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Q)

Set of non zero positive irrational number under complex multiplication for

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

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.