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 $x^{2}+1=0$. In other words, its basic property is $i^{2}=-1$. Of course, since the square root of any real number is positive, $i\notin \mathbb {R}$. A priori, it is not even clear whether such an object exists and that it deserves be called 'a number', i.e. whether we can associate with it some natural operations as addition or multiplication. Admitting for a moment that the positive answer is given for granted, we define

$\mathbb {C} =\{a+bi|a,b\in \mathbb {R} \}$

Aside on notation: There is a well established tradition in mathematics of adopting notation that is suggestive, even if it is, in some ways, unnatural or awkward. For example, if complex numbers are ordered pairs of real numbers, why not represent them as pairs, i.e., use $(a,b)$ rather than $a+bi$? Thee are several ways of answering this question. One is that our notation tends to guide our thinking, and writing $x=x+0i$ emphasizes the idea that the real number x is a complex number, whereas writing $(x,0)$ for the same number suggests that, as a complex number, x is something fundamentally different (perhaps it is). A second, and rather different, reason for using the notation $a+bi$ is that it suggests a parallel with another part of mathematics. In elementary number theory, we learn to perform arithmetic modulo a number base. for example, we may write

$4+5\equiv 2(mod7)$

to indicate that when we add 4 and 5 and then divide the result by 7, the remainder is 2. We can do something similar with polynomials in a single variable x. We know that $(x+1)(x+2)=x^{2}+3x+2$, but $x^{2}+3x+2=1\cdot (x^{2}+1)+(3x+1)$, so when we divide by $x^{2}+1$, the remainder is $2x+1$. And by the same token,

$(1+i)(2+i)=2+3i+i^{2}=1+3i$

so, when we add or multiply complex numbers, we are just doing modular arithmetic! Of course, there are also times when we wish to focus on the geometric or analytic aspects of complex numbers rather than the algebraic ones, but there is a tendency to want to retain the same notation where possible, and there is no question but that mathematical notation also tends to be dictated by tradition and historical accident.

Working with Complex Numbers

Basic Operations

We 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b = Im (c)} . If the imaginary part of a complex number is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} , the number is said to be real, and we write Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} instead of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a + 0i} . We thus identify Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} with a subset (and, in fact, a subfield) of $\mathbb {C}$.

Going a bit further, we can introduce the important operation of complex conjugation. Given an arbitrary complex number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z = x + iy} , we define its complex conjugate to be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{z} = x - iy} . Using the identity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a + b)(a - b) = a^2 - b^2} we derive the important formula

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z \bar{z} = x^2 + y^2}

and we define the modulus of a complex number z to be

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |z| = \sqrt{z \bar{z}}}

Note that the modulus of a complex number is always a real number.

The modulus (also called absolute value) satisfies three important properties that are completely analogous to the properties of the absolute value of real numbers

1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |z| \ge 0} and $|z|=0$ if and only if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z = 0}
2. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |z_1 z_2| = |z_1| |z_2|}
3. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |z_1 + z_2 | \le |z_1| + |z_2|}

The last inequality is known as the triangle inequality.

The Complex Exponential

Recall that in real analysis, the ordinary exponential function may be defined as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exp x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots}

The same series may be used to define the complex exponential function

$\exp z=1+z+{\frac {z^{2}}{2!}}+{\frac {z^{3}}{3!}}+\ldots$

(where, of course, convergence is defined in terms of the complex modulus, instead of the real absolute value).

Notation: The expressions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exp z} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^z} mean the same thing, and may be used interchangeably.

The complex expomential has the same multiplicative property that holds for real numbers,namely

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{z_1 z_2} = e^{z_1} e^{z_2}}

The complex exponential function has the important property that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{i\theta} = \cos \theta + i \sin \theta}

as may be seen immediately by substituting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z = i\theta} and comparing terms with the usual power series expansions of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin \theta} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos \theta} .

The familiar triginometric identity

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin^2 \theta + \cos^2 \theta = 1}

immediately implies the important formula

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |e^{i\theta}| = 1} , for any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta \in \mathbb{R}}

Geometric Interpretation

Since a complex number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z = x + iy} corresponds (essentially by definition) to an ordered pair of real numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x, y)} , it can be interpreted as a point in the plane (i.e., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^2)} . When complex numbers are represented as points in the plane, the resulting diagrams are known as Argand diagrams, after Robert Argand.

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 $\mathbb {C}$. This result is known as the fundamental theorem of algebra. There are many proofs of this theorem. Many of the simplest depend crucially on complex analysis. To illustrate, we consider a proof based on Liouville's theorem: If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(z)} is a polynomial function of a complex variable then both Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(z)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/p(z)} will be holomorphic in any domain where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(z) \not= 0} . But, by the triangle inequality, we know that outside a neighborhood of the origin Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |p(z)| > |p(0)|} , so if there is no Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_0 } such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(z_0) = 0} , we know that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/p(z)} is a bounded entire (i.e., holomorphic in all of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{C}} ) function. By Liouville's theorem, it must be constant, so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(z)} must also be constant.

There are also proofs that do not depend on complex analysis, but they require more algebraic or topological machinery. The starting point here is that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} is a real closed field (i.e., an ordered field containing positive square roots and in which odd degree polynomials always do posess a root). The starting point is to note that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{C} = \mathbb{R}[i]} is the splitting field of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2 + 1} , so if we can show that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{C}} has no finite extensions. We are done. Suppose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K/\mathbb{C}} is a finite normal extension with Galois group G. A Sylow 2-subgroup H must correspond to an intermeiate field L, such that L is an extension of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} of odd degree, but we know no such extensions exist. This contradiction establishes the theorem.

As an aside, it is interesting to note that avoiding the methods of one branch of mathematics (complex analysis), requires the use of more advanced methods from another branch of mathematics (in this case, field theory).

Notational Variants

This article follows the usual convention in mathematics (and physics) of using Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} as the imaginary unit. Complex numbers are frequently used in electrical engineering, but in that discipline it is usual to use Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} instead, reserving Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} for electrical current. This usage is found in some programming languages, notably Python.