Complex number/Citable Version

The complex numbers $\mathbb {C}$ are numbers of the form a+bi, obtained by adjoining the imaginary unit i to the real numbers (here a and b are reals). The number i can be thought of as a solution of the equation $x^{2}+1=$. In other words, its basic property is $i^{2}=1$. Of course, since the square root of any real number is positive, $i\notin \mathbb {R} </amth>.''Apriori'',itisnotevenclearwhethersuchanobjectexistsandthatit''deserves''becalled'anumber',i.e.whetherwecanassociatewithitsomenaturaloperationsasadditionormultiplication.Admittingforamomentthatthepositiveanswerisgivenforgranted,wedefine<math>\mathbb {C} =\{a+bi|a,b\in \mathbb {R} \}$

We then define addition and multiplication in the obvious way, using $i^{2}=-1$ to rewrite results in the form $a+bi$:

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

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

To handle division, we simply note that $(c+di)(c-di)=c^{2}+d^{2}$, so

${\frac {1}{c+di}}={\frac {c-di}{c^{2}+d^{2}}}$

and, in particular,

${\frac {a+bi}{c+di}}={\frac {(ac+bd)+(bc-ad)i}{c^{2}+d^{2}}}$

It turns out that with addition and multiplication defined this way, $\mathbb {C}$ satisfies the axioms for a field, and is called the field of complex numbers. If $c=a+bi$ is a complex number, we call $a$ the real part of $c$ and write $a=Re(c)$. Similarly, $b$ is called the imaginary part of $c$ and we write $b=Im(c)$. If the imaginary part of a complex number is $0$, the number is said to be real, and we write $a$ instead of $a+0i$. We thus identify $\mathbb {R}$ with a subset (and, in fact, a subfield) of $\mathbb {C}$.

Algebraic Closure

An important property of $\mathbb {C}$ is that it is algebraically closed. This means that any non-constant real polynomial must have a root in $\mathbb {C}$.

A Note on Notation

This article follows the usual convention in mathematics (and physics) of using $i$ as the imaginary unit. Complex numbers are frequently used in electrical engineering, but in that discipline it is usual to use $j$ instead, reserving $i$ for electrical current. This usage is found in some programming languages, notably Python.