Why Are Complex Numbers Numbers but 2D Vectors Usually Aren’t?

The difference is not that a complex number cannot be written as a 2D vector. It can: $a+bi$ corresponds to the pair $(a,b)$. The difference is which operations and roles are part of the structure. Complex numbers have a standard way to multiply and divide them; ordinary vectors in $\mathbb{R}^2$ do not.

So the terms are not mutually exclusive. A complex number can be viewed as a 2D real vector, while a general 2D vector is not automatically a complex number.

What makes something a number here?

There is no single definition that captures every use of the word “number.” In algebra, familiar number systems are sets equipped with specified operations. The real numbers and complex numbers, for example, have addition and multiplication, along with identities $0$ and $1$. Every nonzero element has a multiplicative inverse, so division by a nonzero element is possible.

A vector space is organized differently. Its elements—vectors—can be added, and they can be multiplied by scalars from a chosen number system. But the vector-space rules do not generally define a product of one vector with another.

For example, in $\mathbb{R}^2$, the expression $2(3,4)=(6,8)$ is scalar multiplication. But the vector-space structure by itself does not tell you what $(1,2)(3,4)$ should mean. There are several possible operations one might define, and none is forced by the fact that the objects are vectors.

Complex numbers are also 2D vectors

Write a complex number as $a+bi$, where $a$ and $b$ are real and $i^2=-1$. Its coordinates are $(a,b)$, so complex addition is exactly ordinary vector addition:

$$(a+bi)+(c+di)=(a+c)+(b+d)i.$$

The extra structure appears in multiplication. Using $i^2=-1$ gives

$$(a+bi)(c+di)=(ac-bd)+(ad+bc)i.$$

In pair notation, this is the rule

$$(a,b)(c,d)=(ac-bd,\,ad+bc).$$

This specific multiplication makes the real plane into the complex numbers. It is not an operation supplied by the usual definition of a real vector in $\mathbb{R}^2$; it is additional structure.

A worked example

Take $1+i$ and $2+3i$. As pairs, these are $(1,1)$ and $(2,3)$. Complex multiplication gives

$$(1,1)(2,3)=(1\cdot2-1\cdot3,\;1\cdot3+1\cdot2)=(-1,5),$$

which represents $-1+5i$. The same result follows by expanding $(1+i)(2+3i)$ and using $i^2=-1$.

If you instead multiply the coordinates component by component, you get $(2,3)$. That is a possible operation on pairs, but it is not complex multiplication. In fact, componentwise multiplication makes $(1,0)(0,1)=(0,0)$ even though neither factor is zero, so it does not give the usual field structure of the complex numbers.

Why the names reflect different roles

In linear algebra, “scalar” and “vector” often name roles in a structure, not completely separate kinds of objects. In a real vector space, the scalars come from $\mathbb{R}$. In a complex vector space, they come from $\mathbb{C}$.

At the same time, $\mathbb{C}$ itself is a vector space over $\mathbb{R}$: each complex number is a pair of real coordinates, and real numbers act on those pairs by scalar multiplication. This real vector space has dimension two. But as a vector space over $\mathbb{C}$, $\mathbb{C}$ has dimension one.

This is why mathematicians can accurately say both “a complex number is a scalar” (when it acts on vectors in a complex vector space) and “a complex number is a 2D vector” (when considering it as a real vector). The context determines which description is useful.

Common misunderstandings

• “Vectors cannot be numbers.” They can be given number-like operations. Complex numbers themselves are a standard example of a set that is both a number system and a vector space over the reals.
• “Any pair of real numbers is automatically a complex number.” It can be identified with one, but using that identification as a number system means using the complex addition and multiplication rules, not just treating the pair as an unstructured list of coordinates.
• “There is no way to multiply vectors.” There are many useful products, including dot products and cross products. They have different definitions and purposes; ordinary vector-space structure does not select one as the general multiplication that makes vectors into a number system.
• “The distinction is about how the object looks.” A pair, an arrow, or a symbol can represent the same mathematical object. What matters is the operations and rules attached to it.

A 2D vector is therefore not usually called a number because the word “vector” highlights its role in a vector space, while “number” highlights arithmetic operations. With the complex multiplication rule, a 2D real vector can play both roles.