Comment by ndriscoll
4 days ago
Assuming you want to be able to make statements like ℕ⊆ℚ⊆ℝ (as one normally does), 3.0 is a whole number, 3 is a real number, and 3.0=3=2.9999999...
Being equal to 3, 2.9999... is also a whole number.
Teaching to use '=' in a statement like '10/3=3' is an example of where teachers don't know math in depth and make errors about details that are actually important/later cause confusion. 10/3 is not equal to 3. '=' doesn't mean "answer". Then not accepting 3.0 which is equal to 3 just layers on that confusion. '=' is transitive. If a=b and b=c, then a=c.
Saying 3.0≠3 is a subtlety you really only get into in math when defining these things, and then you immediately redefine them so that 3.0=3 and you don't have to think about it again.
You're continuing to focus on the functional similarities and functional equivalences of 3, 3.0, 6/2, 2.999... etc. You're right, from an arithmetic standpoint, these are all the same value. I've definitely always agreed with this and fully understand it.
But the question wasn't testing that you knew how to divide and round. The question was testing if you understood what the teacher was trying to teach about whole numbers, integers, rational numbers, real numbers, etc.
6/2 as written is not an integer. It is not a whole number. The value it represents can be written as a whole number, I fully agree, but as written it itself is not a whole number. Whole numbers are the set of numbers Z including -3, -2, -1, 0, 1, 2, 3, 4,... I doubt any math teacher, upon teaching "what is a whole number", draws a number line and proceeds to label it -3.999, -6/2, -12/6, -5/5, 0, 5/5, 12/6, 6/2, 3.999..., and on.
The notation was the key part of the question and was a key part of the answer.
The teacher wasn't looking for a value (which is what you're so focused on looking at), they were looking for a notation, a format.
Then the teacher is teaching something that someone more knowledgeable in the subject will later have to unteach. I'm focusing on functional equivalences because that's how math works as practiced by mathematicians. The functional equivalences are the point, and you may not notice it, but you're also relying on those equivalences, which is why you can write "6/2" in the first place. Integers are already equivalence classes of pairs of natural numbers (which is why 2-3=3-4). Rationals are equivalence classes of pairs of integers (which is why 3/1 = 6/2). If you actually try to define any of this stuff in a coherent way, you're immediately forced to deal with equivalence as a central idea.
6/2 is a whole number. 6/2 = 3. 3 is a whole number. They are equal. Usually, they are the exact same mathematical object. It is not merely that they share properties. They are literally definitionally the exact same thing (the same set in ZFC). "n is a whole number" is a proposition. It is true for n=6/2.
If a teacher is teaching that 6/2 is not an integer, unless they are in the middle of constructing the rationals and need to make a distinction between integers and equivalence classes of pairs of integers, then they are wrong. The very first thing you do after you're forced to make that distinction is you make it go away. They shouldn't be teaching the student to hyperfocus on a specific notation or format. That's a bad lesson to teach, and is something a real teacher will need to fix later. Actual mathematics professors are happy to let you write "let <christmas tree>∈ℝ". An intro proofs professor will definitely put something like "-3.999..., -6/2, -12/6, -5/5, 0, 5/5, 12/6, 6/2, 3.999..." on a number line to illustrate the point that these are just different ways to write the same thing. Fluidity in switching through and following different notations without getting distracted is a centrally important mathematical skill.
> unless they are in the middle of constructing the rationals and need to make a distinction between integers and equivalence classes of pairs of integers
Finally, you're starting to understand the context of the question at hand.
I'm also happy you're starting to show you do understand there's a notational difference between 6/2 and 3. That the values are the same the notation is quite different, thus there are some differences. Not functionally, true, but notationally.
The notational difference was the point of this lesson. You may think it'll only be a barrier in the future to point it out like that (maybe it is!), but the notational difference was the lesson.
> Fluidity in switching through and following different notations
If you don't really have an understanding of the notations, you're going to have a hard time being fluid switching between them.
> An intro proofs professor
An intro proofs professor wasn't leading the lesson, it was probably an elementary or middle school math teacher. The point of the lesson is different, the context of the lesson is different.
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