ℝ² vs ℂ

Similarities and differences between ℝ² and ℂ

Quick answer — same space, different structure

At the level of points and topology, $\mathbb{R}^2$ and $\mathbb{C}$ are the same: there is a bijection

$$ \Phi:\mathbb{R}^2\to\mathbb{C},\qquad \Phi(x,y)=x+iy, $$

which is a homeomorphism and an isometry if you use the Euclidean norm on $\mathbb{R}^2$ and the modulus on $\mathbb{C}$. So they share the same geometry (distance, open sets, continuity), the same dimension as real vector spaces (dimension $2$), etc.

But they differ in extra algebraic structure and the consequences of that structure.

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Similarities

• As sets / topological spaces / metric spaces: $\mathbb{R}^2$ ≅ $\mathbb{C}$ (via $(x,y)\mapsto x+iy$). Same Euclidean metric: $\|(x,y)\|=\sqrt{x^2+y^2}=|x+iy|$.
• Real vector space: Both are real vector spaces of dimension 2. Basis in $\mathbb{C}$ (over $\mathbb{R}$) is $\{1,i\}$.
• Linear algebra / inner product: Using the usual dot product on $\mathbb{R}^2$ and $\langle z,w\rangle=\Re(z\overline w)$ on $\mathbb{C}$ they represent the same Euclidean inner product.
• Orientation and area: Same notion of oriented area; multiplication by a complex number $re^{i\theta}$ scales area by $r^2$ and preserves orientation.

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Key differences (what makes $\mathbb{C}$ special)

1. Field vs plain vector space

   • $\mathbb{C}$ is a field: you can multiply complex numbers and divide by nonzero ones.
   • $\mathbb{R}^2$ as usually presented has no canonical multiplication. You can define a multiplication to make it a field isomorphic to $\mathbb{C}$, but $\mathbb{R}^2$ with only coordinate-wise structure is not a field.

2. Complex multiplication = rotations + scalings

   • Multiplication by $a+bi$ corresponds to the linear map on $\mathbb{R}^2$ with matrix

     $$ \begin{pmatrix} a & -b\\[2pt] b & a\end{pmatrix}, $$

     i.e. a composition of rotation and dilation. This structure is not present in a raw $\mathbb{R}^2$ unless you pick this complex multiplication.

3. Scalar fields

   • $\mathbb{R}^2$ is a 2-dimensional vector space over $\mathbb{R}$.
   • $\mathbb{C}$ is a 1-dimensional vector space over $\mathbb{C}$ and a 2-dimensional vector space over $\mathbb{R}$. This difference matters for linear maps: $\mathbb{C}$-linear maps are much more restrictive than $\mathbb{R}$-linear maps.

4. Complex linearity vs real linearity

   • A map $T:\mathbb{C}\to\mathbb{C}$ that is $\mathbb{C}$-linear must satisfy $T(i z)=iT(z)$. In $\mathbb{R}^2$ language those are exactly the real-linear maps with matrices of the form above (complex-scalar matrices).
   • There are many more $\mathbb{R}$-linear maps on $\mathbb{R}^2$ than $\mathbb{C}$-linear maps on $\mathbb{C}$. Example: complex conjugation $z\mapsto\overline z$ is $\mathbb{R}$-linear but not $\mathbb{C}$-linear.

5. Differentiability / holomorphicity

   • If you view functions $f:\mathbb{R}^2\to\mathbb{R}^2$ the usual (real) differentiability is much weaker than complex differentiability. A complex function $f:\mathbb{C}\to\mathbb{C}$ being holomorphic requires the Cauchy–Riemann equations — a very rigid condition. Holomorphic ⇒ infinitely differentiable and analytic; real differentiable need not imply that.

6. Algebraic closure

   • $\mathbb{C}$ is algebraically closed: every nonconstant polynomial (with complex coefficients) has a root in $\mathbb{C}$.
   • $\mathbb{R}^2$ (as a plain 2D real vector space) has no meaning of algebraic closure unless you impose multiplication; with the usual identification $\mathbb{R}^2\cong\mathbb{C}$, that property belongs to $\mathbb{C}$.

7. Order structure

   • $\mathbb{R}$ is an ordered field; $\mathbb{C}$ cannot be given a field order compatible with its operations. $\mathbb{R}^2$ has no natural total order compatible with vector-space structure.

8. Unique complex structure (up to sign)

   • A “complex structure” on $\mathbb{R}^2$ is a linear map $J:\mathbb{R}^2\to\mathbb{R}^2$ with $J^2=-I$. Choosing such $J$ identifies $\mathbb{R}^2$ with $\mathbb{C}$. All such $J$ that are compatible with the Euclidean metric are rotations by $90^\circ$ (and its negative) — so the complex structure is essentially unique (two choices corresponding to $i$ vs $-i$).

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Useful formulas / examples

• Identification: $(x,y)\leftrightarrow x+iy$.

• Addition: $(x,y)+(u,v)=(x+u,y+v)$ corresponds to $(x+iy)+(u+iv)$.

• Multiplication (via identification):

  $$ (x,y)\cdot(u,v)=(xu-yv,\; xv + yu). $$

• Matrix for multiplication by $a+bi$:

  $$ M(a,b)=\begin{pmatrix} a & -b\\ b & a\end{pmatrix}. $$

• Modulus / norm: $|x+iy|=\sqrt{x^2+y^2}$.

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One-sentence summary

$\mathbb{R}^2$ and $\mathbb{C}$ are the same as sets and as real vector spaces / metric spaces, but $\mathbb{C}$ carries extra algebraic (field) and analytic structure (complex multiplication, complex-linear maps, holomorphicity) that makes it strictly richer than the plain 2-dimensional real plane.

Would you like a short table comparing properties side-by-side, or examples showing how complex differentiability (Cauchy–Riemann) restricts a real map?