Math Myth: The rationals are numbers

[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.

 

[pbuk]

Whether you accept something isn't of any relevance.

Well yes, I can agree with that.

This point of view is one of my criticisms. "Because we always did so, it is right."

That was not what I was trying to say: I was trying to ask what was wrong with the way we always did it. I could equally well characterize your criticisim as 'Because we always did so, it is wrong'.

I gave a definition in post #77, where we set $S=\mathbb{Z}^\times$ and ##R=\mathbb{Z}.

But that is not a definition of the rationals, nor even a set that is bijective with the rationals. To get to the rationals from here I need to eliminate the duplicates with an equivalence relation - why is it better to travel in this direction?

If you consider only quotients, how could you not distinguish $\dfrac{1}{1}$ from $\dfrac{12}{12}?$

I am not sure what you mean here, but I can't distinguish any collections of symbols without context.

 

[fresh_42]

I think simple and go with Kronecker (almost):

"Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk"
("God made the integers, all else is the work of man")

I believe that counting is the only valid base to start with. That gives us the semigroup $\mathbb{N}.$ I don't even consider $0$ a natural number. I think that naming something which isn't there is actually a human achievement, an Indian to be precise. Probably accounting in Babylon brought us the next extension to the additive group of integers $\mathbb{Z}$ which are naturally a ring. We were lucky that they are an integral domain, that allows the next extension to a quotient field $\mathbb{Q}$ in a very easy way. Next came the real numbers, which already require some topology to get there. No wonder that the ancient Greeks spoke of irrational numbers. And they had only the algebraic reals which arise from geometry. It took almost 2000 years to get a reasonable definition of all real numbers. The next and in some sense final step are the complex numbers, which are in my opinion again a set of equivalence classes $\mathbb{C}=\mathbb{R}[x]/(x^2+1).$

 

[Infrared]

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

It looks like $\frac{-2}{3}$ and $\frac{2}{-3}$ are distinct elements here. If so, you would definitely need to take equivalence classes. Or does $a/b$ already refer to the equivalence class, not just the pair $(a,b)$?

 

[jbriggs444]

It looks like $\frac{-2}{3}$ and $\frac{2}{-3}$ are distinct elements here. If so, you would definitely need to take equivalence classes.

One could pick out canonical exemplars from each class, for instance requiring the denominator to be strictly positive.

 

[Infrared]

Right, but I'm just reading what is written.

 

[fresh_42]

So the question for @fresh_42 is: are the elements of $\mathbb{Q}_3$ also equivalence classes?

I think so. It is all hidden in the condition $gcd(a,b)=\pm 1$. How do you handle $\dfrac{12}{12}$ in such a case? It is not part of $\mathbb{Q}_3$ but part of $\mathbb{Q}$. So we must identify it with $\dfrac{1}{1}$ to get $\mathbb{Q}_3$ work. It always comes down to equivalences. E.g. in @pbuk's formula as quotients, the simple need to make it well-defined requires to answer what makes $\dfrac{2}{4}$ equal to $\dfrac{3}{6}.$ And the answer is always: because $2\cdot 6 = 3\cdot 4,$ which is exactly the formal definition as $S^{-1}R.$ The equality sign is a convenience, an abbreviation.

 

[dextercioby]

Yes, you are correct. I did intend integers, so indeed $\mathbb Q_3$ is also "not enough".
So there:

Definition 4

$$\mathbb Q_4 := \left\{\frac{a}{b}\vert a\in\mathbb Z, b\in\mathbb N^{*}, \text{gcd} (a,b) = \{\pm 1\}\right\}$$

This should do it.

 

[jbriggs444]

How do you handle $\dfrac{12}{12}$ in such a case?

That seems to be a question of notation, rather than something with much mathematical meat on its bones.

If we allow $\dfrac{1}{1}$ as a numeral that is literally the rational number 1 then we have removed the notation $\dfrac{1}{1}$ as possibly denoting an expression for the rational number 1 divided by the rational number 1 in the field of rational numbers.

Edit to add:

In practice, the expression evaluates to the same thing as the numeral, so it is a distinction without much of a difference.

So I guess I've argued my way around to your position. We have this equivalence class of expressions which we customarily refer to as all being equal (or equivalent) to one another.

 

[pbuk]

E.g. in @pbuk's formula as quotients, the simple need to make it well-defined requires to answer what makes $\dfrac{2}{4}$ equal to $\dfrac{3}{6}.$ And the answer is always: because $2\cdot 6 = 3\cdot 4,$ which is exactly the formal definition as $S^{-1}R.$ The equality sign is a convenience, an abbreviation.

Are you saying that I must accept a category theory foundation before I can do any maths?

 

[fresh_42]

That seems to be a question of notation, rather than something with much mathematical meat on its bones.

If we allow $\dfrac{1}{1}$ as a numeral that is literally the rational number 1 then we have removed the notation $\dfrac{1}{1}$ as possibly denoting an expression for the rational number 1 divided by the rational number 1 in the field of rational numbers.

Of course, there is always the view of the ancient Greeks, i.e. a rational (sic!) number is the ratio of two lengths. But even this leads to equivalence classes since ratios can be equal even if the lengths are not.

Whichever I look at it, I see these classes. However, I don't want to revolutionize teaching, I only want it to be mentioned. Closer to real mathematics and away from algorithms. I find it embarrassing if people in quiz shows are asked to calculate e.g. $3^3$ and then say, that they were always bad at mathematics. Heck, this ain't mathematics. Make it two classes, calculation and mathematics, but do not pretend it was the same.

 

[fresh_42]

Are you saying that I must accept a category theory foundation before I can do any maths?

The principle of equivalence relations can hardly be called category theory. Relations are important, equivalence relations even more. It does no harm to teach it. 1 hour at most. The bargain, however, is much more if it comes to other examples.

 

[pbuk]

The principle of equivalence relations can hardly be called category theory.

But surely if you elevate equivalence to a higher level than equality then that is where you will end up?

 

[atyy]

Following Bourbaki's discussionm it is likely that even $1 \neq 1$ due to physics, even if we allow equivalence classes of symbols at different positions.

 

[trees and plants]

Following Bourbaki's discussionm it is likely that even $1 \neq 1$ due to physics, even if we allow equivalence classes of symbols at different positions.

Can you show me how $1\neq1$ can hold? As i know equality is an equivalence relation, so the reflexive property holds https://en.wikipedia.org/wiki/Equivalence_relation .

 

[trees and plants]

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.

Could you provide an example or more when saying that people associate a single element with the word number, not a set?