Must irrational numbers be real?

If you think so, how do you reconcile the various definitions of irrational?

If you don’t think so, why do we seem to perpetuate this idea with students that irrationals are composed entirely in the real number system...perhaps not by stating that directly, but by using representations such as the ones below?

This next is an extra credit project for a college teacher prep program ... these students obviously don't know their subject all that well and this "teacher" is no better. "Hands On Math: Burn The Textbooks, Shred The Worksheets, Teach Math." is the blog motto.

This is incorrect? |

Are the visual organizers getting in the way of the understanding?

Source:

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@MathCurmudgeon Here's a possible math argument: Are Complex Numbers Irrational? are-complex-numbers-irrational
```

— Matt Enlow (@CmonMattTHINK) January 30, 2014

Interesting question. I could see it definitely starting a good discussion. If an irrational number is defined as a number (intentionally vague) that is not rational, and every rational number is real, then it would follow that any number that is not real must be irrational. That being said, I don't think the above is a good definition of irrational. It would be like defining an odd number as any number (intentionally vague) that is not even. Since every even number is an integer it would follow that any non-integer is odd. Again probably not a good definition. Personally I would define things so that an odd number must be an integer and an irrational number must be real. But this question brings up the great issue of choices we make in definitions and justifying why we make the choices we do. By the way, any idea what the definition of "imaginary" number is in the first visual representation?

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