2022-01-17

Is there a problem in defining a complex number by
$z=x+iy?$

nick1337

Expert

Step 1 There is no explicit problem, but if you are going to define them as formal symbols, then you need to distinguish between the + in the symbol $a+bi$, the + operation from $\mathbb{R}$, and the sum operation that you will be defining later until you show that they can be confused/identified with one another. That is, you define $\mathbb{C}$ to be the set of all symbols of the form $a+bi$ with . Then you define an addition $\oplus$ and a multiplication $\otimes$ by the rule $\left(a+bi\right)\oplus \left(c+di\right)=\left(a+c\right)+\left(c+d\right)i$ $\left(a+bi\right)\otimes \left(c+di\right)=\left(ac-bd\right)+\left(ad+bc\right)i$ where + and - are the real number addition and subtraction, and + is merely a formal symbol. Then you can show that you can identify the real number a with the symbol $a+0i$; and that $\left(0+i\right)\otimes \left(0+i\right)=\left(-1\right)+0i$; etc. At that point you can start abusing notation and describing it as you do, using the same symbol for , and +. So, the method you propose (which was in fact how complex numbers were used at first) is just a bit more notationally abusive, while the method of ordered pairs is much more formal, giving a precise substance to complex numbers as things (assuming you think the plane is a thing) and not just as formal symbols.

Vasquez

Expert

Step 1 There is a completely rigorous way to do the construction you allude to in the last paragraph, namely by means of quotient rings. Indeed, $\mathbb{C}\cong \frac{\mathbb{R}\left[X\right]}{{X}^{2}+1}$ This generalises, for example, we can construct a commutative ring with elements of the form $x+yϵ$ where ${ϵ}^{2}=0$ The ring so constructed is emphatically not a field, but it is sometimes useful for doing symbolic differentiation.

alenahelenash

Expert

Just set and for any real x, and the notation $x+iy$ is just a shorthand for the ordered pairs notation. Of course you could also choose