Algebraic Structure of Complex Numbers

When you come to quadratic equations you will be confronted with an entity (or non-entity) whose name is written this way - -1, and pronounced "square root of minus one."

Mary Everest Boole
Philosophy And Fun Of Algebra,
London: C. W. Daniel, LTD, 1909

Complex numbers are points in the plane endowed with additional structure. We consider the set R2 = {(x, y): x, yR}, i.e., the set of ordered pairs of real numbers. Two such pairs are equal if their corresponding components coincide:

(x1, y1) = (x2, y2) iff x1 = x2 and y1 = y2.

With two operations - addition and multiplication - defined below, the set R2 becomes the set C of complex numbers. R2 considered as a set of complex numbers C is called Argand diagram or Argand plane or Gauss plane.

Addition

(x1, y1) + (x2, y2) = (x1 + x2, y1 + y2).

Multiplication

(x1, y1)(x2, y2) = (x1x2 - y1y2, x1y2 + x2y1).

Addition is defined componentwise in a relatively standard way that extends to spaces of higher dimension. Multiplication on the other hand is peculiar to complex numbers. It only has weakened analogues in R4 (quaternions) and R8 (octonions).

Addition and multiplication of complex numbers inheret most of the properties of addition and multiplication of real numbers:

 z + w = w + z and zw = wz  (Commutativity),
 z + (u + v) = (z + u) + v and z(uv) = (zu)v  (Associativity),
 z(u + v) = zu + zv  (Distributive Law).

Several complex numbers play exclusive roles. For example, the number (0, 0) has the properties of 0:

(x, y) + (0, 0) = (x, y) and
(x, y)(0, 0) = (0, 0).

It is therefore natural to identify it with 0. The symbol is exactly the same as used to identify the "real" 0. We shall see shortly that there is a good reason to think of the two zeros - real and complex - as one and the same number.

Another complex number of consequence is (1, 0). This number plays an important role in multiplication that stems from the following property:

(x, y)(1, 0) = (x·1 - y·0, x·0 + y·1)
 = (x, y).

Among complex numbers (1, 0) behaves like the real unit 1 among the real numbers. Again, there is a good reason to say that the two are one and the same. It is customary to write (1, 0) = 1.

The third number of importance is (0, 1). It has the remarkable feature of having a negative square. More accurately,

(i)
(0, 1)(0, 1) = (0·0 - 1·1, 0·1 + 1·0)
 = (-1, 0).

In engineering sciences, the number (0, 1) is sometimes denoted as j. Elsewhere, it is standard to denote it i: i = (0, 1). Multiplication by i has a curious effect:

(x, y)(0, 1) = (x·0 - y·1, x·1 + y·0)
 = (-y, x).

If you compare two points (x, y) and (y, -x) on the plane (even if for hust a few specific values of a and y) and join the two to the origin, you'll be able to observe that the two segments are perpendicular to each other. Moreover, the latter is obtained from the former by rotation through 90o in the positive (counterclockwise) direction.

The theory of complex numbers can be developed wholy in algebraic terms, see, for example, Landau. Often, however, both on elementary and advanced level drawing from geometric intuition is extremely useful. Numbers (x, 0) correspond to points on the horizontal x-axis. If it were not for the presence of the y-axis, the points on the horizontal number line would be associated with plain real numbers. The point corresponding to (x, 0) would be considered as a real number x. Thus, it is natural to identify the two representaions of the same point:

(1) x = (x, 0).

Note that previously we already did this for x = 0 and x = 1. The identification (1) is also supported algebraically. For, algebraically, x and (x, 0) are indistinguishable:

(x1, 0) + (x2, 0) = (x1 + x2, 0) and
(x1, 0)(x2, 0) = (x1x2, 0),

as if the second component (0) was not present. With (1) in mind, we can write

(x, y)= (x, 0) + (0, y)
 = (x, 0) + (y, 0)(0, 1)
 = x + yi,

which is called the algebraic form of complex number (x, y):

x + yi = (x, y).

With (1), we easily multiply complex numbers by real:

r(x, y) = (r, 0)(x, y) = (rx, ry),

naturally. Which of course introduces nothing new but a convenience of notations. In the algebraic form, the addition and multiplication are redefined as

(x1 + y1i) + (x2 + y2i) = (x1 + x2) + (y1 + y2)i and
(x1 + y1i)(x2 + y2i) = (x1x2 - y1y2) + (x1y2 + x2y1)i.

Without (1), i2 would for ever remain (-1, 0). Taking (1) into account we obtain the famous identity

(2) i2 = (0, 1)2 = (-1, 0) = -1.

The possibility of embedding of the set R of reals into the set of complex numbers C, as defined by (1), is probably the single most important property of complex numbers. For, without (1) and (2), the theory of complex numbers would not deliver the closure to the branch of algebra that drove much of its development, viz., the search for the roots of polynomial equations.

The two parts of the complex number z = x + yi have special notations:

x = Re(z), y = Im(z).

Both real and imaginary parts of a complex number are real and any complex number can be written as

z = Re(z) + i·Im(z).

Re and Im are mnemonics for Real and Imaginary. Neither is less real or more imaginary than the other. In the complex plane the axes also are referred to as real and imaginary, although both are real enough to the extent that the only way to distinguish between the two is by means of orientation: the rotation from the real to the imaginary axis proceeds counterclockwise. Complex numbers for which the real part is 0, i.e., the numbers in the form yi, for some real y, are said to be purely imaginary.

With every complex number (x + yi) we associate another complex number (x - yi) which is called its conjugate. The conjugate of number z is most often denoted with a bar over it, sometimes with an asterisk to the right of it, occasionally with an apostrophe and even less often with the plain symbol Conj as in

Conj(z) = z* = z' = z.

For technical reasons, I prefer using the least common notations, z' or Conj(z):

Conj(x + yi) = (x + yi)' = x - yi.

The importance of the conjugate stems from the following fact:

(x + yi)(x + yi)' = (x + yi)(x - yi) = x2 + y2,

which, for any complex number z = x + yi, is a non-negative real number. The non-negative square root of this number is called the modulus or absolute value of complex number z:

(m) |x + yi| = x² + y²

From the Pythagorean theorem, |z| is the distance from the point represented by z (or the point with complex coordinate z and real coordinates (x, y)) to the origin 0. In general, |z - u| is the distance between points z and u.

The operator Conj is an involution, its square is the identity operator:

Conj(Conj(z)) = z'' = z

and therefore

|z|2 = z·z' = |z'|2.

Also

(zw)' = z'·w',

which easily submits to direct verification. The definition of modulus is consistent with the definition of the absolute value of the real numbers:

|(x, 0)| = |x|.

In addition, it has several important properties:

(m1)|z| ≥ 0, |z| = 0 iff z = 0.
(m2)|zw| = |z|·|w|.
(m3)|z + w| ≤ |z| + |w|.

The latter is known as the triangle inequality. Geometrically, it's a feature inherited from Euclid. It can be obtained from the Argand diagram, i.e. from the identification of complex numbers with points in a plane with a reference to a triangle inequality valid in the Euclidean plane. It can also be derived algebraically. (m1) follows from the definition. (m2) is a consequence of the basic laws

|zw|2= (zw)·(zw)'
 = z·w·z'·w'
 = [z·z']·[w·w']
 = |z|2·|w|2
 = (|z|·|w|)2.

In particular, for a real r > 0, |rz| = r|z|, for any complex z.

Since |z| is the distance to the origin, |z| = 1 is the equation of the unit circle centered at the origin. Points on the unit circle are associated with an angle such that those points and only them have the form (cosa, sina), for some angle a which is not determined uniquely. As a complex number, the points on the unit circle have the form

z = cosa + i·sina.

For any z ≠ 0, |z/|z|| = 1 which means that z/|z| lies on the unit circle and therefore has the above form for some a. If we denote |z| = r, then z can be written as

(3) z = r(cosa + i·sina),

which is known as the trigonometric form of complex number, its polar representation.

If z ≠ 0 and a1 and a2 satisfy (3), then they differ by a factor of 2p:

a1 - a2 = k·2p,

where k an integer. In other words,

a1 = a2 (mod 2p).

There is a unique a[0, 2p), which denotes a half-open interval from 0 to 2p in which 0 is included but 2p is not. This a is called the argument of z denoted arg(z):

z = |z|·(cos(arg(z)) + i·sin(arg(z))),

where arg(z)[0, 2p). Zero is not assigned an argument. Thus the argument of a complex number is a real number in a limited interval. The extended argument Arg(z) of a number z is the set of all real numbers congruent to arg(z) modulo 2p:

 Arg(z)= {a: a = arg(z) (mod 2p))}
   = {arg(z) + k·2p: kN}.

So that

z = |z|·(cos(a) + i·sin(a)), for any aArg(z).

Obviously, the number i = (0, 1) plays a special role in the theory of complex numbers. From the properties of sine and cosine, arg(i) = p/2. Furthermore,

i·(x, y) = (0, 1)·(x, y)
  = (0·x - 1·y, 0·y + 1·x)
  = (-y, x).

Treated as two vectors, (x, y) and (-y, x) are immediately seen to be orthogonal, with the latter obtained from the former by a rotation through p/2 in the positive direction.

(There is a dynamic illustration of the properties of complex numbers and the operations discussed here.)

References

  1. T. Andreescu, D. Andrica, Complex Numbers From A to ... Z, Birkhäuser, 2006
  2. C. W. Dodge, Euclidean Geometry and Transformations, Dover, 2004 (reprint of 1972 edition)
  3. Liang-shin Hahn, Complex Numbers & Geometry, MAA, 1994
  4. E. Landau, Foundations of Analisys, Chelsea Publ, 3rd edition, 1966
  5. C. Zwikker, The Advanced Geometry of Plane Curves and Their Applications, Dover, 2005

Complex Numbers

  1. Algebraic Structure of Complex Numbers
  2. Division of Complex Numbers
  3. Useful Identities Among Complex Numbers
  4. Useful Inequalities Among Complex Numbers
  5. Trigonometric Form of Complex Numbers
  6. Real and Complex Products of Complex Numbers
  7. Complex Numbers and Geometry
  8. Plane Isometries As Complex Functions
  9. Remarks on the History of Complex Numbers
  10. Complex Numbers: an Interactive Gizmo
  11. Cartesian Coordinate System
  12. Fundamental Theorem of Algebra
  13. Complex Number To a Complex Power May Be Real
  14. One can't compare two complex numbers
  15. Riemann Sphere and Möbius Transformation
  16. Problems

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