tripping delusions of having a problem. That's the problem.Your notion that a mathematical expression names a thing, is the problem you need to deal with. — Metaphysician Undercover
This idea allows people like fishfry to argue that "2+2" refers to the same object as "4". — Metaphysician Undercover
As seems to be the case often, you don't seem to be able to express your point very well, and you leave me wondering what you're talking about. — Metaphysician Undercover
have to round off pi — Metaphysician Undercover
to round off at some point, carry it to two decimals, three, whatever — Metaphysician Undercover
if there are issues with similar division problems we simply round things off (like with pi, and some square roots, and other division problems) — Metaphysician Undercover
Sorry, but we haven't resolved that there's an actual problem here (not to me, or to anyone else here that I've seen). — InPitzotl
If turtles are animals, why do they lay eggs? — InPitzotl
Since when does being called a prime have anything to do with being a number? — InPitzotl
The very fact that you even asked this question and actually think it's relevant shows that something is majorly wrong with your "problems". — InPitzotl
We already know what numbers are and what expressions mean. — InPitzotl
Looks like the same point on the number line to me. So where's the actual problem again? — InPitzotl
What exactly are you rounding off to decimal notation...? 1/9 π √2 ... You already acknowledge those numbers that you round off, only to go ahead and deny them. Inconsistent. — jorndoe
Numbers in the abstract are quantities of whatever we may want to examine, where the rules of mathematics are invariant (e.g. division) or otherwise set out. — jorndoe
So, I ended up thinking that you're no longer talking mathematics. — jorndoe
Clearly you haven't got a clue what a number is, yet you keep insisting that such figures represent numbers. — Metaphysician Undercover
And you're immune to it?Denial is one of many possible responses. — Metaphysician Undercover
It was meant to be an analogy... primes are numbers, but not all numbers are prime, was the point. But apparently you're even more messed up than this:What kind of nonsense is this? — Metaphysician Undercover
Utterly wrong. There is a history to the concept of prime numbers... after some time in the development of number theory, it was quite apparent that it would be more useful to exclude one from the definition of primes in particular to avoid having to keep making exceptions for it, especially in the fundamental theory of arithmetic which is heralded as being an especially important theorem. That has nothing to do with considering one as a number though... that ship has long since sailed:They are called "prime numbers". And "one" fulfills all the conditions of "being called a prime", except that it is not a number. — Metaphysician Undercover
...but ultimately it's just a loss of religion. There's no actual deep reason to not consider 1 (and 0) a number, except a bunch of meaningless mumbo jumbo.1 (one, also called unit, and unity) is a number, and a numerical digit used to represent that number in numerals. It represents a single entity, the unit of counting or measurement. For example, a line segment of unit length is a line segment of length 1. — one
You still have no idea what you're talking about... consistent with everything I've been saying for 11 pages, this is a language barrier and you're still confused.This is very clearly not true, as I think everyone else on this thread has admitted, except you. — Metaphysician Undercover
You're referring to the fact that @Michael listed some categories of numbers here; namely, N (the whole numbers), Z (the integers), Q (the rationals), R (the reals), and C (complex numbers). Those are indeed categories, but there are more; beyond C, there are quaternions and octonions. In contrast to R, there are surreals and hyperreals. Take just Z into the complex plan and you get Gaussian Integers. This is not an exhaustive inventory. All of these things have their own kinds of numbers, and we can even make up new kinds of numbers on the fly.There is no clear definition of what a number is, and there are supposed to be different sorts, natural numbers, rational numbers, real numbers.
Vagueness is not transitive. An animal can be anything. My pet is an animal. But my pet cannot be anything; my pet is a cat. A number in general likewise could be just about anything. But 1/9 is a fraction, and 0.(1) is a repeated decimal. Generally discussions of such things are in R, though Q suffices.What the other participants in this thread have indicated is that "number" is just a vague term with no real defining features. — Metaphysician Undercover
That doesn't surprise me, but I gave you a link to it. So I guess a bit more spoon feeding you is in order:Sorry, but I have no idea what your little diagram is supposed to be showing. — Metaphysician Undercover
On the same page:In basic mathematics, a number line is a picture of a graduated straight line that serves as abstraction for real numbers, denoted by . Every point of a number line is assumed to correspond to a real number, and every real number to a point. — number line (wikipedia)
Compare to here.Two numbers can be added by "picking up" the length from 0 to one of the numbers, and putting it down again with the end that was 0 placed on top of the other number. — number line (wikipedia)
The diagram tells you how you're supposed to play the language game with real numbers. 0.(1) is a real number.It's obviously not providing a definition, or any sort of indication as to what a number is. — Metaphysician Undercover
Silly MU, there is no case. You have no jurisdiction, the defendant is a non-entity (language), there is no standing, and there is nothing actionable.So how is that diagram supposed to argue your case?
Utterly wrong. There is a history to the concept of prime numbers... after some time in the development of number theory, it was quite apparent that it would be more useful to exclude one from the definition of primes in particular to avoid having to keep making exceptions for it, especially in the fundamental theory of arithmetic which is heralded as being an especially important theorem. That has nothing to do with considering one as a number though... that ship has long since sailed: — InPitzotl
Most early Greeks did not even consider 1 to be a number,[34][35] so they could not consider its primality. A few mathematicians from this time also considered the prime numbers to be a subdivision of the odd numbers, so they also did not consider 2 to be prime. However, Euclid and a majority of the other Greek mathematicians considered 2 as prime. The medieval Islamic mathematicians largely followed the Greeks in viewing 1 as not being a number.[34] By the Middle Ages and Renaissance mathematicians began treating 1 as a number, and some of them included it as the first prime number.[36] In the mid-18th century Christian Goldbach listed 1 as prime in his correspondence with Leonhard Euler; however, Euler himself did not consider 1 to be prime.[37] In the 19th century many mathematicians still considered 1 to be prime,[38] and lists of primes that included 1 continued to be published as recently as 1956.[39][40]
...but ultimately it's just a loss of religion. There's no actual deep reason to not consider 1 (and 0) a number, except a bunch of meaningless mumbo jumbo.
TL;DR version: That one is not considered prime has nothing to do with the consideration of one being a number. It's just yet another confusion of yours. — InPitzotl
I'm far from unaware of this MU... in fact, we've both gone through this. Here is the post where you said you were "trying to learn the language". And here is the reply I gave you seven days ago. Numbers defined differently is not a problem for math; it's just homonyms... just a feature of languages. To avoid the issues a language speaker just applies context. — InPitzotl
Vagueness is not transitive. An animal can be anything. My pet is an animal. But my pet cannot be anything; my pet is a cat. A number in general likewise could be just about anything. But 1/9 is a fraction, and 0.(1) is a repeated decimal. Generally discussions of such things are in R, though Q suffices. — InPitzotl
That doesn't surprise me, but I gave you a link to it. So I guess a bit more spoon feeding you is in order:
In basic mathematics, a number line is a picture of a graduated straight line that serves as abstraction for real numbers, denoted by RR. Every point of a number line is assumed to correspond to a real number, and every real number to a point.
— number line (wikipedia) — InPitzotl
The diagram tells you how you're supposed to play the language game with real numbers. 0.(1) is a real number. — InPitzotl
That's all there is, except for the fact that there are 11 pages of it. — InPitzotl
You mean obviously I have, which would be generous of you. Because I have not read all, but many, and enough. But as absolutely standard MU m.o., you ignored the question.Obviously you haven't read those pages. — Metaphysician Undercover
...Your so-called history of prime numbers is backward compared to what Wikipedia has to say: — Metaphysician Undercover
...while we're on the subject, what does the very next paragraph say?So, according to Wikipedia — Metaphysician Undercover
Welcome to the year 2020. So what's the problem?Then, in more modern times mathematicians wanted to treat 1 as a number, so they had to include it in the prime numbers and this created a problem. Now they've excluded 1 from the prime numbers, by definition. — Metaphysician Undercover
...so where does that leave you? Do you have the foggiest idea what a number is? Do you make universal, uncategorized statements about numbers?What is a problem is conceited people making the universal, uncategorized statements like "we already know what numbers are", when it's very evident that they haven't the foggiest idea of what a number is. — Metaphysician Undercover
I think you're lost, MU. This is supposed to be a thread about 0.(9)=1.It is not logical to refer to a property of a special type of number (real number) to demonstrate what a number is in general. — Metaphysician Undercover
Well that's really easy MU. Here's the primary motivation, in your words:I apologize for not joining your little game, but I see no reason to restrict our discussion of "numbers" to real numbers. — Metaphysician Undercover
The way to avoid inconsistencies and contradictions that lead to misunderstandings and deceptions (aka, amphibolies/equivocations) where languages have homonyms is to restrict the conversation to applicable shades of meaning. When in a pool hall and someone talks about how to sink the 7 without sinking the 8, English should be regarded as a pool-technique, so it simply means to invoke a spin on the ball... countering a discussion invoking the use of English with debates about how some hypothetical guy from England might sink the 7 is a meaningless distraction. In this context, we're supposed to be talking about 0.(9)=1. 0.(9) is a repeated decimal. Repeated decimals are special cases of fractions, suggesting a treatment of at a minimal Q, though decimals just commonly invoke R. So to avoid misunderstandings and deceptions, to meaningfully talk about Q and R, we should be employing the context of one of these two things.I'm trying to learn the language, and I don't like inconsistency or contradictions within the language I use. Such things lead to misunderstanding and even deception. — Metaphysician Undercover
We've been through this already, application is different from theory — Metaphysician Undercover
"the rules of mathematics" are not invariant — Metaphysician Undercover
"One" only submits to being a multitude when it is applied to a thing which can be divided — Metaphysician Undercover
The question is, philosophically, what is a grapefruit spoon for? Do you see the problem?The question is, philosophically, what is number for? — Gary M Washburn
As a matter of representing numbers, wouldn't most be fine with 9/9 = 9 × (1/9) = 9 × (0.111...) ? — jorndoe
I'll ask it again. What is your point? — tim wood
So what's the problem? — InPitzotl
Do you have the foggiest idea what a number is? — InPitzotl
The way to avoid inconsistencies and contradictions that lead to misunderstandings and deceptions (aka, amphibolies/equivocations) where languages have homonyms is to restrict the conversation to applicable shades of meaning. — InPitzotl
Incidentally MU, even if we don't restrict our discussions to the reals, 2+2 and 4 refer to the same object in the reals, and you claim they don't refer to the same object (again, in case you missed it, "Do you make universal, uncategorized statements about numbers?"). — InPitzotl
TL;DR, we should restrict our discussion to the reals because that's the context within which 0.(9)=1 and 0.(1)=1/9 are meant to be discussed; i.e., it is this context from which the meaning of such things derives. Ranting and raving about what some guy in 300BCE would have called 1 is a meaningless distraction. — InPitzotl
Your previous side-track doesn't really matter much here; it's about the numbers, 1/9 π √2 ... By rounding them off, you've already admitted them. Denying them is hence inconsistent; you wouldn't have anything to round off in the first place. — jorndoe
Saw the word "invariant" and took it for a ride? Having five fingers on each of your two hands means having ten fingers on them, not none, not a dozen. 5 + 5 = 10 = 2 × 5 (and 5 < 10 by the way). Notice how that goes for toes and claws as well? Whether yours or mine or the Pope's? You don't mysteriously get a dozen fingers in that case. That's what's meant by invariance here, + - × /, and what you tried to dismiss with a casual handwave. Oh, also, √2 × √2 = 2 (and 1 < √2 < 2), irrespective of your rounding, so there. — jorndoe
As mentioned, whatever your "One" is, this is something you've added here, much like I added distaste for pizza with pineapple. Your "One" apparently does not figure as the number 1 does in arithmetic. — jorndoe
Has anyone ever persuaded a change of an opinion or belief you've held? — dex
My beliefs change like the weather. But to be honest, I wouldn't say that it's others who persuade me to change. — Metaphysician Undercover
It ought to be clear to you by now, that I do not accept "the reals" as a representation of numbers. Any system of interpretation which ignores the role of "+" within an equation, to claim that "2+2" says the same thing as "4", cannot really be taken seriously. — Metaphysician Undercover
What are you talking about, "problem" and "required"? The fundamental theorem of arithmetic states, in the modern reading, that all positive integers can be represented as a unique product of primes (barring order). That's perfectly phraseable with prime including 1, it's just clumsy: "All positive integers can be represented as a unique products of primes, barring order, excluding from said product the number 1". Both phrases describe the same fact. One is just clumsier.It was only in relatively modern times that mathematicians wanted 1 to be a number, and this created the problem which required an exception to be added into the rule of primality. — Metaphysician Undercover
Pretty much.I provided a... — Metaphysician Undercover
Sort of, but not really. "Number" applies to a lot of things. But that's not a problem; it's actually a benefit. The definition of number should not merely not be nailed down; it should be open. But part of the point of categorizing these numbers is so that we can give particular kinds of numbers names.This is the point I've argued from the beginning of the thread. To know whether the op offers an acceptable representation of numbers, we need a working definition of "number", — Metaphysician Undercover
This is jargon... they refer to the same mathematical object.Clearly "2+2", and "4" do not refer to the same "object" by any conventional definition of "object". — Metaphysician Undercover
Anyone who uses the decimal system to count above 9 shouldn't take your pronouncement seriously.Any system of interpretation which ignores the role of "+" within an equation, to claim that "2+2" says the same thing as "4", cannot really be taken seriously. — Metaphysician Undercover
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