Why is complex numbers considered numbers while vectors quantities? A vector is said to be a quantity because it has both magnitude and direction.  A matrix is also said to be a quantity.  So it seems it is custom to call a combination of numbers a quantity.  Then why is a complex number, also a combination of two numbers (one for the real part, one for the imaginary part), called a number, not a quantity?
 A: I don't think there is a precise mathematical definition or reason - it's more a question of English language than mathematics. But usage suggests we commonly use the word "number" when traditional numbers are included in whatever new type of entity we are talking about. We start with the Natural counting numbers, but when we extend them to positive/negative Integers, the Natural numbers are included. Likewise when we extend them to the Rationals (which are also a combination of two numbers in their usual representation as a fraction $a/b$), the integers are included, the Rationals are included in the Reals, and the Reals are included in the Complex numbers.
In the normal way of looking at things, vectors and matrices don't actually include the Real numbers as a subset of themselves. They are of different types, with incompatible operations. (Although 1D vectors and $1\times 1$ matrices look similar, they can't normally be mixed in the the same way as numbers. You can multiply an $n\times n$ matrix by a Real number but not by a $1\times 1$ matrix.) So we need another word for them.
But the distinction isn't very precise, and depends on context. Back in the late 1800s mathematicians like Grassman, Hamilton, and Clifford tried to extend the concept of numbers to cover geometric concepts in the various "hypercomplex" algebras, and a Clifford algebra in particular includes Reals (as scalars), vectors, Complex numbers, Quaternions, and more, all on an equal footing. In such a context, vectors are indeed considered to be "numbers". So it depends.
A: More formally, a vector is a member of a vector space, a set $V$ with the associated operations of addition and scalar multiplication satisfying certain axioms.  It can easily be shown that $\mathbb{C}$ meets all of the vector space axioms.  Thus, complex numbers are vectors.  This is most obvious when plotting them on a Cartesian plane.
But $\mathbb{C}$ provides additional operations beyond the two required for general vectors.  In particular, you're not limited to multiplying complex numbers by real “scalars”; you can also multiply them by other complex numbers, using the definition $(a+ib)(c+id) = (ac - bd) + i(ad + bc)$.  From this, we can define a multiplicative inverse $\frac{1}{a+ib} = \frac{a-ib}{a^2+b^2}$ of any $a + ib \ne 0$.  Complex multiplication is associative, commutative, and distributive, thus satisfying the field axioms.  This gives $\mathbb{C}$ a commonality with $\mathbb{R}$ (the real numbers) and $\mathbb{Q}$ (the rational numbers), which are also fields, and clearly established as being “numbers”.
