nonnegative complex numbers

I am reading Axler's LA Done Right and there is this sentence that says a complex number is nonnegative iff it has nonnegative root ( page 225, positive operators) Taking contrapositive , a number is negative iff it has a negative square root but -1 for example doesnt have negative square root yet i think is considered negative .(supposing i am not mistaken)

Ps: there is a quite ,almost actually same, question asked here but i could not find answer to my question there .

Solution 1:

I am reading Axler's LA Done Right and there is this sentence that says a complex number is nonnegative iff it has nonnegative root

Taking contrapositive , a number is negative iff it has a negative square root

I agree with your contrapositive; however, under this interpretation, the given statement is false (as you have shown).

The author's intended interpretation of a "nonnegative complex number", though, is precisely $[0,\infty).$ (As such, the author would consider $3+7i$ as neither negative nor nonnegative.) Under this interpretation, here's the correct contrapositive:

a complex number has a negative real part or nonzero imaginary part iff its square root has a negative real part or nonzero imaginary part.

P.S. While I call $7$ a positive (real) number and a nonnegative (real) number, I think it is bad practice to call it a "positive" complex number or a "nonnegative" complex number.