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

The field $\mathbb{C} $ of complex numbers is well-defined by the Hamilton axioms of addition and product between complex numbers, i.e., a complex number $z$ is a ordered pair of real numbers $(x,y)$, which satisfy the following operations $+$ and $\cdot$:

$ (x_1,y_1) + (x_2,y_2) = (x_1+x_2,y_1+x_2) $

$(x_1,y_1)(x_2,y_2) = (x_1x_2-y_1y_2,x_1y_2 + x_2y_1)$

The properties of field follow from them.

My question is: Is there a problem in defining complex numbers simply by $z = x+iy$, where $i² = -1$ and $x$, $y$ real numbers and import from $\mathbb{R} $ the operations? Or is just a elegant manner to write the same thing?

Solution 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 $a,b\in\mathbb{R}$. Then you define an addition $\oplus$ and a multiplication $\otimes$ by the rules

$(a$+$bi)\oplus(c$+$di) = (a+c)$ + $(c+d)i$

$(a$+$bi)\otimes(c$+$di) = (ac - bd)$ + $(ad+bc)i$

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 $(0$+$i)\otimes(0$+$i) = (-1)$+$0i$; etc. At that point you can start abusing notation and describing it as you do, using the same symbol for $+$, $\oplus$, 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".

Solution 2:

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

Solution 3:

Just set $i=(0,1)$ and $x=(x,0)$ for any real $x$, and the notation $x+iy$ is just a shorthand for the ordered pairs notation.

Of course you could also choose $i=(0,-1)$ ........