Complex number/Citable Version: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>Greg Woodhouse
(reference for electron diffraction)
imported>John Stephenson
m (moved Complex number to Complex number/Citable Version: citable version policy)
 
(93 intermediate revisions by 12 users not shown)
Line 1: Line 1:
The '''complex numbers''' <math>\mathbb{C}</math> are numbers of the form ''a+bi'',
{{subpages}}
obtained by adjoining the [[imaginary unit]] ''i'' to the [[real number]]s (here ''a'' and ''b'' are reals).<ref>This article follows the usual convention in [[mathematics]]  and [[physics]] of using <math>i</math> as the imaginary unit. Complex numbers are frequently used in [[electrical engineering]], but in that discipline it is usual to use <math>j</math> instead, reserving <math>i</math> for [[electrical current]]. This usage is found in some [[programming language]]s too, notably [[Python]].</ref> The number ''i'' can be thought of as a solution of the equation <math>x^2+1=0</math>. In other words, its basic property is <math>i^2=-1</math>. 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
'''Complex numbers''' are numbers of the form <math>a+bi</math>, where <math>a</math> and <math>b</math> are [[real number]]s and <math>i</math> denotes a number satisfying <math>i^2=-1</math>.<ref>This article follows the usual convention in [[mathematics]]  and [[physics]] of using <math>i</math> as the imaginary unit. Complex numbers are frequently used in [[electrical engineering]], but in that discipline it is usual to use <math>j</math> instead, reserving <math>i</math> for [[electrical current]]. This usage is found in some [[programming language]]s too, notably [[Python]].</ref> Of course, since the square of any real number is nonnegative, <math>i</math> cannot be a real number. At first glance, it is not even clear whether such an object exists and can be reasonably called a number; for example, can we sensibly associate with <math>i</math> natural operations such as addition and multiplication? As it happens, we can define mathematical operations for these "complex numbers" in a consistent and sensible way and, perhaps more importantly, using complex numbers provides mathematicians, physicists, and engineers with an extremely powerful approach to expressing parts of these sciences in a convenient and natural-feeling way.
 
:<math>\mathbb{C} = \{ a + bi | a, b \in \mathbb{R} \}.</math>


==Historical example ==
==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 equation]]s. This is so even for equations with three [[real number|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
The need for complex numbers might 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 equation|cubic equations]]. Even for equations with three [[real number|real]] solutions, the method they used sometimes required calculations with numbers whose squares are negative. Here is such an example (with modern notation). Let us consider the equation


: <math>x^3=15x+4. \ </math>
: <math>x^3=15x+4. \ </math>


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


: <math>u^3+3u^2v+3uv^2+v^3=15(u+v)+4, \ </math>
: <math>u^3+3u^2v+3uv^2+v^3=15(u+v)+4, \ </math>
Line 18: Line 16:
: <math>u^3+(3uv-15)(u+v)+v^3=4. \ </math>
: <math>u^3+(3uv-15)(u+v)+v^3=4. \ </math>


Now we choose the second condition on <math>u</math> and <math>v</math>, that is <math>3uv-15=0</math>, or <math>uv=5</math>. This implies that <math>u^3</math> and <math>v^3</math> are numbers which sum and product are given by
Now we recall that we did not completely specify <math>u</math> and <math>v</math>; we only required that <math>x=u+v</math>. Hence, we can choose another condition on <math>u</math> and <math>v</math>. We pick this condition to be <math>3uv-15=0</math>, or <math>uv=5</math>, in order to simplify the above equation. This implies that <math>u^3</math> and <math>v^3</math> are numbers whose sum and product are given by


: <math>\begin{cases}
: <math>\begin{cases}
Line 25: Line 23:
\end{cases}</math>
\end{cases}</math>


Now, it is a well-known fact that if a [[second degree]] [[polynomial]] <math>x^2-sx+p</math> has two roots, their sum is <math>s</math> and their product is <math>p.</math><ref>To verify this, one just has to write <math>r_1</math> and <math>r_2</math> for the roots, to expand <math>\left(x-r_1\right)\left(x-r_2\right)</math> and to identify the coefficients.</ref>Hence we may find some values for <math>u^3</math> and <math>v^3</math> by solving the [[quadratic equation]]
It follows from the second equation that <math>v^3 = 4 - u^3</math>. Substituting this in the first equation, we get <math>u^3 (4-u^3) = 125</math>. Hence we may find some values for <math>u^3</math> by solving the equation <math>x(4-x) = 125</math>. Get rid of the brackets and move the number 125 to the left-hand side to get the [[quadratic equation]]


: <math>x^2-4x+125=0. \ </math>
: <math>x^2-4x+125=0. \ </math>


Its [[discriminant]] is <math>\Delta=(-4)^2-4\cdot 125=-484=-22^2</math>, 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.
Its [[discriminant]] is <math>\Delta=(-4)^2-4\cdot 125=-484=-22^2</math>, which is ''negative'', so that the quadratic equation has ''no real solution'': the usual formulae giving the solutions require taking the [[square root]] of the discriminant, which is undefined here.


Well, let us be bold and write <math>\Delta=\left(22\sqrt{-1}\right)^2</math>. Here, the symbol <math>\sqrt{-1}</math> denotes an hypothetical number which square would be <math>-1.</math><ref> Please note that this notation is purely formal and usual properties of the arithmetic (real) square root do not apply. Consider e.g. the following computation <math>-1=\sqrt{-1}\times\sqrt{-1}=</math> <br> <math>=\sqrt{(-1)\times(-1)}=\sqrt{1}=1</math> and the contradiction follows. The point is that the second equality can not be applied. The meaning of the symbol can be understood and the admissible operations can be specified by giving a precise definition in terms of more elementary mathematical objects (see the formal description).</ref><ref>Observe also that the symbol <math>\sqrt{a}</math> (or <math>\sqrt[n]{a}</math>) with <math>a\in\mathbb{C}</math> is sometimes used to denote the set of ''complex roots'' of ''a'', i.e. the set of the solutions of the equation <math>x^2=a</math> (<math>x^n=a</math> respectively). The set contains 2 (''n'', respectively) "equally important" elements and there is no canonical way to distinguish a "representative". Consequently, no computations are performed using this symbol.</ref> 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
Well, let us be bold and write <math>\Delta=\left(22\sqrt{-1}\right)^2</math>. Here, the symbol <math>\sqrt{-1}</math> denotes an hypothetical number whose square would be <math>-1.</math> At this stage, such a number has no meaning (squares 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


: <math>u^3=\frac{4+22\sqrt{-1}}{2}=2+11\sqrt{-1}</math> and <math>v^3=\frac{4-22\sqrt{-1}}{2}=2-11\sqrt{-1}.</math>
: <math>u^3=\frac{4+22\sqrt{-1}}{2}=2+11\sqrt{-1}</math> and <math>v^3=\frac{4-22\sqrt{-1}}{2}=2-11\sqrt{-1}.</math>
Line 37: Line 35:
It remains to find cube roots of these "numbers". A straightforward calculation shows that <math>u=2+\sqrt{-1}</math> and <math>v=2-\sqrt{-1}</math> do the job. For instance, remembering the rule <math>\left(\sqrt{-1}\right)^2=-1</math>, we have
It remains to find cube roots of these "numbers". A straightforward calculation shows that <math>u=2+\sqrt{-1}</math> and <math>v=2-\sqrt{-1}</math> do the job. For instance, remembering the rule <math>\left(\sqrt{-1}\right)^2=-1</math>, we have


: <math>\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;</math>
: <math>\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</math>


: <math>\left(2+\sqrt{-1}\right)^3=8+12\sqrt{-1}-6-\sqrt{-1}=2+11\sqrt{-1}.</math>
::::: <math>=8+12\sqrt{-1}-6-\sqrt{-1}=2+11\sqrt{-1}.</math>


But now, going back to the original cubic equation, we get the ''real'' solution <math>x=u+v=2+\sqrt{-1}+2-\sqrt{-1}=4</math>! One can verify it is indeed a solution, as <math>4^3=64=15\cdot 4+4</math> (and once this solution is found, it is easy to find the two other solutions, which are also real).
But now, going back to the original cubic equation, we get the ''real'' solution <math>x=u+v=(2+\sqrt{-1})+(2-\sqrt{-1})=4</math>. One can verify it is indeed a solution, as <math>4^3=64=15\cdot 4+4</math>. And once this solution is found, it is easy to find the two other solutions <math>-2\pm\sqrt3</math>, which are also real.


The fact that the formal calculations managed to give a real solution suggests that the "number" <math>\sqrt{-1}</math> 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.
The fact that the formal calculations managed to give a real solution suggests that the "number" <math>\sqrt{-1}</math> 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 as pairs of real numbers was given much later by [[William Rowan Hamilton]] in 1837; this construction is explained  [[Complex number#Formal definition|later in this article]].


==Formal definition==
==Working with complex numbers==
Formally, complex numbers are [[ordered pair]]s of real numbers<ref>We follow a popular approach. From another and perhaps more abstract point of view, complex numbers are defined in terms of polynomials, as the quotient <math>\mathbb{C}=\mathbb{R}[X]/\left(X^2+1\right).</math> Cf. also the section "Working with complex numbers" where an example of a quotient of polynomials is discussed.</ref>, i.e.
:<math>\mathbb{C}= \{ (a,b)~|~~ a,b\in \mathbb{R} \}.</math>


To call them 'numbers' we need to introduce some operations on such pairs. So we define
As a first step in giving some legitimacy to the "number" <math>\sqrt{-1}</math>, we will explain how to compute with it. How do you add, multiply and divide expressions with this number? It turns out that this is not that difficult; the main rule to keep in mind is that the square of <math>\sqrt{-1}</math> equals <math>-1</math>.
*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;<ref>that is, the addition (multiplication) is [[commutativity|commutative]]</ref> the sum (product) of three or more elements does not depend on order of operations ('we can suppress the parentheses');<ref>This is called [[associativity]]</ref> the product of a complex number with a sum of two other numbers expands in the usual way.<ref>In other words, multiplication is [[distributivity|distributive]] over addition</ref> In mathematical language this means that with addition and multiplication defined this way, <math>\mathbb{C}</math> satisfies the [[axiom]]s for a [[field]], and is called the field of complex numbers.  


Now we are ready to understand the 'real' meaning of <math>\sqrt{-1}</math> and its usage in the above historical example. Observe that the pairs of type (''a'',0) are identical<ref>i.e. [[isomorphism|isomorphic]], which basically means that the mapping <math>\mathbb{C}\ni (a,0)\mapsto a\in\mathbb{R},</math> which preserves the addition and multiplication.</ref> 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 <math>\sqrt{-1}</math>, 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.
In the remainder of the article, we will use the letter <math>i</math> to denote one solution of the equation <math>i^2 = -1</math>, where we previously used <math>\sqrt{-1}</math>.<ref>Part of the reason for not using <math>\sqrt{-1}</math> is that the symbol <math>\sqrt{a}</math> (or <math>\sqrt[n]{a}</math>) with <math>a\in\mathbb{C}</math> is sometimes used to denote the set of ''complex roots'' of <math>a</math>, i.e., the set of the solutions of the equation <math>x^2=a</math> (<math>x^n=a</math> respectively). The set contains 2 (<math>n</math>, respectively) "equally important" elements and there is no canonical way to distinguish a "representative". Consequently, no computations are performed using this symbol.</ref> With this convention, all complex numbers can be written as <math>a + bi</math>, where <math>a</math> and <math>b</math> are '''real''' numbers. We call <math>a</math> the real part of the complex number and <math>b</math> the imaginary part. Complex numbers whose imaginary part is <math>0</math> are of the form <math>a+0i</math>. In this way, the real number <math>a</math> is considered as the complex number <math>a + 0i</math> whose imaginary part is zero.


==Beyond the notation==
===Basic operations===
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 <math>(a,b)</math> rather than <math>a + bi</math>? There are several ways of answering this question. One is that our notation tends to guide our thinking, and writing <math>x = x +0i</math> emphasizes the idea that the real number ''x'' is a complex number, whereas writing <math>(x, 0)</math> 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 <math>a + bi</math> 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
 
:<math>4 + 5 \equiv 2 \pmod 7</math>
 
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 [[polynomial]]s in a single variable ''x''. We know that <math>(x + 1)(x +2) = x^2 + 3x + 2</math>, but <math>x^2 + 3x + 2 = 1\cdot(x^2 + 1) + (3x + 1)</math>, so when we divide by <math>x^2 + 1</math>, the remainder is <math>3x + 1</math>. And by the same token,
 
:<math>(1 + i)(2 + i) = 2 + 3i + i^2 = 1 + 3i \ </math>
 
so, when we add or multiply complex numbers, we are just doing modular arithmetic!<ref>This example gives also a hint why the complex numbers are sometimes defined in terms of polynomials, as <math>\mathbb{C}=\mathbb{R}[X]/\left(X^2+1\right).</math> </ref>  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==
Addition of complex numbers is straightforward, <math>(a + bi) + (c + di) = (a + c) + (b + d)i.</math> The result is again a complex number.
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 <math>i^2=-1</math>. Algebraic operations become as natural as for the reals.


===Basic operations===
Multiplication is more interesting. Suppose we want to compute <math>(a+bi)(c+di)</math>. Using <math>i^2 = -1</math>, we can rewrite this product in a form which clearly shows it to be another complex number:
 
Addition is straightforward, <math>(a + bi) + (c + di) = (a + c) + (b + d)i \ </math>. More notably, we can rewrite multiplication using <math>i^2 = -1</math> to obtain results in the form <math>a + bi</math>:
:<math>(a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (bc + ad)i. \ </math>
:<math>(a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (bc + ad)i. \ </math>


Line 83: Line 60:
from which it follows that
from which it follows that
:<math>\frac{a + bi}{c + di} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}.</math>
:<math>\frac{a + bi}{c + di} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}.</math>
If <math>z = a + bi</math> is a complex number, we call <math>a</math> the real part of <math>z</math> and write <math>a = Re (z)</math>. Similarly, <math>b</math> is called the imaginary part of <math>z</math> and we write <math>b = Im (z)</math>. If the imaginary part of a complex number is <math>0</math>, the number is said to be real. As mentioned earlier, we write <math>a</math> instead of <math>a + 0i</math> and thus we identify <math>\mathbb{R}</math> with a subset of <math>\mathbb{C}</math>.


Going a bit further, we can introduce the important operation of complex conjugation. Given an arbitrary complex number <math>z = x + iy</math>, we define its complex conjugate to be <math>\bar{z} = x - iy</math>. Using the identity <math>(a + b)(a - b) = a^2 - b^2</math> we derive the important formula
Going a bit further, we can introduce the important operation of complex conjugation. Given an arbitrary complex number <math>z = x + iy</math>, we define its complex conjugate to be <math>\bar{z} = x - iy</math>. Using the identity <math>(a + b)(a - b) = a^2 - b^2</math> we derive the important formula
Line 92: Line 67:
Note that the modulus of a complex number is always a ''nonnegative real'' number.
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
The modulus (also called absolute value) satisfies three important properties that are completely analogous to the properties of the absolute value of real numbers
*<math>|z| \ge 0</math> and <math>|z| = 0</math> if and only if <math>z = 0</math>
*<math>|z| \ge 0</math>; furthermore, <math>|z| = 0</math> if and only if <math>z = 0</math>
*<math>|z_1 z_2| = |z_1| |z_2| \ </math>
*<math>|z_1 z_2| = |z_1| |z_2| \ </math>
*<math>|z_1 + z_2 | \le |z_1| + |z_2|</math>
*<math>|z_1 + z_2 | \le |z_1| + |z_2|</math>
Line 100: Line 75:
Recall that in real analysis, the ordinary [[exponential]] function may be defined as
Recall that in real analysis, the ordinary [[exponential]] function may be defined as


:<math>\exp x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots</math>
:<math>e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots</math>


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


:<math>\exp z = 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \cdots</math>
:<math>e^z = 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \cdots</math>


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


:'''Notation''': The expressions <math>\exp \ z</math> and <math>e^z \ </math> mean the same thing, and may be used interchangeably.
The complex exponential has the same multiplicative property that  holds for real numbers, namely


The complex exponential has the same multiplicative property that  holds for real numbers,namely
:<math>e^{z_1 + z_2} = e^{z_1} e^{z_2} \ </math>
 
:<math>e^{z_1 z_2} = e^{z_1} e^{z_2} \ </math>


The complex exponential function has the important property that
The complex exponential function has the important property that
Line 118: Line 91:
:<math>e^{i\theta} = \cos \theta + i \sin \theta \ </math>
:<math>e^{i\theta} = \cos \theta + i \sin \theta \ </math>


as may be seen  immediately by  substituting <math>z = i\theta</math> and comparing terms with the usual power series expansions of <math>\sin \theta</math> and <math>\cos \theta</math>.
as may be seen  immediately by  substituting <math>z = i\theta</math> and comparing terms with the usual [[Taylor series|power series expansions]] of <math>\sin \theta</math> and <math>\cos \theta</math>.


The familiar [[trigonometry|trigonometric]] identity
The familiar [[trigonometry|trigonometric]] identity
Line 132: Line 105:
:<math>|e^{i\theta}|^2 = e^{i\theta}e^{-i\theta} = e^0 = 1. \ </math>
:<math>|e^{i\theta}|^2 = e^{i\theta}e^{-i\theta} = e^0 = 1. \ </math>


==Geometric interpretation==
===Geometric interpretation===


Since a complex number <math>z = x + iy</math> corresponds (essentially by definition) to an ordered pair of real numbers <math>(x, y)</math>, it can be interpreted as a point in the plane (i.e., <math>\mathbb{R}^2)</math>. When complex numbers are represented as points in the plane, the resulting diagrams are known as [[Robert Argand|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 [[geometry|geometrical]] and [[physics|physical]] problems.
[[Image:Complex_plane3.png|thumb|right|250px|Graphical representation of a complex number and its conjugate]]
Since a complex number <math>z = x + iy</math> is specified by two real numbers, namely <math>x</math> and <math>y</math>, it can be interpreted as the point <math>(x,y)</math> in the plane. When complex numbers are represented as points in the plane, the resulting diagrams are known as [[Robert Argand|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 and as a tool in applying complex numbers to [[geometry|geometrical]] and [[physics|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 [[vector]]s in <math>\mathbb{R}^2</math>. Similarly, if <math>\alpha \in \mathbb{R}</math> is real, multiplication by <math>\alpha</math> is just scalar multiplication. In <math>\mathbb{C}</math> we have
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 [[vector]]s in <math>\mathbb{R}^2</math>. Similarly, if <math>\alpha \in \mathbb{R}</math> is real, multiplication by <math>\alpha</math> is just scalar multiplication. In <math>\mathbb{C}</math> we have


:<math>z_1 + z_2 = (x_1 + iy_1) + (x_2 + iy_2) = (x_1 + x_2) + i(y_1 + y_2) \ </math>


:<math>z_1 + z_2 = (x_1 + iy_1) + (x_2 + iy_2) = (x_1 + x_2) + i(y_1 + y_2) \ </math>
and
and
:<math>\alpha z = \alpha(x + iy) = \alpha x + i\alpha y \ </math>


To put it succintly, <math>\mathbb{C}</math> is a 2-dimensional [[real number|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 coordinates|polar]] form
:<math>\alpha z = \alpha(x + iy) = \alpha x + i\alpha y. \, </math>


:<math>z = r e^{i\theta}</math>
To put it succinctly, <math>\mathbb{C}</math> is a 2-dimensional [[real number|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 coordinates|polar]] form


Here, r is simply the modulus <math>\sqrt{x^2 + y^2}</math> or vector length. The number <math>\theta</math> 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
:<math>z = r e^{i\theta}.</math>


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


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, [[scale|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
:<math>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)}.</math>
[[Image:Graphical_multiplication1.png|thumb|300px|left|Multiplication by <math>i</math> amounts to rotation by 90 degrees]]
In other words, multiplication by a complex number <math>z</math> has the effect of simultaneously scaling by the number's modulus and ''rotating'' by its argument. This is really astounding. For example, to multiply a given complex number <math>z</math> by <math>i</math> we need only to rotate <math>z</math> by <math>\pi/2</math> (that is, 90 degrees). [[Translation (geometry)|Translation]] corresponds to complex addition, [[scale|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


:<math>\frac{1}{z} = \frac{\bar{z}}{|z|^2}</math>
:<math>\frac{1}{z} = \frac{\bar{z}}{|z|^2}.</math>


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) <math>x + iy \mapsto x - iy</math> or, in vector notation, <math>(x, y) \mapsto (x, -y)</math>. 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.
In other words, up to a scaling factor, division of one by <math>z</math> is just complex conjugation. Returning to the representation of complex numbers in rectangular form, we note that complex conjugation is just the transformation (or map) <math>x + iy \;\mapsto\; x - iy</math> or, in vector notation, <math>(x, y)\; \mapsto \;(x, -y)</math>. This is nothing other than reflection in the <math>x</math>-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 [[physics|mathematical physics]]).
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 [[physics|mathematical physics]]).
Line 161: Line 136:
==Algebraic closure==
==Algebraic closure==


An important property of <math>\mathbb{C}</math> is that it is [[algebraically closed]]. This means that any non-constant real [[polynomial]] must have a root in <math>\mathbb{C}</math>. 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 <math>p(z)</math> is a polynomial function of a complex variable then both <math>p(z)</math> and <math>1/p(z)</math> will be [[holomorphic]] in any domain where <math>p(z) \not= 0</math>. But, by the triangle inequality, we know that outside a neighborhood of the origin <math>|p(z)| > |p(0)|</math>, so if there is no <math>z_0 </math> such that <math>p(z_0) = 0</math>, we know that <math>1/p(z)</math> is a bounded entire (i.e., holomorphic in all of <math>\mathbb{C}</math>) function. By [[Liouville's theorem]], it must be constant, so <math>p(z)</math> must also be constant.
An important property of the set of complex numbers is that it is [[algebraically closed]]. This means that any non-constant [[polynomial]] with complex coefficients has a complex root. This result is known as the [[Fundamental Theorem of Algebra]].  
 
There are also proofs that do not depend on [[complex analysis]], but they require more [[algebra|algebraic]] or [[topology|topological]] machinery. The starting point here is that <math>\mathbb{R}</math> 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 <math>\mathbb{C} = \mathbb{R}[i]</math> is the splitting field of <math>x^2 + 1</math>, so if we can show that <math>\mathbb{C}</math> has no finite extensions, then we are done. Suppose <math>K/\mathbb{C}</math> 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 <math>\mathbb{R}</math> 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 <math>y = f(x)</math>, we say ''f'' is differentiable at <math>x_0</math> if the limit
 
:<math>\lim_{h\to 0} \frac{f(x_0 + h) - f(x_0)}{h}</math>
 
exists, and we call the limiting value the derivative of ''f'' at <math>x_0</math>, and the function that assigns to each point x the derivative of ''f'' at ''x'' is called the derivative of ''f'', and is written <math>f'(x)</math> or <math>df/dx</math>. 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 <math>c = f'(x)</math>, then for every (real) number <math>\varepsilon > 0</math>, there is a (real) number <math>\delta</math> such that if <math>|h| < \delta</math>
 
:<math>\left | \frac{f(x + h) - f(x)}{h} - c \right | < \varepsilon</math>
 
This makes perfect sense for functions of a complex variable, but we need to keep in mind that <math>| \cdot |</math> 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 <math>w = f(z)</math> ''u'' and ''v'', (i.e., <math>w = f(z) = u + iv</math>), this requirement can be expressed in terms of the [[partial derivative]]s of ''u'' and ''v'' with respect to ''x'' and ''y'':
 
:<math>\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}</math>
and
:<math>\frac{\partial v}{\partial x} = - \frac{\partial u}{\partial y}</math>
 
These equations are known as the [[Cauchy-Riemann equations]].
 
:'''Note''': These equations are frequently written in the more compact form, <math>u_x = v_y</math> and <math>v_x = - u_y</math>.
 
They may be obtained by noting that if the approach path is on ''x''-axis, <math>\partial f / \partial y = 0</math>, so
 
:<math>\frac{df}{dz} = \frac{1}{2} \left ( \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x} \right )</math>
 
and that on the ''y''-axis, <math>\partial f / \partial x = 0</math>, so
 
:<math>\frac{df}{dz} = \frac{1}{2} \left (-i \frac {\partial u}{\partial y} + \frac{\partial v}{\partial y} \right )</math>
 
These equations have far-reaching implications. To get some idea if why this is so, consider that we can take second derivatives to obtain


:<math>u_{xx} + u_{yy} = 0</math>
This is actually quite remarkable. We started out with the real numbers. There are many polynomials with real coefficients that do not have a real root. We took just one of these, the polynomial <math>x^2+1</math>, and we introduced a new number, <math>i</math>, which is defined to be a root of the polynomial. Suddenly, all non-constant polynomials have a root in this new setting where we allow complex numbers.
and
:<math>v_{xx} + v_{yy} = 0</math>


In other words, ''u'' and ''v'' satisfy [[Laplace's equation]] in 2 dimensions. These functions arise in [[physics|mathematical physics]] as [[scalar potential]]s in, for example, [[fluid dynamics]]. [[Laplace's equation]] is also basic to the study of [[partial differential equation]]s. This is but one indication of the reason for the ubiquity of complex functions in [[physics]].
There are many proofs of the Fundamental Theorem of Algebra. Many of the simplest depend crucially on [[complex analysis]]. But it is by no means necessary to rely on complex analysis here. A proof using [[field theory]] is alluded to at the very end of this article.
 
===Integration===
 
By contrast, the definition of [[integral|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 <math>\gamma</math> defined in that domain
 
:<math>\int_{\gamma}\nolimits f \, dz = 0.</math>
 
It is essential that the domain of definition be simply connected. For example, let
 
:<math>D = \{ z \mid \textstyle\frac 1 2 < |z| < \frac 3 2 \}</math>
 
and let <math>f(z) = 1/z</math>. Then if we define <math>\gamma (t) = e^{it}</math> where ''t'' ranges from 0 to <math>2 \pi</math> (i.e., we take <math>\gamma</math> to be the unit circle), then the integral will ''not'' be 0.
 
It follows that if <math>\gamma_1</math> and <math>\gamma_2</math> are two [[homotopic]] paths joining a pair of points <math>P, Q \in D</math> (intuitively, one can be deformed into the other), then
 
:<math>\int_{\gamma_1}\nolimits f dz = \int_{\gamma_2}\nolimits f \, dz.</math>
 
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]]
 
:<math>f(z) = \frac{1}{2\pi i} \int_{\gamma}\nolimits \frac{f(\zeta) \, d \zeta}{\zeta - z}.</math>
 
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.


==Complex numbers in physics==
==Complex numbers in physics==
Line 237: Line 146:
Complex numbers appear everywhere in mathematical physics, but one area where the role of complex numbers is especially difficult to ignore is in [[quantum mechanics]]. There are a number of ways of formulating the basic laws of quantum mechanics, but here we consider just one: the [[Schrödinger equation]], discovered by Erwin Schrödinger in 1926. In rectangular coordinates, it may be written
Complex numbers appear everywhere in mathematical physics, but one area where the role of complex numbers is especially difficult to ignore is in [[quantum mechanics]]. There are a number of ways of formulating the basic laws of quantum mechanics, but here we consider just one: the [[Schrödinger equation]], discovered by Erwin Schrödinger in 1926. In rectangular coordinates, it may be written


:<math>i\hbar \frac{\partial \psi}{\partial t} = - \frac{\hbar^2}{2m}\Delta\psi + V(x,y,z)\psi</math>,
:<math>i\hbar \frac{\partial \psi}{\partial t} = - \frac{\hbar^2}{2m}\Delta\psi + V(x,y,z)\psi,</math>


where
where


:<math>\Delta = \frac{\partial^2}{\partial^2 x} + \frac{\partial^2}{\partial^2 y} + \frac{\partial^2}{\partial^2 z}</math>  
:<math>\Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}</math>
 
is known as the [[Laplacian operator]] and <math>V(x,y,z)</math> is the potential function. (As a practical example, minus the gradient of the potential function might represent the attractive force  between the nucleus of a hydrogen atom and an electron).
 
==Formal definition==


is known as the [[Laplacian operator]] and <math>V(x,y,z)</math> is the potential function. (As a practical example, the potential function might represent the attractive force per unit mass between the nucleus of a hydrogen atom and an electron). Now, there is some subtlety in the interpretation of <math>\psi</math> because a system can be affected by observation, and the functions \psi we "see" must be ''eigenstates'' of the operator defined by the Schrödinger equation, but when we do measure, say, the position of a particle, the probability of finding it in a small region R is just
This all shows that complex numbers behave very much like real numbers and that they can be very useful, but it does not prove that they exist. In fact, it is quite easy to go wrong when using complex numbers. Consider for instance the following computation:
:<math>-1=\sqrt{-1}\times\sqrt{-1}=\sqrt{(-1)\times(-1)}=\sqrt{1}=1.</math>
This computation seems to show that <math>-1</math> equals <math>1</math>, which is nonsense. The point is that the second equality can not be applied. Positive real numbers satisfy the identity
:<math> \sqrt{a}\times\sqrt{b} = \sqrt{a \times b}, </math>
but this identity does not hold for negative real numbers, whose square roots are not real.


:<math>\int_R\nolimits |\psi|^2 </math>.
One possibility to feel more secure when using complex numbers is to define them in terms of constructs which are better understood. This approach was taken by [[William Rowan Hamilton|Hamilton]], who defined complex numbers as [[ordered pair|ordered pairs]] of real numbers, that is,
:<math>\mathbb{C}= \{ (a,b) \colon a,b\in \mathbb{R} \}.</math>
Such pairs can be added and multiplied as follows
*addition: <math>(a, b) + (c, d) = (a + c, b + d)</math>
*multiplication: <math>(a, b)(c, d) = (ac - bd, bc + ad)</math>
The multiplication may look artificial, but it is inspired by the formula
:<math>(a + bi)(c + di) = (ac - bd) + (bc + ad)i. \ </math>
which we derived before.


It's not hard to see that these functions must be complex waves, but it can be demonstrated experimentally that this must be so. In a famous experiment<ref>[http://physicsweb.org/articles/world/15/9/1]</ref> electron beams split into separate streams that then converge on the same target  turn out to be able to  create a diffraction pattern (just as if it were ripples in water form two patterns of concentric circular waves interfering with one another!) The diffraction pattern shows that we have two waves interfering with one another and not simple particles scattered at random angles.
These definitions satisfy most of the basic properties of addition and multiplication of real numbers, 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;<ref>that is, the addition (multiplication) is [[commutativity|commutative]]</ref> the sum (product) of three or more elements does not depend on order of operations ('we can suppress the parentheses');<ref>This is called [[associativity]]</ref> the product of a complex number with a sum of two other numbers expands in the usual way.<ref>In other words, multiplication is [[distributivity|distributive]] over addition</ref> In mathematical language this means that with addition and multiplication defined this way, <math>\mathbb{C}</math> satisfies the [[axiom|axioms]] for a [[field]] and is called the field of complex numbers.  


==Further reading==
Now we are ready to understand the 'real' meaning of <math>i</math>. Observe that the pairs of type (<math>a</math>,0) are identical<ref>i.e., [[isomorphism|isomorphic]], which basically means that the mapping <math> \mathbb{C}\ni (a,0)\mapsto a\in\mathbb{R},</math> preserves the addition and multiplication.</ref> to the set of reals, so we write <math>(a,0)=a</math>. Observe also that by definition <math>(0,1)(0,1) = (-1,0)=-1</math>. In other words, we can define <math>i</math>, the number satisfying <math>i^2=-1</math>, as the pair (0,1).<ref>although we should be careful about giving this particular definition too much credit: after all, the number <math>(0,-1)</math> has exactly the same property!</ref>


*{{cite book
Another way to define the complex numbers comes from [[field theory]]. Because <math>x^2+1</math> is [[irreducible polynomial|irreducible]] in the [[polynomial ring]] <math>\mathbb{R}[x]</math>, the [[ideal (ring theory)|ideal]] generated by <math>x^2+1</math> is a [[maximal ideal]].<ref>An ideal <math>I = \left(f(x)\right)</math> in a polynomial ring over a field is maximal if and only if <math>f(x)</math> is irreducible over the field.</ref> Therefore, the [[quotient ring]] <math> \mathbb{C}=\mathbb{R}[x]/\left(x^2+1\right)</math> is a [[field (mathematics)|field]]. We can choose the polynomials of degree at most 1 as the representatives for the equivalence classes in this quotient ring. So in a sense, we can imagine that the dummy variable <math>x</math> is the imaginary number <math>i</math>, and the elements of the quotient ring behave exactly the way we expect the complex numbers to behave. For example, <math>x^2</math> is in the same equivalence class as <math>-1</math>, and so <math>x^2=-1</math> in this quotient ring. (As a final comment in this analysis, we could next show that <math>\mathbb{C}</math> has no finite [[field extension|extension]] and must therefore be [[algebraic closure|algebraically closed]].)
|last = Ahlfors
|first = Lars V.
|title = Complex Analysis
|edition = 3rd edition
|date = 1979
|publisher = McGraw-Hill, Inc.
|isbn = 0-07-000657-1}}
*{{cite book
|last = Apostol
|first = Tom M.
|title = Mathematical Analysis
|edition = 2nd edition
|date = 1974
|publisher = Addison-Wesley
|isbn = 0-201-00-288-4 }}
* {{cite book
|last = Conway
|first = John H.
|coauthors = Derek A. Smith
|title = On Quaternions and Octonions: Their Geometry, Arithmetic and Symmetry
|date = 2003
|publisher = A K Peters, Ltd.
|isbn = 1-56881-134-9 }}
*{{cite book
|last = Jacobson
|first = Nathan
|title = Basic Algebra I
|date = 1974
|publisher = W.H. Freeman and Company
|isbn = 0-7167-0453-6 }}
* {{cite book
|last = Williams
|first = Floyd
|title = Topics in Quantum Mechanics
|series = Progress in Mathematical Physics
|date = 2003
|publisher = Birkhäuser
|isbn = 0-8176-4311-7 }}


==Notes and references==
==Notes and references==
{{reflist|2}}
{{reflist|2}}
[[Category:Mathematics Workgroup]]
[[category:CZ Live]]

Latest revision as of 17:11, 25 August 2013

This article has a Citable Version.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
Advanced [?]
 
This version approved either by the Approvals Committee, or an Editor from the listed workgroup. The Mathematics Workgroup is responsible for this citable version. While we have done conscientious work, we cannot guarantee that this version is wholly free of mistakes. See here (not History) for authorship.
Help improve this work further on the editable Main Article!

Complex numbers are numbers of the form , where and are real numbers and denotes a number satisfying .[1] Of course, since the square of any real number is nonnegative, cannot be a real number. At first glance, it is not even clear whether such an object exists and can be reasonably called a number; for example, can we sensibly associate with natural operations such as addition and multiplication? As it happens, we can define mathematical operations for these "complex numbers" in a consistent and sensible way and, perhaps more importantly, using complex numbers provides mathematicians, physicists, and engineers with an extremely powerful approach to expressing parts of these sciences in a convenient and natural-feeling way.

Historical example

The need for complex numbers might 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. Even for equations with three real solutions, the method they used sometimes required calculations with numbers whose squares are negative. Here is such an example (with modern notation). Let us consider the equation

Cardano's method for solving it suggests looking for a solution by writing it as a sum , where another condition on and is to be decided later. Recording this in the equation, we have, once the left member is expanded,

which can be written as

Now we recall that we did not completely specify and ; we only required that . Hence, we can choose another condition on and . We pick this condition to be , or , in order to simplify the above equation. This implies that and are numbers whose sum and product are given by

It follows from the second equation that . Substituting this in the first equation, we get . Hence we may find some values for by solving the equation . Get rid of the brackets and move the number 125 to the left-hand side to get the quadratic equation

Its discriminant is , which is negative, so that the quadratic equation has no real solution: the usual formulae giving the solutions require taking the square root of the discriminant, which is undefined here.

Well, let us be bold and write . Here, the symbol denotes an hypothetical number whose square would be At this stage, such a number has no meaning (squares 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

and

It remains to find cube roots of these "numbers". A straightforward calculation shows that and do the job. For instance, remembering the rule , we have

But now, going back to the original cubic equation, we get the real solution . One can verify it is indeed a solution, as . 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" 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 as pairs of real numbers was given much later by William Rowan Hamilton in 1837; this construction is explained later in this article.

Working with complex numbers

As a first step in giving some legitimacy to the "number" , we will explain how to compute with it. How do you add, multiply and divide expressions with this number? It turns out that this is not that difficult; the main rule to keep in mind is that the square of equals .

In the remainder of the article, we will use the letter to denote one solution of the equation , where we previously used .[2] With this convention, all complex numbers can be written as , where and are real numbers. We call the real part of the complex number and the imaginary part. Complex numbers whose imaginary part is are of the form . In this way, the real number is considered as the complex number whose imaginary part is zero.

Basic operations

Addition of complex numbers is straightforward, The result is again a complex number.

Multiplication is more interesting. Suppose we want to compute . Using , we can rewrite this product in a form which clearly shows it to be another complex number:

To handle division, we simply note that , so

from which it follows that

Going a bit further, we can introduce the important operation of complex conjugation. Given an arbitrary complex number , we define its complex conjugate to be . Using the identity we derive the important formula

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

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

  • ; furthermore, if and only if

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

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

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

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

The complex exponential function has the important property that

as may be seen immediately by substituting and comparing terms with the usual power series expansions of and .

The familiar trigonometric identity

immediately implies the important formula

, for any

Of course, there is no reason to assume this identity. We only need note that , so

Geometric interpretation

Graphical representation of a complex number and its conjugate

Since a complex number is specified by two real numbers, namely and , it can be interpreted as the point in the plane. 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 and 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 . Similarly, if is real, multiplication by is just scalar multiplication. In we have

and

To put it succinctly, 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

Here, r is simply the modulus or vector length. The number is just the angle formed with the -axis, and is called the argument. Now, when complex numbers are written in polar form, multiplication is very interesting

Multiplication by amounts to rotation by 90 degrees

In other words, multiplication by a complex number has the effect of simultaneously scaling by the number's modulus and rotating by its argument. This is really astounding. For example, to multiply a given complex number by we need only to rotate by (that is, 90 degrees). 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

In other words, up to a scaling factor, division of one by is just complex conjugation. Returning to the representation of complex numbers in rectangular form, we note that complex conjugation is just the transformation (or map) or, in vector notation, . This is nothing other than reflection in the -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 the set of complex numbers is that it is algebraically closed. This means that any non-constant polynomial with complex coefficients has a complex root. This result is known as the Fundamental Theorem of Algebra.

This is actually quite remarkable. We started out with the real numbers. There are many polynomials with real coefficients that do not have a real root. We took just one of these, the polynomial , and we introduced a new number, , which is defined to be a root of the polynomial. Suddenly, all non-constant polynomials have a root in this new setting where we allow complex numbers.

There are many proofs of the Fundamental Theorem of Algebra. Many of the simplest depend crucially on complex analysis. But it is by no means necessary to rely on complex analysis here. A proof using field theory is alluded to at the very end of this article.

Complex numbers in physics

Complex numbers appear everywhere in mathematical physics, but one area where the role of complex numbers is especially difficult to ignore is in quantum mechanics. There are a number of ways of formulating the basic laws of quantum mechanics, but here we consider just one: the Schrödinger equation, discovered by Erwin Schrödinger in 1926. In rectangular coordinates, it may be written

where

is known as the Laplacian operator and is the potential function. (As a practical example, minus the gradient of the potential function might represent the attractive force between the nucleus of a hydrogen atom and an electron).

Formal definition

This all shows that complex numbers behave very much like real numbers and that they can be very useful, but it does not prove that they exist. In fact, it is quite easy to go wrong when using complex numbers. Consider for instance the following computation:

This computation seems to show that equals , which is nonsense. The point is that the second equality can not be applied. Positive real numbers satisfy the identity

but this identity does not hold for negative real numbers, whose square roots are not real.

One possibility to feel more secure when using complex numbers is to define them in terms of constructs which are better understood. This approach was taken by Hamilton, who defined complex numbers as ordered pairs of real numbers, that is,

Such pairs can be added and multiplied as follows

  • addition:
  • multiplication:

The multiplication may look artificial, but it is inspired by the formula

which we derived before.

These definitions satisfy most of the basic properties of addition and multiplication of real numbers, 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;[3] the sum (product) of three or more elements does not depend on order of operations ('we can suppress the parentheses');[4] the product of a complex number with a sum of two other numbers expands in the usual way.[5] In mathematical language this means that with addition and multiplication defined this way, satisfies the axioms for a field and is called the field of complex numbers.

Now we are ready to understand the 'real' meaning of . Observe that the pairs of type (,0) are identical[6] to the set of reals, so we write . Observe also that by definition . In other words, we can define , the number satisfying , as the pair (0,1).[7]

Another way to define the complex numbers comes from field theory. Because is irreducible in the polynomial ring , the ideal generated by is a maximal ideal.[8] Therefore, the quotient ring is a field. We can choose the polynomials of degree at most 1 as the representatives for the equivalence classes in this quotient ring. So in a sense, we can imagine that the dummy variable is the imaginary number , and the elements of the quotient ring behave exactly the way we expect the complex numbers to behave. For example, is in the same equivalence class as , and so in this quotient ring. (As a final comment in this analysis, we could next show that has no finite extension and must therefore be algebraically closed.)

Notes and references

  1. This article follows the usual convention in mathematics and physics of using as the imaginary unit. Complex numbers are frequently used in electrical engineering, but in that discipline it is usual to use instead, reserving for electrical current. This usage is found in some programming languages too, notably Python.
  2. Part of the reason for not using is that the symbol (or ) with is sometimes used to denote the set of complex roots of , i.e., the set of the solutions of the equation ( respectively). The set contains 2 (, respectively) "equally important" elements and there is no canonical way to distinguish a "representative". Consequently, no computations are performed using this symbol.
  3. that is, the addition (multiplication) is commutative
  4. This is called associativity
  5. In other words, multiplication is distributive over addition
  6. i.e., isomorphic, which basically means that the mapping preserves the addition and multiplication.
  7. although we should be careful about giving this particular definition too much credit: after all, the number has exactly the same property!
  8. An ideal in a polynomial ring over a field is maximal if and only if is irreducible over the field.