# real vs complex numbers

Can someone write REAL numbers in rectangular form as well? And if so, is it useful?

For example: On the complex plane, $(x + yi)$ is $x$ units on the Real $x$ axis and y units on the Imaginary y axis. If I say, $(x + y)$ is that the same as $x$ units on the Real $x$ axis and y units on the Real $y$ axis?

Also, can I say $(x + y + yi)$? Is that $x$ units on the Real $x$ axis, $y$ units on the Real $y$ axis, and $y$ units on the Imaginary $y$ axis?

• If you're only thinking about the complex numbers as pairs of real numbers (i.e. without looking at the algebraic structure), then certainly you can talk about triple of real numbers; these triples will not be represented as a complex number though. The use of $+$ here is not being used in lieu as a comma or anything of the sort. It is an algebraic operation, and sending two real number $x$ and $y$ to another real number $z$ that just happens to be denoted by $x+y$. Use something like $(x,y)$. – Hayden Jan 29 '15 at 23:38
• I'm confused, I thought rectangular form means + is being us as a comma? Like there are 6 dimensions (axes), an imaginary one that is paired with each real one? – Nova Jan 29 '15 at 23:40
• For complex numbers, $x+yi$ is representing a complex number. However, we can model the complex numbers as the set of all pairs of real numbers, $\mathbb{R}^2$, where the bijection between $\mathbb{C}$ and $\mathbb{R}^2$ is given by sending $x+yi$ to $(x,y)$. However, in principle they are not the same. With this in mind, we can view the $+$ in $x+yi$ as a sort of formal sum (analogous to a comma, in a way), but when you say something like $(x+yi)+(x'+y'i)$, this isn't the case, because the middle $+$ is actually meaningful: it is the addition of complex numbers.... – Hayden Jan 29 '15 at 23:46
• ...and is an actual operation that outputs a new complex number $(x+x')+(y+y')i$. Here $+$ means a couple different things again! In $x+x'$ and $y+y'$ the $+$ is referring to the addition of real numbers, whereas the middle $+$ is more of the formal variety as in $x+yi$. So when you say $x+y+yi$, you're really meaning the complex number with real part $x+y$ and imaginary part $y$. – Hayden Jan 29 '15 at 23:48
• Hayden I think I almost understand. I think the repetitive use of y is confusing me. So, lets take the complex number A + Bi. We can graph it on the complex plane with A units on the "x" axis and B units on the "y" axis. Now, can I graph A + Bi on the real plane as well? So if I take the point (1,1) it would be (1,1) on the real plane. What if I take (A + Bi,1)? Will that be (A + Bi) units on the x axis and 1 unit on the y axis? – Nova Jan 29 '15 at 23:58

## 1 Answer

Any function $f:{\rm I\!R}^2 \rightarrow {\rm I\!R}$ that is at least injective could be said to provide a way to "write REAL numbers in rectangular form". Usefulness would depend on what purpose this function fulfills. The example you gave:

$$f(x,y) = x+y$$

Is surjective, but not injective. So given (x+y) you cannot uniquely determine $x$ and $y$.

The second example you give is a function $f:{\rm I\!R}^2 \rightarrow \mathbb{C}$, which is both surjective and injective, hence bijective. Also it it additive so: $$f(x,y)=(x+y)+yi$$ $$f(w,t)=(w+t)+ti$$ Hence: $$f(x,y)+f(w,t)=(x+y)+yi+(w+t)+ti = f(x+w,y+t)$$

Therefore it maps x units on the Real x axis, y units on the Real y axis, and y units on the Imaginary y axis. If my understanding of this phrase is accurate.