They are both irrational.
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Yes, irrational numbers are neither odd nor even.
No. It can be odd or even, or a rational number, or irrational.
They have only one thing in common. By adding 1 to either an even number or an odd number makes them opposite. (even becomes odd and odd becomes even).
Every integer (whole numbers, including positives, negatives, and zero) is either odd or even. No number is both odd and even. Other numbers, such as fractions that don't reduce to an integer, irrational numbers, and pure imaginary numbers are neither odd nor even. Zero is an even number and is not an odd number.
No. In fact, no odd integer is irrational. They are all rational.