Here's something that the students might easily grasp and that could also be entertaining:
What's "natural" about the natural numbers?
What's "rational" about the rational numbers?
What's "real" about the real numbers?
Every one of these different sets of numbers is a mathematical idealization of something encountered in the "real" world. None of them is ...
I'll take the dissenting opinion: don't use the phrase "Real Number" until you're prepared to teach what an imaginary number is.
"Today in science, we're going to be using an optical microscope to
look at culture slides."
... which immediately begs the question from any remotely curious student:
"Wait, so there are non-optical microscopes?"
And you ...
I recall having been taught different classes of numbers (in maths at school) way before we were introduced to complex numbers. Main reason was to distinguish
finally ... real numbers
It's reasonable to teach students things like the coverage of numbers on the number line:
Why are real numbers continuous while ...
You should not avoid use of the term real numbers. This is a term-of-art in mathematics, and it is important for students to learn the correct jargon. However, this technical term should be introduced as such—emphasize that "real" in mathematics does not mean the same thing that "real" means in everyday vernacular English.
For 95% of high school students, this sort of thing is of no interest. But:
The 5% do need to be served well and helped to achieve their potential.
The 95% may find such things confusing if they are never explained, so it makes sense to offer them at least some brief explanation.
Even the 5% are in no position at this point to understand fully what is meant ...
I wouldn't. You feel the gap because you know what's coming. But qualifying like that before the students have the context leads to bafflement.
It's sort of a routine issue where people who already know the stuff want to present it perfect (complete). But this is not pedagogically sound. Suited to careful math explanation, but not to learning.
Regarding all possible numbers on the continuous number line, in my opinion you're overthinking this. The vast majority of students aren't concerned with what the numbers are called, but how to solve the problems they have for homework and on tests. As for the use of real being unnecessary in the absence of knowing about complex numbers, I disagree. For them,...
In my opinion if some numbers are "imaginary" it doesn't mean they don't exist.
It is needed to distinguish somehow between the two kinds of numbers, the "real" that we easily see im our daily life and that make sense to everyone, and the "imaginary" numbers (or complex in general) that are there in life, real of course with no doubt, but they're not easy ...