Imaginary part of the following complex number.

Question

I am completely new to Complex numbers, and having problem with the following sum from my textbook:

If z = –3–i , find Re(z) and Im(z).

I noticed that the above complex number is not in the standard form i.e., a+ib, so I wrote it as (–3)+(–i*1).

My textbook says Im(z) is –1, why wouldn't it be +1? Or is it that I should write the number as (–3)+(i*–1)? But then again, the multiplicative identity is always +1, isn't it?

Answer 1

Your expression "$-i\cdot1$" is not in the required form "$ib$". For that, you would have to rewrite

$$-i\cdot 1=i\cdot(-1)$$

so then you see that $\operatorname{Im} z$ is indeed $-1$.

(Recall that "$-(x\cdot y)$", "$(-x)\cdot y$", and "$x\cdot(-y)$" all mean the same thing).

Answer 2

The imaginary part of a complex number is defined to be the (real) coefficient of the imaginary unit $i.$ In this case the real coefficient of $i$ is clearly $-1.$ Thus, $\Im z=-1,$ as your text claims.