Classifying Numbers in a Set...Part 1

[mathdad]

See attachment for math work.

Thanks.

 

[Dr.Peterson]

Can a real number be neither rational nor irrational?

Which answer is wrong?

 

[mathdad]

Can a real number be neither rational nor irrational?

Which answer is wrong?

I asked my friend the same question. Here is his reply and quote:

"No, a real number cannot be neither rational nor irrational. This is because the definitions of rational and irrational numbers are mutually exclusive amd collectively exhaustive in the set of real numbers."

He went on to say this:

"Let’s break that down:

• Mutually Exclusive: A number cannot be both rational and irrational at the same time. They are distinct categories.
• Collectively Exhaustive: Every real number must fall into one of these two categories: rational or irrational."

Dr. Peterson, you say?

 

[Dr.Peterson]

Why did you have to ask anyone (either human or AI)? Your own diagram shows that every number is one or the other. If a number is not rational, then by definition it is irrational, which means "not-rational".

My point was that one of your answers was wrong, because your lists of rational and irrational omit one number. Where does that number go?

 

[mathdad]

Why did you have to ask anyone (either human or AI)? Your own diagram shows that every number is one or the other. If a number is not rational, then by definition it is irrational, which means "not-rational".

My point was that one of your answers was wrong, because your lists of rational and irrational omit one number. Where does that number go?

I don't ask AI anything. AI is more often wrong than correct. I asked my friend Mark who has a math degree but often too busy to help out.

The decimal number 2.151515. . . is rational and should be on the list with the other rational numbers. Since 15 endlessly repeats, is it correct to place a bar over the first 15 in place of writing 2.151515. . .?

 

[Dr.Peterson]

I don't ask AI anything. AI is more often wrong than correct. I asked my friend Mark who has a math degree but often too busy to help out.

That's right; I mentioned AI only because the style was reminiscent of ChatGPT's style, so it was possible you'd done that.

Actually, it may be correct more than half the time; the problem, as I put it, is that it is better at appearing correct than at being correct, so it can be hard to tell without already knowing something.

The decimal number 2.151515. . . is rational and should be on the list with the other rational numbers.

Correct.

Since 15 endlessly repeats, is it correct to place a bar over the first 15 in place of writing 2.151515. . .?

Yes, [imath]2.151515\dots=2.\overline{15}[/imath].

 

[mathdad]

That's right; I mentioned AI only because the style was reminiscent of ChatGPT's style, so it was possible you'd done that.

Actually, it may be correct more than half the time; the problem, as I put it, is that it is better at appearing correct than at being correct, so it can be hard to tell without already knowing something.

Correct.

Yes, [imath]2.151515\dots=2.\overline{15}[/imath].

Thank you for your reply and for being attentive to my textbook questions.