复数乘积定义的意义探究：为何采用$(x_1x_2-y_1y_2,x_1y_2+x_2y_1)$形式？

Great question—you’re right that complex numbers are essentially 2D vectors with a special multiplication rule, and it’s totally reasonable to ask why we landed on that specific rule instead of others that also satisfy $i^2 = -1$. Let’s unpack this:

First, the meaning behind the standard multiplication rule #

The definition $(x_1x_2 - y_1y_2, x_1y_2 + x_2y_1)$ isn’t arbitrary—it checks several critical boxes that make complex numbers useful and consistent with the math we already know:

• Backward compatibility with real numbers: When you multiply two real numbers (where $y_1 = y_2 = 0$), this rule simplifies to $(x_1x_2, 0)$—exactly regular real multiplication. Since complex numbers are supposed to extend the real number system, this compatibility is non-negotiable.
• Geometric intuition: If you represent complex numbers in polar form ($r(\cos\theta + i\sin\theta)$), multiplying two complex numbers corresponds to multiplying their magnitudes and adding their angles. This is a super powerful geometric operation—think rotating a vector by some angle and scaling its length. It’s used everywhere, from engineering signal processing to physics modeling of rotational motion.
• Algebraic consistency: This multiplication rule turns the set of complex numbers into a field. That means it follows all the familiar algebraic rules we rely on: commutative law ($ab = ba$), associative law ($(ab)c = a(bc)$), distributive law over addition, and every non-zero complex number has a multiplicative inverse. These properties make complex numbers a robust system for solving equations and doing algebra.

Why your proposed rule doesn’t work (even though it satisfies $i^2=-1$) #

Let’s take your example: $(-y_2, x_1y_2 + x_2y_1)$. You’re correct that this does give $i^2 = -1$ (if $i=(0,1)$, then $i*i = (-1, 0)$). But it fails in key ways that make it impractical:

• Breaks real number compatibility: Multiply two real numbers like $(2,0)$ and $(3,0)$ using this rule, and you get $(-0, 20 + 30) = (0,0)$—which is nothing like real multiplication ($2*3=6$). This would mean real numbers don’t behave like real numbers within this system, which defeats the purpose of extending them.
• No commutative property: Try multiplying $(1,0)$ and $(0,1)$:
  • $(1,0)(0,1) = (-1, 11 + 0*0) = (-1,1)$
  • $(0,1)(1,0) = (-0, 00 + 1*1) = (0,1)$
    The results are different! Without commutative multiplication, most of our standard algebraic tools (like rearranging terms in equations) fall apart.
• No useful geometric interpretation: This operation doesn’t map to any intuitive geometric transformation on the plane, so it wouldn’t have the practical applications that make complex numbers so valuable.

At the end of the day, we choose the standard complex multiplication rule because it’s the simplest way to get a system that extends real numbers, has meaningful geometry, and follows the algebraic rules we need to do useful math.

内容的提问来源于stack exchange，提问作者Daniel Li