Why can complex numbers and vectors be written in form of $a+bi$?

Question

Why can complex numbers and vectors be written in form of $a+bi$, when you can't do the same with coordinates and lines? Coordinates have to be written in form of ($x,y$), and lines as $y=mx+b$.

How is writing complex numbers and vectors in form of $a+bi$ compatible with the other rules of math? You can't write a coordinate or a line on a number plane like the above!

By the way, can the people answering this question try their best to not use any complicated math symbols and notation? I'm still a beginner and have trouble understanding them.

Answer 1

This is not the most mathematically rigorous answer, but perhaps it will help with understanding.

The complex numbers are a two dimensional real vector space. We have the real axis a.k.a. the $x$-axis and the imaginary axis a.k.a. the $y$-axis. The point is that we have a way of thinking of a complex number $a+bi$ as a pair of real numbers $(a,b)$ and vice versa. Part of your confusion might be coming from the fact that the complex numbers are also a one dimensional complex vector space.

As far as compatibility with ''the rules of math," there are a lot of different possible rules. Generally, addition of vectors will usually look like $(a,b)+(c,d)=(a+c,b+d)$. This behaves well with how we add complex numbers. Multiplication is the subtler point. We have $$(a+bi)(c+di)=ac-bd+(ad+bc)i.$$ Likewise, we could define $$(a,b)(c,d)=(ac-bd,ad+bc).$$ This gives us a way of multiplying ordered pairs. If we replaced $i$ with $\sqrt{-2}$ we would have $$(a,b)(c,d)=(ac-2bd,ad+bc).$$ In some ways this choice is just as good as $i$. But in other ways it is not.

It might be good to look up what a basis for a vector spaces is.

A line will usually contain a lot of ordered pairs. For example, the line $y=x$ has points $(1,1),(2,2),$ and so forth. If I consider this line over the complex numbers, I get points like $(i,i), (1+i,1+i), (3,3)$, and so forth. Though $(1,1)$ and $(2,2)$ can be thought of as complex numbers $1+i$ and $2+2i$, it is easier to think of them as just ordered pairs of real numbers that satisfy $y=x$. The point here is if I ask for complex solutions to $y=x$, I need a pair of complex numbers.

Answer 2

You could write coordinates as $x+yi$ or complex numbers as $(a,b)$, though you might risk confusing people. This correspondence is the Argand diagram on the complex plane

One of the properties that complex numbers have is that you can add them and multiply them, so in coordinate form you would have $(a,b)+ (c,d) = (a+c,b+d)$ and $(a,b)\times (c,d) = (ac-bd,ac+bd)$.

But generally with coordinates you do not have either property, while with vectors you only have addition. It all depends on the algebraic structure you are working with

Answer 3

A piece of paper with two drawn axes and chosen units is a handy facility for visualizing data structures of the form $(a,b)$ or $a+bi$, where $a$ and $b$ are real numbers. Note that "datawise" $(a,b)$ and $a+bi$ are the same thing, but we present them typographically differently, because we have different purposes in mind. The typographical picture $(a,b)$ is used when $(a,b)$ is just a pair of real data, and no further computing is intended. The same notation is used when we plan two add such pairs or multiply them with a given factor $\lambda$ "componentwise". Geometrically this amounts to vector addition ("parallelogram of forces") and scaling of the picture by the factor $\lambda$. The typographical picture $a+bi$ however indicates to the reader that we are willing to apply $+$, $-$, $\times$, and $:$ to such objects, according to the rules of complex numbers.