There are mathematical concepts that mathematicians call "imaginary numbers" these are a multiple of the square root of minus 1. Infinity is not a real number either.
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Yes, all natural numbers are real numbers. Natural numbers are a subset of real numbers, so not all real numbers are natural numbers.
All rational numbers are real numbers.
Rational numbers form a proper subset of real numbers. So all rational numbers are real numbers but all real numbers are not rational.
All irrational numbers are real, but not all real numbers are irrational.
The result is all real numbers.