Yes indeed. There are infinitely many 0 is Pi and others too root2 etc etc
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No. For example, -root(2) + root(2) is zero, which is rational.Note that MOST calculations involving irrational numbers give you an irrational number, but there are a few exceptions.
Let q be a non-zero rational and x be an irrational number.Suppose q*x = p where p is rational. Then x = p/q. Then, since the set of rational numbers is closed under division (by non-zero numbers), p/q is rational. But that means that x is rational, which contradicts x being irrational. Therefore the supposition that q*x is rational must be false ie the product of a non-zero rational and an irrational cannot be rational.
If you multiply an irrational number by ANY non-zero rational number, the result will be irrational.
Such a product is always irrational - unless the rational number happens to be zero.
The product of two rational numbers, as in this example, is always RATIONAL.However, if you mean 10 x pi, pi is irrational; the product of a rational and an irrational number is ALWAYS IRRATIONAL, except for the special case in which the rational number is zero.