Infinitely many.
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Next to any rational number is an irrational number, but next to an irrational number can be either a rational number or an irrational number, but it is infinitely more likely to be an irrational number (as between any two rational numbers are an infinity of irrational numbers).
When added to a rational number, any irrational number will produce an irrational number.also, when added to an irrational number, any rational number will produce an irrational number.
The product of 0 and an irrational is 0 (a rational), the product of a non-zero rational and any irrational is always irrational.
Yes. Any irrational number can be divided by itself to produce 1, which is a rational number.
Rational