Today I’ll collect a few basic properties of complex numbers.
First off, they form a vector space over the reals. We constructed them as an algebra — the quotient of the algebra of polynomials by a certain ideal — and every algebra is a vector space. So what can we say about them as a vector space? The easiest fact is that it’s two-dimensional, and it’s got a particularly useful basis.
To see this, remember that we have a basis for the algebra of polynomials, which is given by the powers of the variable. So here when we throw in the formal element , its powers form a basis of the ring . But we have a relation, and that cuts things down a bit. Specifically, the element is the same as the element .
Given a polynomial in the “variable” , we can write it as
We can peel off the constant and linear terms, and then pull out a factor of :
Now this factor of can be replaced by , which drops the overall degree. We can continue like this, eventually rewriting any term involving higher powers of using only constant and linear terms. That is, any complex number can be written as , where and are real constants. Further, this representation is unique. This establishes the set as a basis for as a vector space over .
Now the additive parts of the field structure are clear from the vector space structure here. We can write the sum of two complex numbers and simply by adding the components: . We get the negative of a complex number by taking the negatives of the components.
We can also write out products pretty simply, since we know the product of pairs of basis elements. The only one that doesn’t involve the unit of the algebra is . So in terms of components we can write out the product of the complex numbers above as .
Notice here that the field of real numbers sits inside that of complex numbers, using scalar multiples of the complex unit. This is characteristic of algebras, but it’s worth pointing out here. Any real number can be considered as the complex number . This preserves all the field structures, but it ignores the order on the real numbers. A small price to pay, but an important one in certain ways.
We also mentioned the symmetry between and . Either one is just as valid as a square root of as the other is, so if we go through consistently replacing with , and with , we can’t tell the difference. This leads to an automorphism of fields called “complex conjugation”. It sends the complex number to its “conjugate” . This preserves all the field structure — additive and multiplicative — and it fixes the real numbers sitting inside the complex numbers.
Studying this automorphism, and similar structures of other field extensions forms the core of what algebraists call “Galois theory”. I’m not going there now, but it’s a huge part of modern mathematics, and its study is ultimately the root of all of our abstract algebraic techniques. The first groups were automorphism groups shuffling around roots of polynomials.