I Math Myth: The rationals are numbers

[fresh_42]

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

That was a guess. What do you associate if you hear the word number? Something like $7$ or something like $\left\{\dfrac{a}{b}=7\,|\,a\in \mathbb{Z},b\in \mathbb{Z}^\times\right\}?$

 

[trees and plants]

That was a guess. What do you associate if you hear the word number? Something like $7$ or something like $\left\{\dfrac{a}{b}=7\,|\,a\in \mathbb{Z},b\in \mathbb{Z}^\times\right\}?$

You mean the use of the number in non scientific language? In math i think rationals are numbers. But i do not think a definition of a number mathematically is easy to make. There are of course examples of sets with numbers.

Somewhere i read i think that things can be defined by giving examples not necessarily definitions.

Excuse me if i make any mistake.

 

[Dale]

That was a guess. What do you associate if you hear the word number? Something like $7$ or something like $\left\{\dfrac{a}{b}=7\,|\,a\in \mathbb{Z},b\in \mathbb{Z}^\times\right\}?$

I think that people do associate the rationals with the entire equivalence class. They don’t know the term and they wouldn’t be able to recognize it written as you did, but the concept is there. People easily recognize 12/12 of a pizza and 8/8 of a pizza as being distinct from each other but both equivalent to 1 pizza.

 

[stevendaryl]

I think that there is something not quite about right saying "we have to be talking about equivalence classes in order to say $\frac{12}{2} = \frac{1}{1}$. If you think of $\frac{x}{y}$ as an expression denoting the value of a binary function on the arguments $x,y$, then there is no implication that there are any equivalence classes involved. If I say that $sin(\pi) = sin(2\pi)$, that doesn't mean I'm dealing with equivalence classes. It just means that the function $(x,y) \rightarrow \frac{x}{y}$ is not one-to-one.

There is often an ambiguity when we talk about an expression such as $\frac{x}{y}$ whether we're talking about the expression or the value denoted by the expression. When asking whether $\frac{12}{12} = \frac{1}{1}$, we're talking about the denotation. When asking "What is the denominator of $\frac{2}{3}$?" we're talking about the expression.

 

[fresh_42]

I think that people do associate the rationals with the entire equivalence class. They don’t know the term and they wouldn’t be able to recognize it written as you did, but the concept is there. People easily recognize 12/12 of a pizza and 8/8 of a pizza as being distinct from each other but both equivalent to 1 pizza.

People will consider it as equal value (sic!) of pizza, but they will clearly see the difference.

The concept of being equivalent isn't too hard to teach students. I simply do not see, why we insist to introduce it as equality. Teach it right and add: "From now one we will write $1=\dfrac{12}{12}$ because we are only interested in values, not in representations. That's why we write $=$ and not $\equiv,$ since nobody wants to count lines.

Such an approach is only marginally more work than what is taught anyway, but it would be closer to math and farther from calculations.

Edit: Nobody performs the double slit experiment and avoids the word interference.

 

[stevendaryl]

The concept of being equivalent isn't too hard to teach students. I simply do not see, why we insist to introduce it as equality. Teach it right and add: "From now one we will write $1=\dfrac{12}{12}$ because we are only interested in values, not in representations.

Well, if we used $=$ only for representations, and not values, then we would have very little occasion to use it. You could write 1=1, 2=2, 3=3, 1/2 = 1/2, but that's very boring mathematics.

 

[atyy]

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 .

Generically one would expect $a \neq b$ since $a$ and $b$ are different. $1\neq1$ is simply the same principle. The two $1$s which appear identical to you are with high probability not identical, since they are not made of exactly the same number of atoms. Or if you are reading on a computer screen, there are also very likely non-uniformities in the display. Thus microscopically $1\neq1$.

Basically, symbols that may appear identical to you are not physically identical.

 

[dextercioby]

I think it's all getting muddy now.

[...]Teach it right and add: "From now one we will write $1=\dfrac{12}{12}$ because we are only interested in values, not in representations. That's why we write $=$ and not $\equiv,$ since nobody wants to count lines.
[...]

Let me add my comment on this part. I do not know if I am right, from the perspective of the most advanced mathematics, but there is a clear distinction between $=$ and $\equiv$.

$$ 3+4 = 7 \tag{1} $$ simply means that the result of the internal operation on $\mathbb N$ for certain chosen elements is identical (completely replaceable) with another particular element of $\mathbb N$ (this would be your "we are only interested in values"),

while

$$ \sin (x+y) \equiv \sin x \cos y + \sin y \cos x \tag{2} $$

means that for whatever elements of $\mathbb R$, the exact element (the value) in $\mathbb R$ in the LHS is identical (we would shift to using the same sign as in the previous paragraph) to the element in the RHS. For $(1)$ we reserve the terminology "equality", while for the second, the terminology "identity".

 

[Infrared]

@PeroK I don't think I really see the distinction that you're making.

In (1), you have a sum of two natural numbers is equal to a third. In (2), you have that a sum of two functions (of two variables) is equal to a third. Besides the addition taking place in two different monoids ($\mathbb{N}$ versus $\mathbb{R}^{\mathbb{R}\times\mathbb{R}}$), what is the key distinction?

 

[atyy]

Myth: Natural numbers are numbers. They are NOT numbers. They are sets!

 

[jbriggs444]

Myth: Natural numbers are numbers. They are NOT numbers. They are sets!

No, no. It's both a floor wax and a dessert topping.

[And it's less filling and still tastes great].

 

[PeroK]

It's certainly true that $1 == \frac{12}{12}$, as we can see from this Python code:

if 1 == 12/12:
print("Fresh 42 is wrong!")
else:
print("Fresh 42 is right!")

 

[WWGD]

We may have to leave it up to Bill Clinton: " It depends on what 'is' is."

 

[mathman]

This thread is absurd!

 

[ShayanJ]

The way I see it, every little detail in mathematics is decided based on the question "so what?", "what does it lead to?". And I don't just mean something that is useful in engineering, physics, etc. I'm also including those aspects that are just in there so that mathematicians can explore more abstract and sophisticated ideas.
So this is how I judge this question. Yeah, that maybe true technically, but what does it lead to? How does it change anything? Why should we care?
I think this is how it should be decided. If someone can come up with a situation in which this actually makes a difference in some theorem or mathematical procedure, then I'm ready to accept it.
Otherwise it might be the first case of "too pedantic even for a pure mathematician" that I have ever seen!

 

[Office_Shredder]

This isn't too pedantic for a pure mathematician, defining the field of fractions for an integral domain is pretty standard undergrad/intro graduate course stuff.

 

[mathman]

The original statement is absurd. "numbers" is a word to defined. In all mathematical discourse rationals fit the definition.

 

[weirdoguy]

The original statement is absurd.

Most of these myths are absurd in one way or another, but I guess it was intended to be a little bit controversial.

 

[S.G. Janssens]

I am afraid the subtleties that the original statement tries to address are already obvious to mathematicians, while in its current form it probably mostly confuses readers that have not seen such subtleties before. Either way, the point is missed.

 

[pinball1970]

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 $\dfrac{12}{12}?$

I have read some of the thread but not every single post so this may have cropped up.

12/12 is not equal to one has been discussed

How about the identically equal sign? one extra bar? Can that not distinguish?

So 12/12 =1 but 12/12 is identically equal to 12/12?

12/12≡12/12?

 

[trees and plants]

How can you prove that $\frac{12}{12}\neq 1$? Show me a proof if you know.What is your definition of equality?

 

[jbriggs444]

The sets are not equal

In mathematics, a set is a collection of distinct elements.[1][2][3] The elements that make up a set can be any kind of things: people, letters of the alphabet, numbers, points in space, lines, other geometrical shapes, variables, or even other sets.[4] Two sets are equal if and only if they have precisely the same elements.[5]

12/12 has 12 elements 1 has 1, could you use that?

I thought that 12/12 had infinitely many elements, each element being an ordered pair. Each ordered pair would naturally be equivalent to every other ordered pair in the set under our standard equivalence relation for the rationals. An ordered pair, following the Kuratowski definition would be a two-element set {a,{a,b}}. In the case of this particular equivalence class, each ordered pair would collapse to {a,{a}}.

So the infinite set would be something like { {1,{1}}, {2,{2}}, {3,{3}}, ... {12,{12}}, ... }

We would likely adopt the notation where 0 = {}, 1 = {0}, 2 = {0,1}, etc. So, for instance, 12 would be a twelve element set.

Of course, this is assuming that "12/12" is to be interpreted as a ratio of naturals. If it is to be a ratio of signed integers, there are additional layers to the construction.

Meanwhile, we all understand that none of this folderol is relevant to what someone means when they write 12/12=1.

 

[Mark44]

The thread seems to have run its course, and is now closed.