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Imaginary numbers are not intrinsically rational or irrational.

Of course, all real numbers are either rational or Irrational Numbers.

Imaginary numbers are not real numbers.

Imaginary numbers have a real part and an imaginary part, sometimes written like z=x+i y.

The two parts, i.e. the x and the y, are real numbers. As real numbers, they are either rational or irrational. Its just that the two parts of a complex number may both be either rational or irrational or one may be rational and the other irrational. One could always make up a new name for these cases, but right now there is no such classification.

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Q: Are imaginary numbers rational or irrational?
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