Question

2. Write two rational numbers which are their own reciprocals.
- Which rational number is the multiplicative identity for rational numbers ?
Express
as a rational number with denominator.​

Answer 1

Answer:

1 is the only real number (and if I’m not mistaken in my brief moment on this question, the only complex number) that is equal to its multiplicative inverse. This is because 1 is the multiplicative identity in complex space.

EDIT: As someone pointed out in comments (thanks!), this answer was incorrect. -1 is equal to its multiplicative inverse as well. That’s what I get for answering so late at night.

Answer 2

Answer:

That depends which binary operator we are working with. The Rational Numbers[1] ( Q ) form a Group[2] under addition and the Rational Numbers with zero excluded ( Q∖{0} ) form a Group under multiplication[3] .

Unsurprisingly, these have different identity elements. For  (Q,+)  it’s  0 , for  (Q∖{0},×)  it’s  1 .

By definition, the two identity elements are their own inverse’s, i.e.

0+0=0  

1×1=1  

However (as pointed out in a comment) for the multiplicative Group we also have:

−1×−1=1