In the complex plane, is the imaginary axis defined by $ \{\text{Im}(z): z\in\mathbb{C}\} $ (in which case it only involves the coefficient of $ i $), or is it more along the lines of $ \{z\in\mathbb{C} : \text{Re}(z) = 0\} $ (in which case it includes $ i $ along with its coefficient)? I know that it represents the latter, but I thought maybe the diagram itself suppresses the $ i $ as a way of representing it implicitly (and many texts seem to do this).
So in other words, can you label the imaginary axis as $ \dots, -3i, -2i, -i, 0, i, 2i, 3i, \dots $ as you go along it, or do you always exclude the $ i $ next to each number? Or is there no universal convention, kind of like how each text differs on whether or not $ 0 $ is included in the set $ \mathbb{N} $?
Thanks in advance.
Best Answer
The presentation of the Complex plane in diagrammatic form is a matter of convention, since there is a 1-1 correspondence between $\mathbb{C}$ and $\mathbb{R^{2}}$. However the two forms that are given in the question do correspond to different conventions and uses in the educational context:
When the x and y axes are labelled $Re(z)$ and $Im(z)$ the coordinates must be just the coefficients. This is the usual form of presenting a diagram in the complex plane to mathematically experienced users.
When the diagram is being used to geometrically explain complex number properties themselves, then 1, $i$, $-i$, -1 etc will need to be placed on the diagram itself. So when initially explaining rotation and complex numbers clarity is increased when $i$ is explicit in the diagram. For example in explaining geometrically why $\sqrt{i} = +\frac{1}{\sqrt2}(1 + i)$ and $-\frac{1}{\sqrt2}(1 + i)$ it would be best to have $i$ on the diagram.