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 of any real number is nonnegative, i cannot be a real number. A priori, it is not even clear whether such an object exists and that it can be called a number, i.e. whether we can associate with it some natural operations as addition or multiplication. Assuming, for a moment, that the answer is "yes", we may rewrite the first sentence above in symbols as

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

Historical example

The need for complex numbers may have appeared for the first time during the sixteenth century, when Italian mathematicians like Scipione del Ferro, Niccolò Fontana Tartaglia, Gerolamo Cardano and Rafael Bombelli tried to solve cubic equations. This is so even for equations with three real solutions, as the method they used sometimes requires calculations with numbers which squares are negative. Here is such an example (with modern notation). Let us consider the equation

$x^{3}=15x+4\$.

Cardano's method for solving it suggests looking for a solution by writing it as a sum $x=u+v$, where some other condition on $u$ and $v$ will be decided later. Reporting this in the equation, we get, once the left member is expanded,

$u^{3}+3u^{2}v+3uv^{2}+v^{3}=15(u+v)+4\$,

which can be written as,

$u^{3}+(3uv-15)(u+v)+v^{3}=4\$.

Now we choose the second condition on $u$ and $v$, that is $3uv-15=0$, or $uv=5$. This implies that $u^{3}$ and $v^{3}$ are numbers which sum and product are given by

${\begin{cases}u^{3}v^{3}=125\\u^{3}+v^{3}=4\end{cases}}$.

Now, it is a well-known fact that if a second degree polynomial $x^{2}-sx+p$ has two roots, their sum is $s$ and their product is $p.$Hence we may find some values for $u^{3}$ and $v^{3}$ by solving the quadratic equation

$x^{2}-4x+125=0\$.

Its discriminant is $\Delta =(-4)^{2}-4\cdot 125=-484=-22^{2}$, which is negative, so that the quadratic equation has no real solution: the usual formulae giving the solutions require to take the square root of the discriminant, which is undefined here.

Well, let us be bold and write $\Delta =\left(22{\sqrt {-1}}\right)^{2}$. Here, the symbol ${\sqrt {-1}}$ denotes an hypothetical number which square would be $-1.$ At this stage, such a number has no meaning (square of real numbers are always nonnegative), but we use it in a purely formal way. Using this symbol, we can write the "solutions" to the quadratic equation as

$u^{3}={\frac {4+22{\sqrt {-1}}}{2}}=2+11{\sqrt {-1}}$ and $v^{3}={\frac {4-22{\sqrt {-1}}}{2}}=2-11{\sqrt {-1}}$.

It remains to find cube roots of these "numbers". A straightforward calculation shows that $u=2+{\sqrt {-1}}$ and $v=2-{\sqrt {-1}}$ do the job. For instance, remembering the rule $\left({\sqrt {-1}}\right)^{2}=-1$, we have

$\left(2+{\sqrt {-1}}\right)^{3}=2^{3}+3\cdot 2^{2}{\sqrt {-1}}+3\cdot 2\left({\sqrt {-1}}\right)^{2}+\left({\sqrt {-1}}\right)^{3}$; $\left(2+{\sqrt {-1}}\right)^{3}=8+12{\sqrt {-1}}-6-{\sqrt {-1}}=2+11{\sqrt {-1}}$.

But now, going back to the original cubic equation, we get the real solution $x=u+v=2+{\sqrt {-1}}+2-{\sqrt {-1}}=4$! One can verify it is indeed a solution, as $4^{3}=64=15\cdot 4+4$ (and once this solution is found, it is easy to find the two other solutions, which are also real).

The fact that the formal calculations managed to give a real solution suggests that the "number" ${\sqrt {-1}}$ may have some sense. But to really give it a legitimate status, one has to construct a new set of numbers, containing the real numbers, but also other numbers whose squares may be negative real numbers. This will be the set of complex numbers. A rigorous construction of this set was given much later by Carl Friedrich Gauss in 1831.

Formal definition

Formally, complex numbers are ordered pairs of real numbers, i.e.

$\mathbb {C} =\{(a,b)~|~~a,b\in \mathbb {R} \}.$

To call them 'numbers' we need to introduce some operations on such pairs. So we define

• addition (a, b) + (c, d) = (a + c, b + d)
• multiplication (a, b)(c, d) = (ac - bd, bc + ad)

While this definition can look arbitrary and artificial at the first sight, it turns out to be very natural one. In particular, the basic properties of the usual operations are preserved and we can employ many formulas from the elementary algebra we are accustomed to. More specifically, the sum (or the product) of two numbers does not depend on the order of terms; the sum (product) of three or more elements does not depend on order of operations ('we can suppress the parentheses'); the product of a complex number with a sum of two other numbers expands in the usual way. In mathematical language this means that with addition and multiplication defined this way, $\mathbb {C}$ satisfies the axioms for a field, and is called the field of complex numbers.

Now we are ready to understand the 'real' meaning of ${\sqrt {-1}}$ and its usage in the above historical example. Observe that the pairs of type (a,0) are identical to the set of reals, so we write (a, 0)=a. Observe also that by definition (0,1)(0,1) = (-1,0)=-1. It follows that ${\sqrt {-1}}$, the hypothetical number whose square root gives -1, is well defined as (0,1). It is so characteristic that we usually denote it by i and call it the imaginary unit. Now, the historical example computation is fully justified: since the basic operations on complex numbers behave like the usual addition and multiplication, we can find the roots of a second degree polynomial with the well-known formulas.

Beyond the notation

Next step is to observe that any complex number (a,b) can be expressed as a+bi and, conversely, any sum of this type represents a complex number. The notation using the imaginary unit i is called the algebraic form or rectangular form or yet Cartesian form of a complex number. It is both traditional and commonly used to perform computations. The importance of such formally trivial rewriting is difficult to overestimate and we discuss it in more details.

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 simply as pairs, i.e., use $(a,b)$ rather than $a+bi$? There 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

$4+5\equiv 2{\pmod {7}}$

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 $3x+1$. 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.

Another reason, this time of practical nature, is elaborated in the next section.

Working with complex numbers

It is not very compelling to compute using the ordered pair definition, especially when it concerns the product of two complex numbers. It turns out that the imaginary unit comes in handy with its property $i^{2}=-1$. Algebraic operations become as natural as for the reals.

Basic operations

Addition is straightforward, $(a+bi)+(c+di)=(a+c)+(b+d)i\$. More notably, we can rewrite multiplication using $i^{2}=-1$ to obtain results in the form $a+bi$:

$(a+bi)(c+di)=ac+adi+bci+bdi^{2}=(ac-bd)+(bc+ad)i.\$

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

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

from which it follows that

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

If $z=a+bi$ is a complex number, we call $a$ the real part of $z$ and write $a=Re(z)$. Similarly, $b$ is called the imaginary part of $z$ and we write $b=Im(z)$. If the imaginary part of a complex number is $0$, the number is said to be real. As mentioned earlier, we write $a$ instead of $a+0i$ and thus we identify $\mathbb {R}$ with a subset of $\mathbb {C}$.

Going a bit further, we can introduce the important operation of complex conjugation. Given an arbitrary complex number $z=x+iy$, we define its complex conjugate to be ${\bar {z}}=x-iy$. Using the identity $(a+b)(a-b)=a^{2}-b^{2}$ we derive the important formula

$z{\bar {z}}=x^{2}+y^{2}$

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

$|z|={\sqrt {z{\bar {z}}}}$

Note that the modulus of a complex number is always a nonnegative 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

• $|z|\geq 0$ and $|z|=0$ if and only if $z=0$
• $|z_{1}z_{2}|=|z_{1}||z_{2}|\$
• $|z_{1}+z_{2}|\leq |z_{1}|+|z_{2}|$

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

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

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 $\exp \ z$ and $e^{z}\$ mean the same thing, and may be used interchangeably.

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

$e^{z_{1}z_{2}}=e^{z_{1}}e^{z_{2}}\$

The complex exponential function has the important property that

$e^{i\theta }=\cos \theta +i\sin \theta \$

as may be seen immediately by substituting $z=i\theta$ and comparing terms with the usual power series expansions of $\sin \theta$ and $\cos \theta$.

The familiar trigonometric identity

$\sin ^{2}\theta +\cos ^{2}\theta =1\$

immediately implies the important formula

$|e^{i\theta }|=1$, for any $\theta \in \mathbb {R}$

Of course, there is no reason to assume this identity. We only need note that ${\overline {e^{i\theta }}}=e^{-i\theta }$ so,

$|e^{i\theta }|^{2}=e^{i\theta }e^{-i\theta }=e^{0}=1\$

Geometric interpretation

Since a complex number $z=x+iy$ corresponds (essentially by definition) to an ordered pair of real numbers $(x,y)$, it can be interpreted as a point in the plane (i.e., $\mathbb {R} ^{2})$. When complex numbers are represented as points in the plane, the resulting diagrams are known as Argand diagrams, after Robert Argand. The geometric representation of complex numbers turns out to be very useful, both as an aid to understanding the properties of complex numbers, but also as a tool in applying complex numbers to geometrical and physical problems.

There are no real surprises when we look at addition and subtraction in isolation: addition of complex numbers is not essentially different from addition of vectors in $\mathbb {R} ^{2}$. Similarly, if $\alpha \in \mathbb {R}$ is real, multiplication by $\alpha$ is just scalar multiplication. In $\mathbb {C}$ we have

$z_{1}+z_{2}=(x_{1}+iy_{1})+(x_{2}+iy_{2})=(x_{1}+x_{2})+i(y_{1}+y_{2})\$

and

$\alpha z=\alpha (x+iy)=\alpha x+i\alpha y\$

To put it succintly, $\mathbb {C}$ is a 2-dimensional real vector space with respect to the usual operations of addition of complex numbers and multiplication by a real number. There doesn't seem to be much more to say. But there is more to say, and that is that the multiplication of complex numbers has geometric significance. This is most easily seen if we take advantage of the complex exponential, and write complex numbers in polar form

$z=re^{i\theta }$

Here, r is simply the modulus ${\sqrt {x^{2}+y^{2}}}$ or vector length. The number $\theta$ is just the angle formed with the x-axis, and is called the argument. Now, when complex numbers are written in polar form, multiplication is very interesting

$z_{1}z_{2}=(r_{1}e^{i\theta _{1}})(r_{2}e^{i\theta _{2}})=r_{1}r_{2}e^{i(\theta _{1}+\theta _{2})}$

In other words, multiplication by a complex number z has the effect of effect of simultaneously scaling by the numbers' modulus and rotating by its argument. This is really astounding. Translation corresponds, to complex addition, scaling to multiplication by a real number, and rotation to multiplication by a complex number of unit modulus. The one type of coordinate transformation that is missing from this list is reflection. On the other hand, there is an arithmetic operation we have not considered, and that is division. Recall that

${\frac {1}{z}}={\frac {\bar {z}}{|z|^{2}}}$

In other words, up to a scaling factor, division by z is just complex conjugation. Returning to the representation of complex numbers in rectangular form, we note that complex conjugation is just th transformation (or map) $x+iy\mapsto x-iy$ or, in vector notation, $(x,y)\mapsto (x,-y)$. This is nothing other than reflection in the x-axis, and any other reflection may be obtained by combining that transformation with rotations and translations.

Historically, this observation was very important and led to the search for higher dimensional algebras that could "arithmetize" Euclidean geometry. It turns out that there are such generalizations in dimensions 4 and 8, known as the quaternions and octonions (also known as Cayley numbers). At that point, the process stops, but the ideas developed in this process have played an important role in the development of modern differential geometry and mathematical physics).

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 $p(z)$ is a polynomial function of a complex variable then both $p(z)$ and $1/p(z)$ will be holomorphic in any domain where $p(z)\not =0$. But, by the triangle inequality, we know that outside a neighborhood of the origin $|p(z)|>|p(0)|$, so if there is no $z_{0}$ such that $p(z_{0})=0$, we know that $1/p(z)$ is a bounded entire (i.e., holomorphic in all of $\mathbb {C}$) function. By Liouville's theorem, it must be constant, so $p(z)$ 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 $\mathbb {R}$ 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 $\mathbb {C} =\mathbb {R} [i]$ is the splitting field of $x^{2}+1$, so if we can show that $\mathbb {C}$ has no finite extensions, then we are done. Suppose $K/\mathbb {C}$ is a finite normal extension with Galois group G. A Sylow 2-subgroup H must correspond to an intermediate field L, such that L is an extension of $\mathbb {R}$ 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).

What about complex analysis?

So far, with one notable exception, we have only made use of algebraic properties of complex numbers. That exception is, of course, the complex exponential, which is an example of a transcendental function. As it happens, we could have avoided the use of the exponential function here, but only at the cost of more complicated algebra. (The more interesting question is why we would want to avoid using it!)

Differentiation

But we now turn to a more general question: Is it possible to extend the methods of calculus to functions of a complex variable, and why might we want to do so? We recall the definition of one of the two fundamental operations of calculus, differentiation. Given a function $y=f(x)$, we say f is differentiable at $x_{0}$ if the limit

$\lim _{h\to 0}{\frac {f(x_{0}+h)-f(x_{0})}{h}}$

exists, and we call the limiting value the derivative of f at $x_{0}$, and the function that assigns to each point x the derivative of f at x is called the derivative of f, and is written $f'(x)$ or $df/dx$. Now, does this definition work for functions of a complex variable? The answser is yes, and to see why, we fix x and unravel the definition of limit. If the limit exists, say $c=f'(x)$, then for every (real) number $\epsilon >0$, there is a (real) number $\delta$ such that if $|h|<\delta$

$\left|{\frac {f(x+h)-f(x)}{h}}-c\right|<\epsilon$

This makes perfect sense for functions of a complex variable, but we need to keep in mind that $|\cdot |$ represents the modulus of a complex number, not the real absolute value.

This seemingly innocuous difference actually has far reaching implications. Recall that the complex plane has two real dimensions, so there are many ways that h can approach 0: successive values of h may be points on the x-axis, points on the y-axis, some other line through the origin, it may spiral in, or take any of a number of paths, but the definition requires that the limit be the same number in every case. This is a very strong requirement! Fortunately, it turns out to be sufficient to consider just two of the possible "approach paths": a sequence of values along the x-axis and a sequence of values along the y-axis. If we call the real and imaginary parts (respectively) of $w=f(z)$ u and v, (i.e., $w=f(z)=u+iv$), this requirement can be expressed in terms of the partial derivatives of u and v with respect to x and y:

${\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}}$

and

${\frac {\partial v}{\partial x}}=-{\frac {\partial u}{\partial y}}$

These equations are known as the Cauchy-Riemann equations.

Note: These equations are frequently written in the more compact form, $u_{x}=v_{y}$ and $v_{x}=-u_{y}$.

They may be obtained by noting that if the approach path is on x-axis, $\partial f/\partial y=0$, so

${\frac {df}{dz}}={\frac {1}{2}}\left({\frac {\partial u}{\partial x}}+i{\frac {\partial v}{\partial x}}\right)$

and that on the y-axis, $\partial f/\partial x=0$, so

${\frac {df}{dz}}={\frac {1}{2}}\left(-i{\frac {\partial u}{\partial y}}+{\frac {\partial v}{\partial y}}\right)$

These equations have far-reaching implications. To get some idea if why this is so, consider that we can take second derivatives to obtain

$u_{xx}+u_{yy}=0$

and

$v_{xx}+v_{yy}=0$

In other words, u and v satisfy Laplace's equation in 2 dimensions. These functions arise in mathematical physics as scalar potentials in, for example, fluid dynamics. Laplace's equation is also basic to the study of partial differential equations. This is but one indication of the reason for the ubiquity of complex functions in physics.

Integration

By contrast, the definition of integration in complex analysis involves no surprises. Path integrals and integrals over regions are defined just as they are in the calculus of functions of two real variables. What is different is that the Cauchy-Riemann equations imply that integrals of complex functions have some very special properties. In particular, if a function f is differentiable (in the sense explained above) in a simply connected domain (intuitively, a domain having no "holes" in it), then for any closed curve $\gamma$ defined in that domain

$\int _{\gamma }\nolimits fdz=0$

It is essential that the domain of definition be simply connected. For example, let

$D=\{z\mid \textstyle {\frac {1}{2}}<|z|<{\frac {3}{2}}\}$

and let $f(z)=1/z$. Then if we define $\gamma (t)=e^{it}$ where t ranges from 0 to $2\pi$ (i.e., we take $\gamma$ to be the unit circle), then the integral will not be 0.

It follows that if $\gamma _{1}$ and $\gamma _{2}$ are two homotopic paths joining a pair of points $P,Q\in D$ (intuitively, one can be deformed into the other), then

$\int _{\gamma _{1}}\nolimits fdz=\int _{\gamma _{2}}\nolimits fdz$

This is commonly expressed by saying that the integrals are path independent, and this is just the condition for the existence of a scalar potential!

Finally, we note that integrals in domains containing singularities (such as 1/z in the above example) can be computed using Cauchy's integral formula

$f(z)={\frac {1}{2\pi i}}\int _{\gamma }\nolimits {\frac {f(\zeta )d\zeta }{\zeta -z}}$

This result lies at the heart of many applications of complex analysis to disciplines ranging from number theory to physics. Its importance would be difficult to overestimate.

Further Reading

• Ahlfors, Lars V. (1979). Complex Analysis, 3rd edition. McGraw-Hill, Inc.. ISBN 0-07-000657-1. 
• Apostol, Tom M. (1974). Mathematical Analysis, 2nd edition. Addison-Wesley. ISBN 0-201-00-288-4. 
• Conway, John H.; Derek A. Smith (2003). On Quaternions and Octonions: Their Geometry, Arithmetic and Symmetry. A K Peters, Ltd.. ISBN 1-56881-134-9. 
• Jacobson, Nathan (1974). Basic Algebra I. W.H. Freeman and Company. ISBN 0-7167-0453-6.