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

Answered question

2022-01-17

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

Answer & Explanation

nick1337

nick1337

Expert2022-01-19Added 777 answers

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 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 C to be the set of all symbols of the form a+bi with a, bR. Then you define an addition and a multiplication by the rule (a+bi)(c+di)=(a+c)+(c+d)i (a+bi)(c+di)=(acbd)+(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)(0+i)=(1)+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

Vasquez

Expert2022-01-19Added 669 answers

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, CR[X]X2+1 This generalises, for example, we can construct a commutative ring with elements of the form x+yϵ where ϵ2=0 The ring so constructed is emphatically not a field, but it is sometimes useful for doing symbolic differentiation.
alenahelenash

alenahelenash

Expert2022-01-24Added 556 answers

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)

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?