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Q: Does the set of rational numbers have a multiplicative inverse?

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All rational numbers, with the exception of zero (0), have a multiplicative inverse. In fact, all real numbers (again, except for zero) have multiplicative inverses, though the inverses of irrational numbers are themselves irrational. Even imaginary numbers have multiplicative inverses (the multiplicative inverse of 5i is -0.2i - as you can see the inverse itself is also imaginary). Even complex numbers (the sum of an imaginary number and a real number) have multiplicative inverses (the inverse of [5i + 2] is [-5i/29 + 2/29] - similar to irrational and imaginary numbers, the inverse of a complex number is itself complex). The onlynumber, in any set of numbers, that does not have a multiplicative inverse is zero.

Yes. The multiplicative identity for the rational numbers is 1 (also can be written as 1/1).

Yes, it is 1.

On the set of all real numbers ZERO has no multiplicative inverse. For other sets there may be other numbers too, so please define your set!

In a set S, the multiplicative inverse of a non-zero element x is an element of the set, y, such that x*y = y*x = i, the identity element of S. For the set of numbers, the multiplicative identity is 1 and the multiplicative identity is also denoted by 1/x or x^-1.

The multiplicative identity is a property of a set of numbers, not of an individual number in the set. 1 is the multiplicative identity for the set of all integers, rationals or reals etc. Individual elements of the set do have a multiplicative INVERSE and for 2, this is 1/2 or 0.5

Yes.

The multiplicative inverse of an element x (in a set S) is an element, y, of the set such that x*y = y*x = 1 where 1 is the multiplicative identity. y is denoted by x^(-1). For the set of numbers, the inverse of x is 1/x.

No. It has a different additive inverses for each element.

Assuming that you mean pi, and not pie, it is not a rational number.The set of rational numbers is a field and this means that for every non-zero rational number, there exists a multiplicative inverse in the setand also, due to closure, the product of any two rational numbers is a rational number.Now suppose 7*pi were rational.7 is rational and so there is its multiplicative inverse, which is (1/7).(1/7) is also rational so (1/7)*(7*pi) is rationalBut by the associative property, this is (1/7*7)*pi = 1*pi = pi.But it has been proven that pi is irrational. Therefore the supposition must be wrong ie 7*pi is not rational.

The multiplicative inverse of a non-zero element, x, in a set, is an element, y, from the set such that x*y = y*x equals the multiplicative identity. The latter is usually denoted by 1 or I and the inverse of x is usually denoted by x-1 or 1/x. y need not be different from x. For example, the multiplicative inverse of 1 is 1, that of -1 is -1.The additive inverse of an element, p, in a set, is an element, q, from the set such that p+q = q+p equals the additive identity. The latter is usually denoted by 0 and the additive inverse of p is denoted by -p.

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