Why is 1/i equal to −i?

 Why is 1/i equal to −i?

There are multiple ways of writing out a given complex number, or a number in general. Usually we reduce things to the "simplest" terms for display -- saying 0 is a lot cleaner than saying 1−1 for example.

The complex numbers are a field. This means that every non-0 element has a multiplicative inverse, and that inverse is unique.

While 1/i=i−1 is true (pretty much by definition), if we have a value c such that c∗i=1 then c=i−1.

This is because we know that inverses in the complex numbers are unique.

As it happens, (−i)∗i=−(i∗i)=−(−1)=1. So −i=i−1.

As fractions (or powers) are usually considered "less simple" than simple negation, when the calculator displays i−1 it simplifies it to 

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