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

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2022-01-17

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

Answer & Explanation

nick1337

nick1337

Expert

2022-01-19Added 573 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

Expert

2022-01-19Added 457 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

Expert

2022-01-24Added 366 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)

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