Math Myth: The rationals are numbers

[Greg Bernhardt]

From @fresh_42's Insight
https://www.physicsforums.com/insights/10-math-things-we-all-learnt-wrong-at-school/

Please discuss!

They are not. They are equivalence classes. My favorite example is, that it makes a huge difference whether you carry home a pie from the bakery or $12/12$ pieces of a pie. The amount of pie and the prizes would be the same, their appearance is not. Of course, we treat $1=\frac{12}{12}$ the same because we are interested in its value, however, they are only equal because $1\cdot 12 = 12 \cdot 1.$ It becomes clearer in its general form:

$$\dfrac{a}{b}\sim\dfrac{c}{d}\Longleftrightarrow a\cdot d= b\cdot c$$

'$\sim$' is strictly speaking an equivalence relation. It gathers really many quotients under one name

$$1=\left\{1,\dfrac{2}{2},\dfrac{-3}{-3},\dfrac{12}{12},\ldots\right\}$$

and the same is true for all other quotients. We take them as the same and write '$=$' instead of '$\sim$' because we are only interested in their values. But $1\neq \dfrac{12}{12}.$ You can literally see that it is different: $5$ symbols instead of $1.$

 

[PeroK]

This, IMO, confuses the mechanics of constructing the rational numbers with the finished product. It's a bit like saying that a car is not a car but a collection of parts. It may be a collection of parts, but it's still a car and that's what a car is.

There should be no fundamental problem that one thing can be represented in two different ways. You could define a function such as $f(x) = \cos^2 x + \sin^2 x$ or as $f(x) = 1$. There's no need to say that either is not a function because you can use two different sets of symbols to represent the same thing.

 

[Paul Colby]

Seems the same might be said of addition and subtraction rather than multiplication and division. $3=1+2=1+1+1$ so integers aren't numbers either?

 

[S.G. Janssens]

It's good to be aware of these things, but two things that may be confusing: What are the "numbers" in the title? And what is the domain and co-domain of the equivalence relation? It seems to me that the former confusion may be avoided by answering the second question.

 

[fresh_42]

This, IMO, confuses the mechanics of constructing the rational numbers with the finished product. It's a bit like saying that a car is not a car but a collection of parts. It may be a collection of parts, but it's still a car and that's what a car is.

Of course, it's debatable and I have had this discussion. However, it is a matter of one's perspective rather than a right or wrong question. E.g. I have the quotient building in rings via multiplicative subsets in mind, in which case the rational numbers or any quotient field of an integral domain are just examples. It makes apparently a difference whether you look from below, $\mathbb{Z}$, or from above, $\mathbb{R}$.

The comparison with the car doesn't reflect this background.

And the complex numbers are equivalence classes, too, namely $p(x)+q(x)\cdot (x^2+1)$

 

[PeroK]

Of course, it's debatable and I have had this discussion. However, it is a matter of one's perspective rather than a right or wrong question.

The title of your Insight is "things that we learned wrong at school".

 

[fresh_42]

The title of your Insight is "things that we learned wrong at school".

Well, the syllabus at school is $\mathbb{N} \longrightarrow \mathbb{Z} \longrightarrow \mathbb{Q}$, ergo from below, ergo equivalence classes.

 

[The Fez]

And the winner of the overly pedantic award goes to... Fresh_42! You can't just take a word that's been used for thousands of years, like "number" and arbitrarily apply rigorous mathematical theory. I look forward to your Biology thread: MYTH: Almonds are walnuts are nuts.

 

[martinbn]

Of course, it's debatable and I have had this discussion. However, it is a matter of one's perspective rather than a right or wrong question. E.g. I have the quotient building in rings via multiplicative subsets in mind, in which case the rational numbers or any quotient field of an integral domain are just examples. It makes apparently a difference whether you look from below, $\mathbb{Z}$, or from above, $\mathbb{R}$.

The comparison with the car doesn't reflect this background.

And the complex numbers are equivalence classes, too, namely $p(x)+q(x)\cdot (x^2+1)$

The point is that whatever the constraction is, then a rational number is an element of that set. It seems that you reserve the word "number" for something tha excludes the reational numbers. Why?

 

[fresh_42]

The point is that whatever the constraction is, then a rational number is an element of that set. It seems that you reserve the word "number" for something tha excludes the reational numbers. Why?

I am only saying that $1\neq \dfrac{12}{12}.$

 

[martinbn]

I am only saying that $1\neq \dfrac{12}{12}.$

That is a strange thing to say!

 

[fresh_42]

That is a strange thing to say!

Because we are used to something doesn't make it right.

 

[martinbn]

Because we are used to something doesn't make it right.

Why isn't it right? Each side of $1=\frac{12}{12}$ stands for the same equivalence class!

 

[fresh_42]

Why isn't it right? Each side of $1=\frac{12}{12}$ stands for the same equivalence class!

Because equality is always an equivalence relation, but an equivalence relation doesn't have to be equality.

 

[jbriggs444]

I am only saying that $1\neq \dfrac{12}{12}.$

I think that most of us who have seen the constructions for the signed integers, the rationals, the reals, etc have at least briefly accepted the prejudice that the rationals "are" equivalence classes of ordered pairs...

But there is nothing in the way we ordinarily use the set of rational numbers that requires that its elements be particular fancy objects. Any set with appropriately many distinguishable objects will do. The structure is the thing. Not the nature of the elements.

If we choose to replace $1_\text{rational}$ within the set of rational numbers with the signed integer $1_\text{signed_integer}$ or with natural number $1_\text{natural_number}$ then that is fine. Then we could say that $1 = \frac{12}{12}$ without having to quibble about having overloaded the numeric literal 1.

 

[martinbn]

Because equality is always an equivalence relation, but an equivalence relation doesn't have to be equality.

It doesn't have to be in general, but here it is, because that is what "=" means for rational numbers.

 

[fresh_42]

I think that most of us who have seen the constructions for the signed integers, the rationals, the reals, etc have at least briefly accepted the prejudice that the rationals "are" equivalence classes of ordered pairs...

But there is nothing in the way we ordinarily use the set of rational numbers that requires that its elements be particular fancy objects. Any set with appropriately many distinguishable objects will do. The structure is the thing. Not the nature of the elements.

If we choose to replace $1_\text{rational}$ within the set of rational numbers with the signed integer $1_\text{signed_integer}$ or with natural number $1_\text{natural_number}$ then that is fine. And then we could say that $1 = \frac{12}{12}$ without having to quibble about having overloaded the numeric literal 1.

Sure, but one has to read it all:

So before you get excited or even angry about what is to come, please keep in mind to take it with a big grain of salt and try to feel entertained, not schooled.

Source https://www.physicsforums.com/insights/10-math-things-we-all-learnt-wrong-at-school/

And if it had been done right, then I wouldn't have to write this post 5 minutes ago:
https://www.physicsforums.com/threads/sets-being-equal-and-equivalent.1002451/#post-6485347

Science is growing at a speed nowadays, that it is already impossible to keep up to date in a single science. Especially in mathematics, there are so many branches and special constructions. It is impossible to know even all of them under the subject "abstract algebra". Yet, we still teach mathematics at school as we did a century ago. The gap between school and science grows by the day, the taught subjects do not. The didactic concept

• 1st year: 2-3 ? not allowed
• second year: 2-3 = -1
• third year: 4/10 ? cannot be done
• fourth year: 4/10 = 0.4
• etc.
• university: in order to define a real number, we first look at Cauchy 0-sequences ...

of impossibilities which later on turn out to be possible, if not even necessary, is an insult to kids' intelligence.

 

[martinbn]

Sure, but one has to read it all:And if it had been done right, then I wouldn't have to write this post 5 minutes ago:
https://www.physicsforums.com/threads/sets-being-equal-and-equivalent.1002451/#post-6485347

Science is growing at a speed nowadays, that it is already impossible to keep up to date in a single science. Especially in mathematics, there are so many branches and special constructions. It is impossible to know even all of them under the subject "abstract algebra". Yet, we still teach mathematics at school as we did a century ago. The gap between school and science grows by the day, the taught subjects do not. The didactic concept

• 1st year: 2-3 ? not allowed
• second year: 2-3 = -1
• third year: 4/10 ? cannot be done
• fourth year: 4/10 = 0.4
• etc.
• university: in order to define a real number, we first look at Cauchy 0-sequences ...

of impossibilities which later on turn out to be possible, if not even necessary, is an insult to kids' intelligence.

That is an over-generalization. I wasn't thought that way. For example my teachers insisted that the solution to the problem $x^2+2x+2=0$ is "The equation has no real solutions." Similarly for all the bullet points.

 

[Mark44]

I am only saying that $1\neq \dfrac{12}{12}.$

I don't buy the whole premise in this thread. 1 and $\frac{12}{12}$ are located at exactly the same point on the number line. Your example in post #1 of 12 slices of a pie that is cut up into twelfths is of course different with the stipulation that we're talking about a physical object divided into equal parts, but without that context, no reasonable person would insist that 1 and $\frac{12}{12}$ are different. To do so is just being pedantic.

 

[Paul Colby]

Okay, I think I see the whole $(1,1)\ne (12,12)$ as truth for ordered pairs. Even if treating rationals as equivalence classes over such ordered pair is a slick as snot approach, is it an essential one?

 

[vela]

• 1st year: 2-3 ? not allowed
• second year: 2-3 = -1
• third year: 4/10 ? cannot be done
• fourth year: 4/10 = 0.4
• etc.
• university: in order to define a real number, we first look at Cauchy 0-sequences ...

of impossibilities which later on turn out to be possible, if not even necessary, is an insult to kids' intelligence.

I think negative numbers aren't usually taught until fifth or sixth grade. And they wouldn't say 4/10 can't be done, but the result would be a quotient of 0 and a remainder of 4.

That is an over-generalization. I wasn't thought that way. For example my teachers insisted that the solution to the problem $x^2+2x+2=0$ is "The equation has no real solutions." Similarly for all the bullet points.

I don't think the problem is uncommon, though, especially in the early grades. When my first-grade teacher brought up subtraction, several classmates loudly declared "you can't subtract a bigger number from a smaller one," and I recall the teacher agreeing. The kid sitting next to me was surprised when I told him that that wasn't really true. My dad had taught me I couldn't solve a problem like 2-3 yet but I would be able to once I learned about negative numbers. I suspect many parents and teachers don't really recognize the difference between "can't be done" and "don't know enough to solve."

 

[fresh_42]

Okay, I think I see the whole $(1,1)\ne (12,12)$ as truth for ordered pairs. Even if treating rationals as equivalence classes over such ordered pair is a slick as snot approach, is it an essential one?

It is the question, whether you are interested in mathematics, or just in calculations. It is how they naturally evolve, namely as quotient field of the integral domain of integers. Such an approach would introduce the general and important concept of equivalence relations, and it comes at zero costs.

The deliberate decision to hide these contexts is in my opinion inappropriate paternalism. I'm no friend of the concept that others decide what is too difficult for me, or what I have to learn and whatnot. It is a personal offense in my mind, a matter of personal freedom. Teach it right and carry on with what you intended to do. It takes at most one hour.

 

[Office_Shredder]

It is the question, whether you are interested in mathematics, or just in calculations. It is how they naturally evolve, namely as quotient field of the integral domain of integers. Such an approach would introduce the general and important concept of equivalence relations, and it comes at zero costs.

Important is a bit of a stretch. What fraction of adults care about this distinction for either their professional career or major hobbies? 0.01%? Fewer?

Lots of people really do just care about the calculations. You can decry the lack of deeper understanding, but the average engineer barely understands their own field to begin with, let alone getting a minor in pure mathematics. I would rather they spend their brainpower on more important things.

I suspect what you really wish was just that people were smarter. I don't disagree that elementary school education could be improved, but this doesn't seem like the low hanging fruit.

 

[Paul Colby]

The deliberate decision to hide these contexts is in my opinion inappropriate paternalism. I'm no friend of the concept that others decide what is too difficult for me, or what I have to learn and whatnot. It is a personal offense in my mind, a matter of personal freedom. Teach it right and carry on with what you intended to do. It takes at most one hour.

Well, from my own personal experience, I couldn't agree more. I recall being that bewildered teenager being asked to factor silly seemingly uninteresting polynomials for no apparent reason. I very likely would have responded to a deeper mathematical discussion than I was provided at the time.

On the flip side, how many in the class would have tuned out even further after a more mathematically honest discussion of algebra? I'm not sure this should even be the concern that it is. Many of those that just learned the steps likely couldn't reproduce them now anyway.

It is the question, whether you are interested in mathematics, or just in calculations.

Unclear it's that black or white.

 

[fresh_42]

Important is a bit of a stretch. What fraction of adults care about this distinction for either their professional career or major hobbies? 0.01%? Fewer?

This isn't a valid argument in my mind. The gap between school and scientific research becomes larger and larger, yet, we don't provide a good base of knowledge. And do you think I ever actually used 2,6,8-trihydroxypurin, endoplasmic reticulum, or even Ohm's law actively in life? If such an argument was valid, then calculation, read, and write would be enough to learn. Our teenagers even ignore the physics they learned and seemingly forgot when driving cars.

 

[Office_Shredder]

Google suggests there are literally 72x as many people who have the title electrical engineer as have the title mathematician, and I bet ohm's law is more useful to people who are not electrical engineers in general than knowing the rationals can be defined as an equivalence relation is useful to people who are not.

 

[fresh_42]

Google suggests there are literally 72x as many people who have the title electrical engineer as have the title mathematician, and I bet ohm's law is more useful to people who are not electrical engineers in general than knowing the rationals can be defined as an equivalence relation is useful to people who are not.

This was only an example, and yes, physics is naturally very much closer to real-life than other fields. I could have chosen an example from optics, but I wear glasses. And I'm very happy with Ohm's law. I use it all the time. I'm not defending the equivalence classes, by all means, I'm only saying that it would be easy to do so. School stuff is usually above what people actually need. If that wasn't so, why don't we teach history much more than we do? This would really be of great advantage in real life. What is Shakespeare good for?

 

[Mark44]

Important is a bit of a stretch. What fraction of adults care about this distinction for either their professional career or major hobbies? 0.01%? Fewer?

This isn't a valid argument in my mind. The gap between school and scientific research becomes larger and larger, yet, we don't provide a good base of knowledge.

Office_Shredder's argument seems valid to me. If 99.99% of people have no use for some concept, how can you conclude that the concept is truly important? If you believe that it's crucial for people to believe that $\frac {12}{12} \ne 1$, because this is somehow "important," I'm afraid you're on a Don Quixote-esque quest and tilting at windmills.

 

[lavinia]

I think that most of us who have seen the constructions for the signed integers, the rationals, the reals, etc have at least briefly accepted the prejudice that the rationals "are" equivalence classes of ordered pairs...

But there is nothing in the way we ordinarily use the set of rational numbers that requires that its elements be particular fancy objects. Any set with appropriately many distinguishable objects will do. The structure is the thing. Not the nature of the elements.

If we choose to replace $1_\text{rational}$ within the set of rational numbers with the signed integer $1_\text{signed_integer}$ or with natural number $1_\text{natural_number}$ then that is fine. Then we could say that $1 = \frac{12}{12}$ without having to quibble about having overloaded the numeric literal 1.

i think the case of the rationals can obscure the pedagogical point because one can think of division as a geometric operation for instance by using a ruler and compass.

But the idea of the rationals as equivalence classes is purely algebraic. And this is the point. And this is not trivial or pedantic. The method of equivalent classes is far more general than geometric methods of dividing whole numbers and applies in situations where there is nothing else to do.

Suppose one has a purely algebraic system, elements with a commutative law of addition, an identity element, and a commutative law of multiplication that distributes over addition. One might ask whether there is an algebraic notion of fractions of these numbers even though there is no intrinsic notion of division.

This leads to a problem. Suppose the number system has zero divisors i.e. there are two elements a and b such that ab=0. For instance, in the integers mod 6, 2⋅3 = 0. Then 1/1 = a/a=b/b = (a/a)(b/b) = ab/ab=0/0 so 0/0 equals 1/1. Then for any ratio c/d= (c/d)(0/0) = 0/0 = 1/1. So if one wants to have multiplication there can only be one equivalence class and the attempt fails.

If there are no zero divisors i.e the number system is an integral domain, then one does get a mutiplication on the equivalence classes and in fact, one gets a field just as with the rationals. In algebra this is called the field of fractions of the integral domain.

For example, the ring of polynomials in one variable over the complex numbers is an integral domain and its field of fractions is all formal ratios of these polynomials. Interestingly, one can not think of a ratio of polynomials in general as a ratio of two functions since if the denominator is not a constant, it will have a root.
Still one can divide any two polynomials in the field of fractions.

Note also that 1 is not equal to a/a where a is some element of the integral domain (as @fresh_42 rightly emphasizes) since 1 is not even an element of the field of fractions. One can only say that the equivalence class of (1,1) equals the equivalence class of (a.a).

 

[fresh_42]

Office_Shredder's argument seems valid to me. If 99.99% of people have no use for some concept, how can you conclude that the concept is truly important?

I'm sure the same can be said about many subjects taught at school. I mentioned a few above. These equivalence classes are certainly an exaggeration. But they stand for an attitude. An attitude that prevents kids to learn more elaborated stuff. Other STEM fields are less concerned. If I remember what I had to learn in biology and chemistry, then those equivalence classes are peanuts in comparison. They taught real science. At least here, but not in mathematics. Mathematics is still mostly calculation (and few triangles).

 

[Mark44]

I am only saying that $1\neq \dfrac{12}{12}.$

IMO, it's sophistry to say that they are unequal.

 

[fresh_42]

IMO, it's sophistry to say that they are unequal.

I think it challenges to think about the use of symbols in general, and specifically equality. It can be viewed as a door opener to abstract algebra or the theory of formal languages, or just logic. It is a provocation, not sophistry, because it asks for more information. $1=\dfrac{12}{12}$ does only teach how to cancel quotients.

 

[mathman]

From @fresh_42's Insight
https://www.physicsforums.com/insights/10-math-things-we-all-learnt-wrong-at-school/

Please discuss!

They are not. They are equivalence classes. My favorite example is, that it makes a huge difference whether you carry home a pie from the bakery or $12/12$ pieces of a pie. The amount of pie and the prizes would be the same, their appearance is not. Of course, we treat $1=\frac{12}{12}$ the same because we are interested in its value, however, they are only equal because $1\cdot 12 = 12 \cdot 1.$ It becomes clearer in its general form:

$$\dfrac{a}{b}\sim\dfrac{c}{d}\Longleftrightarrow a\cdot d= b\cdot c$$

'$\sim$' is strictly speaking an equivalence relation. It gathers really many quotients under one name

$$1=\left\{1,\dfrac{2}{2},\dfrac{-3}{-3},\dfrac{12}{12},\ldots\right\}$$

and the same is true for all other quotients. We take them as the same and write '$=$' instead of '$\sim$' because we are only interested in their values. But $1\neq \dfrac{12}{12}.$ You can literally see that it is different: $5$ symbols instead of $1.$

$1=\frac{12}{12}$ is mathematics. 1 pizza not equal to 12 slices is not.

 

[martinbn]

I still don't get it. Why is it wrong to say that 1=12/12!?

 

[fresh_42]

I still don't get it. Why is it wrong to say that 1=12/12!?

It is as right or wrong as it is to say $-2=5$. But in that case we write $-2=5\;(7)$ or $-2=5\mod 7$ or $-2\equiv 5$. As soon as we are in the quotient field of the integers, we do not distinguish between representatives anymore. E.g. we could write $[1]=\left[\dfrac{12}{12}\right]$ or even $1=\dfrac{12}{12}$ after we said it is an abbreviation.