I Math Myth: The rationals are numbers

[Dale]

Is there a widely accepted official definition of "number". If so then the question of whether or not the rationals are numbers has nothing to do with whether or not they are also an equivalence class. The question of whether or not they are numbers is determined only by whether or not they fit the definition of number.

That said, I don't know the official definition of number. Does anyone else?

 

[mathman]

Is "one" different from "1" or "I"(Roman)?

 

[fresh_42]

Throughout mathematics there are various entities referred to as “numbers”; in modern mathematics it would be more accurate to refer to anyone of various types called “number systems”, and simply define a number to be a term of such a type.

A numeral on the other hand is a syntactic representation of a number, part of a system of numeration.

It is interesting to try to describe the general conditions under which something comes to be designated as a “number”. After all, such number systems tend to form commutative rings or at least commutative rigs, but not all commutative rigs are considered to consist of “numbers”, or at least that is not how the language is used in practice.

The root notion, known to the great ancient civilizations and particularly ancient Hellenic civilization, is that of (the system of) natural numbers, and in some way or other each of the various number systems are extensions of that one.

https://ncatlab.org/nlab/show/number

 

[fresh_42]

Is "one" different from "1" or "I"(Roman)?

Yes. One can see it. In order to be the same, you have to provide context.

 

[dextercioby]

How do you define rational numbers? As a set, of course, but a set of what?

1st definition: A number $p$ is called a rational number iff is an element of this set:

$$\mathbb Q_1 :=\left\{\frac{a}{b}\vert a\in\mathbb Z, b\in \mathbb Z\setminus\{0\}\right\}$$

According to this definition: $\left(\frac{1}{1}=:\right) 1_{\mathbb Q_1} \neq \frac{12}{12}_{\mathbb Q_1}$

2nd definition: Let us endow $\mathbb Q_1$ with an equivalence relation, because there are solid reasons to believe $\mathbb Q_1$ has too many elements for the purpose of "clean" mathematics. For this we exploit the fact that $\mathbb Z$ is closed under multiplication.

$$\forall p,q\in\mathbb Q_1, p\equiv \frac{a}{b}, q\equiv \frac{c}{d}$$

$$ \frac{a}{b}\sim \frac{c}{d} \Leftrightarrow a\cdot d =_{\mathbb Z} b\cdot d$$

From here we simply define:

$$ \mathbb Q_2 := \mathbb Q_1 / \sim $$

So we obviously have that $\left[\frac{1}{1}\right]_{\mathbb Q_2} = \left[\frac{12}{12}\right]_{\mathbb Q_2}$.

So to say simply that $1=12/12$ is true only by ignoring the very definition of $\mathbb Q_2$ by scrambling the definition of $\mathbb Q_1$. What @fresh_42 is saying is that the standard mathematics education system (from elementary 1st grade to the end of high-school) simply uses improperly defined mathematical objects and perhaps even the highest level of mathematics education attained (college/university, before specializing to a PhD) does not properly define "rational numbers".

 

[Dale]

How do you define rational numbers?

First, how do you define numbers?

 

[dextercioby]

As elements of particular sets, i.e. sets with particular properties of their elements.

 

[Office_Shredder]

Why can't we just say the rational numbers are the smallest field of characteristic 0. We know the integers embed into it, and by axioms of a field 1=12/12.

You might ask how we know this field exists (I.e. there exists one that embeds into all others). It's pretty easy to prove your $\mathbb{Q}_2$ is it, so we're done. But I haven't defined the field as an equivalence class, it just exists, contains integers, and let's you divide by integers as well. Then saying $1\neq 12/12$ is like saying $9\neq 3^2$ because they're only equal after you apply evaluation of functions which you haven't done yet.

The construction of the rationals as equivalence classes is a convenient way to prove everything is on the up and up, but most people don't spend any time thinking about Peano's construction of the integers and it doesn't seem to be an issue either. This is like arguing that we need to teach everyone that 2 is the successor of 1, not just, you know, 2.

 

[Dale]

As elements of particular sets, i.e. sets with particular properties of their elements.

By that definition the rationals certainly are numbers.

 

[dextercioby]

By that definition the rationals certainly are numbers.

Sure, that is the definition of $\mathbb Q_1$ in my post. But because of it's void of the equivalence relation, it has "redundant elements", i.e. it has both $\frac{1}{1}$ and $\frac{14}{14}$ as distinct elements.

 

[Office_Shredder]

The set of rational numbers is a set, and 1/1 is an element of it.

The tuple (1,1) you have in $\mathbb{Q}_1$ most people would say is not a number

 

[dextercioby]

So the set of "rational numbers" is properly the set of all "distinct results of division by non-zero in the integers", so you

a) First take $\mathbb Z$ and define the operator $/$ which is applied to any two elements (tuple), with exception of 0 being the second element of the tuple. Then
b) Define a "rational number" as being the element $q := a/b$, so that $q \cdot b \equiv a$ and here the multiplication of $q$ by an integer is defined as a repeated addition/subtraction, just like in $\mathbb Z$.
c) Eliminate all duplicates produced by step b).

This way 1,6 = 8/5 and 1,6 = 16/10 are counted only once, i.e. 8/5 and 16/10 define one number.

 

[Dale]

Sure, that is the definition of $\mathbb Q_1$ in my post. But because of it's void of the equivalence relation, it has "redundant elements", i.e. it has both $\frac{1}{1}$ and $\frac{14}{14}$ as distinct elements.

But regardless of that it is not a myth that the rationals are numbers by that definition.

Of course, with that definition colors are numbers as are animals and many other things I would not normally associate with numbers.

 

[fresh_42]

A funny side note: $1$ and $\dfrac{12}{12}$ are not equal, in the sense, that someone not knowing the context, cannot see any equality. However, they are of equal value, equal valence. Just saying.

Again, I don't think that a discussion about words is or even can be meaningful. A discussion, why we teach algorithms instead of mathematics seems overdue to me.

 

[martinbn]

A discussion, why we teach algorithms instead of mathematics seems overdue to me.

May be a separate thread. A lot can be said in that regard.

 

[jbriggs444]

Sure, that is the definition of $\mathbb Q_1$ in my post. But because of it's void of the equivalence relation, it has "redundant elements", i.e. it has both $\frac{1}{1}$ and $\frac{14}{14}$ as distinct elements.

Ahh. I did not get from that post that $\frac{x}{y}$ was intended as an otherwise uninterpreted notation for an ordered pair of integers.

 

[fresh_42]

If we have a ring $R$ and a multiplicative closed set $1\in S$, then
$$
S^{-1}R = (R\times S)/\sim \text{ where } (r,s)\sim (p,t) \Longleftrightarrow \exists \,u\in S\, : \,(rt-ps)u=0
$$

 

[Dale]

The set of rational numbers is a set, and 1/1 is an element of it.

The tuple (1,1) you have in $\mathbb{Q}_1$ most people would say is not a number

Do you have a definition of “number” that we can use to claim that? I am not a mathematician, but the definition in post 67 seems overly broad to me. By that definition the Q1 rationals are numbers, but so are colors and animals

 

[martinbn]

Do you have a definition of “number” that we can use to claim that? I am not a mathematician, but the definition in post 67 seems overly broad to me. By that definition the Q1 rationals are numbers, but so are colors and animals

I don't think you will get a satisfactory answer, because it is not the case that there is something called a number, and then depending on its properties it gets aditional desripstion, as a natural number. It is more the other way around. You have definitions of natural numbers, complex numbers, Gauss numbers and so on. Then a number is an element of any of those sets.

 

[fresh_42]

Do you have a definition of “number” that we can use to claim that? I am not a mathematician, but the definition in post 67 seems overly broad to me. By that definition the Q1 rationals are numbers, but so are colors and animals

I think the closest you can come is to accept number as a name for what can be counted, the natural numbers. From there on there is a natural way up to the complex numbers, so we call everything in between, plus the complex numbers number.

nLab as quoted in post no. 63 explains the difficulties.

 

[Dale]

I think the closest you can come is to accept number as a name for what can be counted, the natural numbers. From there on there is a natural way up to the complex numbers, so we call everything in between, plus the complex numbers number.

That isn’t really a definition of “number” but by that heuristic then the rationals would be numbers and colors would not. It is reasonably satisfactory.

 

[stevendaryl]

John Conway has given a definition of "number" in terms of two-person games. A game is, set-theoretically, defined as either the empty set ($0$) or a pair $(L, R)$ where L and R are both sets of games. To understand this as a game, you imagine two players, called "Left" and "Right" who alternate turns. On Right's turn, he picks one game out of the set R, and then Left must play that game. A player loses if it's his turn, and the game is the empty set (so he has no next move). So there are 4 types of games. (Assume that each player always makes the best move possible)

1. Left wins, no matter who starts.
2. Right wins, no matter who starts.
3. The first player wins.
4. The second player wins.

Let's call a game "positive" if it is in category 1, "negative" if it is in category 2, and 0 if it is in category 4.

Now, we can obviously flip a game from a win for Left to a win for Right by just switching L and R all the way down. Call that the "negative" of a game. So for example,

If $G = 0$, then $-G = 0$
If $G = (\{ g^L_1, g^L_2 ...\}, \{ g^R_1, g^R_2 ...\})$, then $-G = (\{ -g^R_1, -g^R_2 ...\}, \{ - g^L_1, -g^L_2 ...\})$.

We can describe the "sum" of two games $G_1 + G_2$ as follows: The two players are playing two games in parallel. At every move, the player has a choice of making a move in the first game, or the second. Taking a move in a game means replacing that game by a simpler game, until eventually it becomes the empty game. At the point, only one of the two games is left, so players must continue in that one.

Now, we can say that $G_1 \gt G_2$ if $G_1 + (-G_2)$ is positive.

Finally, we are in a position to define a "number". A number is defined recursively as:
$0$ is a number
$(L, R)$ is a number if for all $g^L \in L$ and $g^R \in R$, $g^L \lt g^R$.

According to this definition,
The empty game corresponds to the number $0$.
The game $(L,R)$ where $L = \{ 0 \}$ and $R = \{\}$ is the number 1.
The game $(L,R)$ where $L = \{0, 1\}$ and $R = \{\}$ is the number 2.
The game $(L,R)$ where $L = \{0, 1, 2\}$ and $R = \{\}$ is the number 3.
Etc.

This notion of number includes almost everything:

The natural numbers.
The integers.
The rational numbers.
The reals.
Transfinite ordinals.
Infinitesimals.

http://www.cs.cmu.edu/afs/cs/academic/class/15859-s05/www/lecture-notes/comb-games-notes.pdf

 

[fresh_42]

That isn’t really a definition of “number” but by that heuristic then the rationals would be numbers and colors would not. It is reasonably satisfactory.

There is simply no mathematical object that is a number. It is a term that belongs to the common language. It requires an additional name to become mathematics: natural number, real number, p-adic number etc. Perhaps you mean cipher when you say number.

 

[Dale]

There is simply no mathematical object that is a number. It is a term that belongs to the common language.

Then it is clearly wrong to claim that the rationals are not numbers. In the common language they are numbers and the term number is a common language term.

 

[fresh_42]

Then it is clearly wrong to claim that the rationals are not numbers. In the common language they are numbers and the term number is a common language term.

By that argument, you have left mathematics. Rationals is short for rational numbers, and with this adjective, they become a mathematical object. How you write them, define them, or otherwise classify them is a different topic. I see them as elements of $(\mathbb{Z}^\times)^{-1}\mathbb{Z}$, others do not want to distinguish the representatives of a given class, which by the way is more than strange:

I bet that everybody who claims that $1=\dfrac{12}{12}$ is also a person who would not accept $\dfrac{12}{12}$ as a correct answer of an exercise, and who would demand to write it in canceled form.

 

[martinbn]

By that argument, you have left mathematics. Rationals is short for rational numbers, and with this adjective, they become a mathematical object. How you write them, define them, or otherwise classify them is a different topic. I see them as elements of $(\mathbb{Z}^\times)^{-1}\mathbb{Z}$, others do not want to distinguish the representatives of a given class, which by the way is more than strange:

I bet that everybody who claims that $1=\dfrac{12}{12}$ is also a person who would not accept $\dfrac{12}{12}$ as a correct answer of an exercise, and who would demand to write it in canceled form.

You lost the bet. I do accept answers like that , and i tell my students that i do accept such answers.

 

[Mark44]

I bet that everybody who claims that $1=\dfrac{12}{12}$ is also a person who would not accept $\dfrac{12}{12}$ as a correct answer of an exercise, and who would demand to write it in canceled form.

You lost the bet. I do accept answers like that , and i tell my students that i do accept such answers.

Same here. I would also accept 12/12 as an answer, as long as the problem wasn't "Simplify the rational number $\frac{12}{12}$."

 

[Dale]

By that argument, you have left mathematics.

Sure, but as you already established the statement “the rationals are numbers” never was a mathematical statement to begin with.

I bet that everybody who claims that $1=\dfrac{12}{12}$ is also a person who would not accept $\dfrac{12}{12}$ as a correct answer of an exercise, and who would demand to write it in canceled form.

I also would accept all of those as an answer. I would even accept $12^0$. I would probably mark $-e^{i\pi}$ wrong but then would give the points back when the student complained

 

[fresh_42]

Sure, but as you already established the statement “the rationals are numbers” never was a mathematical statement to begin with.

Yes, but I also said:

And, yes, I used rhetorical methods, because I wrote a pamphlet and not an article.

But if we really continue to debate on this Wittgenstein level, then let me add:

1. Rationals is short for rational numbers

2. From there [$\mathbb{N}$] on there is a natural way up to the complex numbers, so we call everything in between, plus the complex numbers number.

Hence, despite being hidden behind common language due to the purpose of the text, the used names can be re-translated into mathematics.

It was a headline, not an abstract!

 

[Dale]

It was a headline, not an abstract!

Sure. I have no problem with it being a headline not an abstract. And we can certainly expand it as “the rational numbers are numbers”. That headline is true, hence not a myth.

As you show later they are also an equivalence class and 12/12 and 3/3 are different equivalent elements of that equivalence class. But none of that implies that the headline is in fact a myth. They are both numbers and an equivalence class.

I have no objection to your math whatsoever. Only the headline.