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# Set of non zero positive irrational number under complex multiplication for

Set of non zero positive irrational number under complex multiplication form group?

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There are four properties required for a set with an operation to be a group. If SS is the set and ∗∗ is the operation, then these are the properties:

• For all a,bin S, a∗bis also in S. (closure)
• For all a,b,c in S, (a∗b)∗c=a∗(b∗c). (associativity)
• 1 is in S, where, for all a in S, 1∗a=a∗1= a. (identity)
• For all a in S, some a−1 is in S, such that a∗a^−1=a^−1∗a=1. (inverse)

The set of irrational numbers is not closed under addition or multiplication: for instance,

- √2+2=0

(-√2)( √2)=-2

The real numbers are a group, if addition is used as the operation. (Only the nonzero real numbers with multiplication are a group.)

This proves that the irrational numbers are not closed but that the real numbers are a group,half the elements of a commutative group (like the real numbers with addition) will not be a closed subset, unless it it is the whole group.