Omitting $i$ in calculations

Question

Is it possible in various calculations related to the complex plane which also include analytic geometry , calculating distances etc, to omit $i$ and treat the imaginary axis as simply the cartesian $y$ axis with purely real values like we do regularly with all Real $x,y$ axis?

Answer 1

The short answer is YES. In topology, $\mathbb{C}$ is equivalent to $\mathbb{R}^2$, so everything you do in $\mathbb{C}$, you can also do it in $\mathbb{R}^2$ by equivalent way.

However, there are nice field properties on $\mathbb{C}$, which means you can treat $(a,b)$ as a single number $a+ib$ to do calculation.

In another word, similar to $\mathbb{R}$, $\mathbb{C}$ is a field, but $\mathbb{R}^2$ is a vector space in general. You can transplant operations on $\mathbb{R}$ to $\mathbb{C}$ easily, while opeartions on vector field are limited.

Answer 2

You can treat the complex numbers as the matrices of the form $$\begin{pmatrix}x&-y\\y&x\end{pmatrix}$$

Then complex multiplication and addition corresponds to the related matrix multiplication and addition.

It's not clear what you mean by "treat the imaginary axis as simply the cartesian y axis" when talking about computations. You can certainly define the complex numbers as the ring $(\mathbb R^2, +,\times)$ with the operations defined as $$(x_1,y_1)+(x_2,y_2)=(x_1+x_2,y_1+y_2)$$ and $$(x_1,y_1)\times(x_2,y_2)=(x_1x_2-y_1y_2, x_1y_2+y_1x_2)$$ That is one way that mathematicians tend to rigorously define complex numbers, in fact...

It's a bit of a misnomer to say you are working "without $i$" here. You have just change notation, but you've really just hidden $i$.