The Imagination Unit, Part 2

The Standard Imaginary Unit

As mentioned earlier,
the relativistic mass M of an ordinary particle in motion can be
related to the same particle's rest-mass, m, by the equation;

M = Rm = m / [(1 - [(v/c)^2])^(1/2)] .

Consider again a tachyon of mass tM, with correspondingly the same
amount of moving mass. The tachyonic mass, tM, can be represented by
describing it as an imaginary analog of M, written;
tM = -iM ,
where i is the standard imaginary-unit, defined;
i = (-1)^(1/2) , so that i^2 = -1 .

Note that the minus-sign accompanying i, in this definition of tM, is
absolutely mandatory for presenting a rigorous definition of tM.

In such cases, the imaginary-unit (i) is used in a purely algebraic
sense, as an operator that, when multiplied to a real number, is
understood - by convention - to imply that the given real number
is to be viewed as an imaginary number. To go any further on this
topic, however, it is necessary to lay some groundwork, so that later
statemements will be readily understood. [Readers sufficiently
familiar with the theory behind complex and imaginary numbers, as
presented below, may wish to proceed to the next post.]

The imaginary-unit comes about as a natural consequence of
considering certain numbers that cannot be categorized as "real".
For instance, no real number x is such that x^2 = -1.

We can, however, imagine another kind of number, i, defined
specifically as the square-root of -1, so that i^2 = -1.
Thus, if X is a positive real number, and we want to find the square-
root of its negative, then we can always write;
(-X)^(1/2) = [(-1)X]^(1/2) = ... = i[X^(1/2)] .
For example,
(-25)^(1/2) = [(-1)(25)]^(1/2) = [(-1)^(1/2)][25^(1/2)] = i5 .

Now, all of the sums of real and imaginary numbers form a set called
the "complex numbers", which, obviously, includes the set of all real
numbers and the set of all imaginary numbers.

To explain how complex numbers work, let x and y denote real
numbers. Then let iy denote an imaginary number, and let z denote
the sum of x and iy, according to the equation;
z = x + iy .

Here, z is a complex number, while x is referred to as the "real-
number part" or "real component" of z, and y is referred to as
the "imaginary-number part" or "imaginary component" of z.

We can also represent this using function notation, where Re is a
function of z that gives a real number, Re(z), and Im is a function
of z that gives an imaginary number, Im(z), defined;
z = x + iy = Re(z) + Im(z) ,
where Re(z) = x , and Im(z) = iy .

Consequently, if x is nonzero but iy = 0, then z is real.
On the other hand, if iy is nonzero but x = 0, then z is referred to
as a "pure imaginary".
Of course, whenever z = 0, then one of the following mutually
exclusive cases must be true;
Case 1: x = 0 and y = 0 simultaneously, or
Case 2: iy = -x , where x and y are each nonzero.

Interestingly, because complex numbers are essentially the same as
ordered pairs of numbers, then the following definitions hold for
almost all complex numbers.

The absolute-value |z| of a standard complex number z, and which
absolute-value is called the "modulus" of z, is a real number that
can be obtained using the Pythagorean theorem;
|z| = |x + iy| = [(x^2) + (y^2)]^(1/2) .
Letting z denote a complex number defined as a sum, so that
z = x + iy ,
and letting z* denote another complex number defined as the
corresponding difference, so that
z* = x - iy ,
where z* employs the same values of x and y as does z,
we say that z* is the "conjugate" of z.
The product of z and z* is the square of the modulus of z, according
to the following proof;
z*z = (x - iy)(x + iy) = (x^2) - xiy + xiy - [(iy)^2]
= (x^2) - [(-1)(y^2)] = (x^2) + (y^2) = |z|^2 .
The ratio, z/Z, of two complex numbers, z and Z, is a real number
obtained by multiplying the numerator and denominator by the
conjugate of the denominator, which is the same as dividing the
product Z*z by the squared modulus of Z, denoted;
z/Z = (Z*z)/(Z*Z) = (Z*z)/(|Z|^2) .

One tremendously useful application of complex numbers is their
appearance in the solutions to certain very important equations,
which should be covered briefly as follows.

An equation of the form
a(x^2) + bx + c = 0
is referred to as a "quadratic equation", in standard form, where x
is a variable, and a, b, and c are arbitrary constants.
Equations of this form are used to solve so many real-world problems
that a full accounting of them would fill an encyclopedia. So, we
need not go into numerous examples here, although at least one
example would perhaps be appropriate.

If, in the given equation, the constant "a" is half the acceleration
g due to gravity near the surface of the Earth, and "x" is changed to
time t, with "b" as the initial velocity v of a falling object,
dropped from an initial height H, reaching a lower height h in the
time t, and we let c = h - H (because we will need a negative
value for this difference, arrising from the fact that the height of
the object is decreasing), then we can write a quadratic equation in
standard form, as follows;
(1/2)g(t^2) + vt + (h - H) = 0 .
When rearranged to isolate h, this equation gives us the height, h,
after the time t has elapsed;
h = -(1/2)g(t^2) - vt + H .
This, then, is an excellent example of how quadratic equations crop
up in real-life situations; in this case, should we need to know the
height of a falling object at some time during its fall.

We move on to point out how complex numbers come into play in the
context of quadratics.

Again, let us assume that there is a quadratic equation in standard
form, denoted;
a(x^2) + bx + c = 0 ,
as described above.
Here, let
s = d^(1/2) ,
where
d = (b^2) - 4ac ,
to establish a convenient abbreviation.
Such an equation has a solution x that can be obtained as follows.
Possibility 1 is; x = (-b + s)/(2a) ,
Possibility 2 is; x = (-b - s)/(2a) ,
where s = d^(1/2) = [(b^2) - 4ac]^(1/2) .
The difference d, in the term s, is called the "discriminant" of the
quadratic equation, and, due simply to the fact that s is the square-
root of a difference, then it is sometimes allowed that s could be
the square-root of a negative number (i.e., the term s could be an
imaginary number).
In particular, if d is positive, then s is real, and therefore x
comes in two distinct and real versions, called the "roots" of the
quadratic equation, corresponding to "-b + s" and "-b - s" .
That is, if d is positive, then it is said that the quadratic
equation has "two distinct real roots".
However, if d = 0, then s = 0, so that
-b + s = -b - s = -b ,
and there is only one real root, called a "double root", because it
satisfies both possibilities given for x above.
Such a root is readily obtained by writing;
x = -b/(2a) .
Alternatively, if d is negative, then s is an imaginary number, and
the given quadratic equation has no real roots. In such cases, the
quadratic equation can be referred to as "irreducible", in venues
where only distinct real roots and/or double roots are considered.
Otherwise, for negative determinants, the possiblities for x can be
denoted as follows.
Possiblity 1 is; x = (-b + si)^(1/2) ,
Possibility 2 is; x = (-b - si)^(1/2) ,
where
si = [d^(1/2)][(-1)^(1/2)] = [(-1)d]^(1/2) = (-d)^(1/2) .
This shows how the imaginary-unit, i, can be introduced in the
context of quadratic equations.

The invention of complex numbers, which hinge on the notion of
imaginary numbers, the basic understanding of which, in turn, is made
clear by the definition and applications of the standard imaginary-
unit, i, has provided us with very useful mathematical tools; for
example, in providing various means of solving quadratic equations
that have negative determinants.

Algebraically, of course, complex numbers obey a special set of
rules, as follows.
Let A, B, C, and D denote real numbers.
Then the following relationships hold true.

A + Bi = C + Di if and only if A = C and D = B .

(A + Bi) + (C + Di) = (A + C) + (B + D)i .

(A + Bi) - (C + Di) = (A - C) + (B - D)i .

(A + Bi)(C + Di) = (AC - BD) + (AD + BC)i .

(A + Bi)/(C + Di) = [(AC + BD)/[(C^2) + (D^2)]]
+ [(BC - AD)/[(C^2) + (D^2)]]i .

Graphically, we have yet another set of rules; as follows.

Consider the standard x,y-plane, and let an ordinary point P be
plotted on the plane;
P = (x,y) .
If we change y to yi, so that the y-axis becomes an imaginary axis,
then P becomes the point indicated by plotting the complex number z
as a point in this plane, so that
z = (x,yi) .
That is, a complex number z, defined by the formula denoted;
z = x + yi = (x,yi) ,
can be represented by a point in a plane that is formed by using the
real and imaginary number-lines as the coordinate axes for the plane.
Such a plane is called the "complex plane", and, with respect to this
plane, the complex number z can always be denoted by the ordered-pair
(x,yi).

Now, because complex numbers are actually ordered-pairs of numbers,
then they can also be used to represent vectors in the plane.
And here is how that is done.
If we stipulate that the point z is at the location indicated by the
arrow of a directed line-segment from the origin O to z, within the
complex plane, then the modulus |z| of z can be interpreted as the
magnitude of a vector represented by this directed line-segment;
explained as follows.

Let "r" denote the magnitude (length) of the vector, and let "a"
indicate the angle the vector makes with the x-axis. Then r is
defined formally;
r = |z| = ((x^2) + [(yi)^2])^(1/2)
= [(x^2) + (-1)(y^2)]^(1/2) = [(x^2) - (y^2)]^(1/2) ,
and we can specify z using the two variables, r and a, called "polar
coordinates", so that
z = x + yi = (x,yi) = (r,a) .
In that case, knowing (from trigonometry) that "r" and "a" are
related to x and y in the standard plane by the identities;
x = r(cosa) and y = r(sina) ,
we can, by substitution, determine a trigonometric representation of
z, with respect to the complex plane, and write;
z = r[(cosa) + i(sina)] ,
which is called the "polar form" of the complex number z.

We must remember, of course, that r is also the modulus of z.

Further, angle "a" is commonly referred to as the "amplitude" of z.

Another useful application of the imaginary-unit is in the
representation of sinusoidal waves; explained as follows.

Consider the graph of a sine-wave in the x,y-plane, with a period T
and wavelength L, and where the sine-wave is pictured as propagating
along the x-axis to the right, so that y is the amplitude of the wave
(its distance above or below the x-axis) at a given instant of time
t, making "y" a function f both of x and of t, denoted;
y = f(x,t) .
If v is the speed of the wave, then the frequency F, period T, and
wavelength L are related using the following formula;
F = 1/T = v/L .

Here, let A be a constant, called the "central maximum", which is the
maximum value of y.
Since a sine-wave can be used to represent a steady oscillation, a
perfect circular orbit, or other such harmonic motion, then we can
introduce another constant K of the motion, called the "wave number",
and relate it to the value of Pi (approximated as 3.14), so that
2(Pi) corresponds exactly to one cycle, according to the formula;
K = 2(Pi)L = 2(Pi)/(Tv) .

Now, any central maximum A approaching the y-axis from the left will
be located a distance D (on the x-axis) from the y-axis, at time t.
However, since the values of K and of D always vary proportionally
with respect to each other, then D can be obtained by introducing a
quantity k, referred to as the "phase constant", the "phase delay",
or simply the "phase", and by defining D as the ratio of k over K,
denoted;
D = k/K .
Then the sine-wave can be represented "on paper" by plotting the
following formula graphically;
y = f(x,t) = A cos[K(x - vt) + k] .

On the other hand, since uniform circular motion can be represented
as the number of radians swept-out per unit time, using the angular
frequency w, defined;
w = 2(pi)F = Kv ,
so that
K(x - vt) = Kx - Kvt = Kx - wt ,
then, alternatively, we can also write;
y = A cos(Kx - wt + k) .

Unfortunately, dealing with sinusoidal waves using trigonometric
functions can get tedious. The more efficient way to do the same
thing is to convert to complex notation, as follows.

From trigonometry, we have the following relationship, using the base
e of natural logarithms;
e^(iV) = cos(V) + i[sin(V)] ,
for any arbitrary or "dummy" variable V.
Thus, letting
V = Kx - wt + k ,
we can write;
e^[i(Kx - wt + k)] = cos(Kx - wt + k) + i[sin(Kx - wt + k)]
where the real (Re) and imaginary (Im) components can be defined;
cos(Kx - wt + k) = Re(e^[i(Kx - wt + k)]) , and
i[sin(Kx - wt + k)] = Im(e^[i(Kx - wt + k)]) .
Suppose, however, that only the real component is needed, or,
otherwise, the imaginary component is zero. Then we can define y
using only the real component, as follows;
y = A cos(Kx - wt + k) = Re(Ae^[i(Kx - wt + k)]) .
Next, we can introduce a new funtion y', defined;
y' = A[e^(ik)]e^[i(Kx - wt)] = A'e^[i(Kx - wt)] ,
where
A' = Ae^(ik) ,
so that the phase k can be temporarily "absorbed" into a more compact
representation, wherein the real component is denoted;
y = Re(y') .
This sort of representation makes for a much faster form of notation
when many waves are to be handled (which is a common task in physics
and engineering). It is referred to as "complex notation", and is
used primarily because it is quicker and easier to deal with
exponents than to manipulate sine and cosine functions. And it has
been explained here as another example of how the standard imaginary-
unit, i, has practical applications in real-world situations.

Having learned something about imaginary numbers, we can proceed to
the introduction of a new imaginary-unit; one that implies a wholly
new kind of operation.

[Continued in next post.]