Math Myth: The rationals are numbers

[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.

 

[fresh_42]

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.

Well, I do not want to blame others, but my original headline had been: 10 things we all learned wrong at school.

 

[Dale]

Well, I do not want to blame others, but my original headline had been: 10 things we all learned wrong at school.

But “the rational numbers are numbers” is not wrong. They indeed are numbers and they are also an equivalence class.

 

[fresh_42]

But “the rational numbers are numbers” is not wrong. They indeed are numbers and they are also an equivalence class.

It is wrong in the sense that they are representatives of equivalence classes and as such stand for entire sets, not numbers, because people associate a single element with the word number, not a set.

 

[Greg Bernhardt]

Only the headline.

It was my idea to break them out for dedicated discussion and added the prefix. If there is a more appropriate prefix let me know.

 

[fresh_42]

Many things we call elements are actually equivalence classes: elements in finite fields (modulus), real numbers (Cauchy 0-sequences, Dedekind cuts), complex numbers (factor ring, Riemann sphere). Equality is a very specific equivalence relation, and a rare one, too.

 

[fresh_42]

It was my idea to break them out for dedicated discussion and added the prefix. If there is a more appropriate prefix let me know.

This shouldn't be necessary, because Wittgenstein is a forbidden topic. I intended to provoke the question: Why do biologists teach biology, physicists physics, and chemists chemistry, but mathematicians teach triangles in the sand and calculations, simple algorithms which are counting in my opinion, but certainly not mathematics.

It wasn't intended to make a philosophical discussion about the meaning of words out of it. The subject that matters is: Why do we pretend to teach mathematics if it is actually calculating?

 

[jbriggs444]

This shouldn't be necessary, because Wittgenstein is a forbidden topic. I intended to provoke the question: Why do biologists teach biology, physicists physics, and chemists chemistry, but mathematicians teach triangles in the sand and calculations, simple algorithms which are counting in my opinion, but certainly not mathematics.

It wasn't intended to make a philosophical discussion about the meaning of words out of it. The subject that matters is: Why do we pretend to teach mathematics if it is actually calculating?

Here in the U.S. in the 1960's (when I was in elementary school) there was this thing called the "New Math" where they tried teaching real math. Instead of getting us to memorize our "times tables" and do long multiplication with pencil and paper, they showed us Venn Diagrams and talked about numeric representation with non-decimal bases.

From where I sat in my desk, the whole thing was a waste of time. Every year, we'd spend two weeks at the beginning of the year doing those New-Mathy things. And then we'd be right back doing calculations.

I had a heck of a time memorizing the multiplication tables. I'd keep trying to do arithmetic rather than just spouting the memorized result. [Nine times seven is seven less than ten times seven, so the answer must be 63]. This was decently fast, but not top-of-the-class fast. So my mother, herself an elementary school teacher, drilled me until I'd simply memorized the table instead. Got a decent calculation speed-up out of that].

Finally, I got to college and in my second year took a 400 series course "Advanced Calculus". Turned out to be a course in what I now know to have been real analysis. Did the whole Peano Axiom, construct the real numbers thing. That was the most enjoyment I'd ever had in a math course. So much that had always been pretty obvious was placed on a rigorous footing. [And some stuff that I thought I had grasped had to be re-learned -- the nature of infinite sets, for instance].

 

[Dale]

It is wrong in the sense that they are representatives of equivalence classes and as such stand for entire sets, not numbers, because people associate a single element with the word number, not a set.

I am not sure that people in fact don’t associate the entire set with the number. I mean, if I order a large pizza I don’t demand to see the manager if they give me 8/8 of a large pizza or 12/12 of a large pizza or 6/6 of a large pizza. I recognize the entire set as being one large pizza.

 

[fresh_42]

Here in the U.S. in the 1960's (when I was in elementary school) there was this thing called the "New Math" where they tried teaching real math. Instead of getting us to memorize our "times tables" and do long multiplication with pencil and paper, they showed us Venn Diagrams and talked about numeric representation with non-decimal bases.

From where I sat in my desk, the whole thing was a waste of time. Every year, we'd spend two weeks at the beginning of the year doing those New-Mathy things. And then we'd be right back doing calculations.

Finally, I got to college and in my second year took a 400 series course "Advanced Calculus". Turned out to be a course in what I now know to have been real analysis. Did the whole Peano Axiom, construct the real numbers thing. That was the most enjoyment I'd ever had in a math course. So much that had always been pretty obvious was placed on a rigorous footing. [And some stuff that I thought I had grasped had to be re-learned -- the nature of infinite sets, for instance].

We had this nonsense, too. However, it gave me one of the nicest tutorials I ever had: a grandma who wanted to learn "set theory" to help her grandchildren with their homework.

I admit that this approach was not very well prepared. There is nothing wrong with Venn diagrams, as long as they are taught at an appropriate age and, say, take no longer than maximal a week.

 

[pbuk]

Many things we call elements are actually equivalence classes: elements in finite fields (modulus), real numbers (Cauchy 0-sequences, Dedekind cuts), complex numbers (factor ring, Riemann sphere).

I don't accept this, it is like looking back through the wrong end of the telescope. We define the set of rationals a priori*, we don't define an equivalence class and then say 'this equivalence class has a representative set which we can call $\mathbb Q$'.

Or have I missed something in the last 40 years since I learned this stuff?

*$\mathbb Q: \forall x \in \mathbb Q (\exists a, b \in \mathbb N: x = \frac{a}{b})$

 

[martinbn]

It is wrong in the sense that they are representatives of equivalence classes and as such stand for entire sets, not numbers, because people associate a single element with the word number, not a set.

These sets are elements of another set. The set of equvalent classes. So each of them is a single element.

 

[fresh_42]

Whether you accept something isn't of any relevance. This point of view is one of my criticisms. "Because we always did so, it is right." Your "definition" isn't one. It is not even well-defined. I gave a definition in post #77, where we set $S=\mathbb{Z}^\times$ and $R=\mathbb{Z}.$ If you consider only quotients, how could you not distinguish $\dfrac{1}{1}$ from $\dfrac{12}{12}?$

 

[fresh_42]

These sets are elements of another set. The set of equvalent classes. So each of them is a single element.

All I want is to recognize this fact! We do not teach it to my best knowledge, @Infrared 's experience aside.

 

[dextercioby]

@fresh_42 This is a follow-up to my previous post here. No. 65 from page 3

IIRC, in Romania (I believe 5th grade, i.e. 11-12 y.o.) we were given a 3rd definition of „rational numbers”.

Definition 3

$$\mathbb Q_3 :=\left\{\frac{a}{b}\vert~ a\in\mathbb Z, b\in \mathbb Z\setminus\{0\}, \text{gcd}(a,b) =\{\pm 1\}\right\}$$

So the question for @fresh_42 is: are the elements of $\mathbb Q_3$ also equivalence classes? Cause teaching people about $\mathbb Q_1$ is utterly wrong, if $\mathbb Q_3$ is available.

 

[mathman]

It seems to me the issue is what does "=" mean. Mathematical definition means that $1=\frac{12}{12}$ is a valid statement. As images they are different.