0
$\begingroup$

I can plot a point on a $2d$-plane which is a complex number like $2+i$ where real part is $2$ and immaginary part is $1$ , so I would call this a vector with $2$ components.

I can also plot a point $(x_1,x_2)$ on a $2d$-plane that has only real components like $x_1=2$ and $x_2=1$.

Can I say now $\mathbb{C} = \mathbb{R}^2$?

  • 1
    You have achieved a mapping between $\mathbb C$ and $\mathbb R^2$. That's not the same as saying they are equal. – 2017-01-09
  • 0
    Reference topic titles : Argand plane, Gauss-Argand plane, complex plane. – 2017-01-09

2 Answers 2

1

In geometric view the $\mathbb{C}$ plane is the same $\mathbb{R}^2$, so we can consider points in $\mathbb{C}$ plane like $\mathbb{R}^2$ vectors, but in analytic view, $\mathbb{C}\cong\mathbb{R}^2$, or are isomorph, which means that much of rules in $\mathbb{R}^2$ could be converted to $\mathbb{C}$ rules.

  • 0
    So scalr multiplication and addition is of course same for both of them – 2017-01-09
  • 0
    for example inner product isn't defined for complex numbers, but we have it for vectors. also, scalar multiplication and addition is the same for both of them. – 2017-01-09
2

In general it holds that $$\mathbb{C} \cong \mathbb{R}^2$$ or in words: The complex plane $\mathbb{C}$ is isomorphic to the real plane $\mathbb{R}^2$.

  • 0
    Why can't I say that these 2 planes are the same. I find both very alike – 2017-01-09
  • 1
    The complex plane has more properties. You can multiply two complex numbers to get another one - you can't do that for points in ${\mathbb{R}^{2}}$ so they are not exactly the same thing. They do share the same properties under addition so they are isomorphic in that sense as vector spaces. – 2017-01-09
  • 0
    @Oleg Sure, from a geometric point it is obvious. But they are **different algebraic objects**. I suppose you did not encounter abstract algebra yet? – 2017-01-09
  • 0
    @Paul I think it is possible to define a multiplication on $\mathbb{R}^2$ such that we have a field homomorphism. – 2017-01-09