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No.

It is not even closed. sqrt(3)*sqrt(3) = 3 - which is rational.

Q: Is the set of irrational numbers a group under the operation of multiplication?

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In maths, the term there are two main meanings to the word inverse - both of which are very closely related. Simple answer in the last three paragraphs. A binary operation, defined on a group of numbers is a rule that tells you how to combine two numbers to get a third. Each binary operations (@) has an identity element, generally denoted by i, such that: x@i = x = i@x for all x in the group. Then, for each element x, there is an element in the group, denoted by x-1 (or the inverse of x) such that x@x-1 = i = x-1@x All this may sound rather technical. So here it is in simpler terms: two everyday examples of binary operation are addition and multiplication. The identity for addition is 0. The identity for multiplication is 1. The inverse of x, under addition, is -x. Under multiplication it is 1/x (not defined for x = 0). These give rise to inverse binary operations: subtraction for addition and division for multiplication.

To any set that contains it! It belongs to {sqrt(30)}, or {45, sqrt(30), pi, -3/7}, or irrational numbers, or real numbers between -6 and 6, or all real numbers or complex numbers, etc.

By adding up all the numbers in the group and dividing by the number of numbers in the group.

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All the elements in a group must be invertible with respect to the operation. The element 0, which belongs to the set does not have an inverse wrt multiplication.

I believe it is because 0 does not have an inverse element.

No. The inverses do not belong to the group.

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They make up the Real numbers.

It belongs in the irrational group of numbers.

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Some would say that there is no intersection. However, if the set of irrational numbers is considered as a group then closure requires rationals to be a proper subset of the irrationals.

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Pi is an Irrational number, which is one of the two subcategories of real numbers.

Pi is both an irrational number and a transcendental number.