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An imaginary number i is defined as the square root of -1, so if you have something like the square root of -2, the answer would be i root 2, and that would be considered an irrational non-real number.

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Not quite. The fact that irrational coefficients can be used, in conjunction with i to create complex numbers (or parts of complex numbers) does not alter the fact that all Irrational Numbers are real numbers.

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Q: Are there irrational numbers that are not real numbers?
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