A beautiful Math Maze

©Fernando Caracena 2013

Before continuing the trek through physics, we have to pause and fill in some mathematics that I learned piecemeal, but which is much more fun to learn as a block of associated concepts. I will guide you through this maze of mathematics so that you can witness its beauty.

Euler's Exponential  and Logarithms

Since ancient times mathematicians have sought to make calculations easier through one mechanism or other. One such device, is the logarithm, which is based on the notion that it is easier to add two numbers than to multiply them. It is based on the follow property numbers, such as x and y, represented by a number (p) raised to a corresponding power (a and b, respectively):

x=pa ,                                                                                       (1a)

y=pb,                                                                                        (1b)

x*y=pa+b.                                                                                  (1c)

The procedure is easy to write conceptually, but the task involved in making it so involves going from the exponential back to ordinary numbers. The reverse of  operations (1) is expressed by the following set of equations:

a=logp(x),                                                                                  (2a)

b=logp(y),                                                                                  (2b)

logp(x*y) = a+b,                                                                         (2c)


logp(x*y) = logp(x) + logp(y).                                                      (2d)

The constant, p, is called the base of the logarithm, which can be any number selected, as long as it remains fixed for the entire calculation.

Note the identity

x ≡ pLogp(x).                                                                                     (3)

In the past, going back and forth between a variable and its logarithm was accomplished by looking up values in a table of logarithms as functions of input numbers, and then interpolating between the values of the logarithm to the answer in the same table to extract the corresponding number. The whole procedure has been rendered moot by the advent of the digital computer, where it is no longer necessary to do hand calculations, unless you find yourself shipwrecked on a desert island and have found an old book of math tables. However, the concepts developed by necessity lead into some interesting mathematical territory that is conceptually useful although no longer necessary.

Derivatives of a number (a) raised to a variable power (x) A beautiful Math Maze

f(x) = ax ,                                                                                            (4a)b

df(x)/dx = lim Δx → 0 [f(x+Δx) - f(x)]/Δx,                                              (4b)


df(x)/dx = ax Q,                                                                                   (4c)


Q = lim Δx → 0 (aΔx -1)/Δx  .                                                                    (4d)

Notice that the derivative of ax gives back the same function ax, but multiplied by a factor that has been labeled Q, which is obtained by a limiting procedure as a function of a. The procedure is easily approximated in Python, see section of python code.

Fig. 1a. A plot of the factor (Q) vs the exponential base (a).

The results are displayed in Fig. 1a, where Q is plotted as a function of the base, a. Note that at some value between 2 and 3, the value of Q passes through 1. This special constant, called e, is one discovered by the great Swiss mathematician, Leonhard Euler who discovered a number of constants in mathematics called Euler's numbers.



Fig. 1b. A plot of a set of points representing the natural logarithm of a versus a on top of a plot of the Q-factor versus a.

Variable exponentials involving e as their base yield functions that have derivatives that repeat the function, which is Euler's exponential function

dex/dx = ex.                    (5a)

The corresponding logarithm, with a base of e, is called the natural logarithm

ln(x) ≡ loge(x) ,              (5b)


and the inverse identity is

x ≡ eln(x).                                                                                                     (5c)

Next, we close the loop to show what Q is:

f=ex ln(a) ,                                                                                                     (6a)

df(x)/dx=f(x) ln(a)                                                                                    (6b)

Q(a)=ln(a).                                                                                                (6c)

In Fig. 1b the points corresponding to to the natural logarithm, ln(a), are plotted on top of the Q(a) vs. a curve, which graphically demonstrates the identity between Q(a) and ln(a). Python uses the base e always in computing natural logarithms and contains the Euler number e, which it will print out on command (print "e =", e) as follows:

e = 2.71828182845.                                                                                    (7)

It is all built into the mathematics of the code. The code incorporates much more mathematics than a lot of users.

Next we will derive an identity involving the derivative of a natural logarithm of a function, called the logarithmic derivative. In (5c) write f(x) instead of x to generate the following identity

f(x)=eln(f(x)                                                                                                  (8a)

After taking the derivative of both sides

df(x)/dx = dln(x)/dx eln(f(x)

df(x)/dx = f(x) dln(x)/dx ,

and dividing both sides by f(x), an identity for the logarithmic derivative pops out.

(df(x)/dx)/f(x) = dln(f(x))/dx .                                                               (8b)

What fun! But there is more to the beautiful mathematical maze.

Imaginary and complex numbers

In physics as in mathematics, we tend to resist new concepts for a long time before we can accept them because they seem too fantastic. The concept of the imaginary number is perhaps one of these that resisted full implementation for millenia.  Some mathematician decided to label something impossible by the letter i as follows:

i=√(-1)                                                                                                      (9a)


i2=-1.                                                                                                         (9b)

This is called the imaginary number. A real number that is added to another, which is multiplied by i, is called a complex number.

Complex numbers involve a combination of a real and an imaginary part

z = x +i y                                                                                                (10a)

where  all numbers are real except i. The complex conjugate of a complex number is generated by changing the sign of all terms involving i, viz, i → -i. Complex conjugation is represented by the dagger symbol,,


z = x -i y      .                                                                                          (10b)

Multiplication of a complex number with its complex conjugate yields the square of its absolute value:

|z|2 =z z                                                                                                  (10c)


|z|2 = z z                                                                                                  (10d)

which results in

|z|2 = x2 + y2.                                                                                            (10e)

The amplitude of a complex number is therefore given by the following expression:

|z| = √(x2 + y2).                                                                                         (10e)


Fig.2. A unit complex number.

The complex plane represents an internal vector space of two dimensions. Like unit vectors, on can define a unit complex number as a complex number that is divided by its magnitude,


uz = z/|z|.                      (11)

Such a unit complex number is characterized by an angle that it makes counter clockwise form the real axis in the complex plane (Fig. 2), which by analogy with ordinary vectors results from the polar representation of x and y components,

x = |z|cos(θ),                                                                                                (12a)

y= |z|sin(θ).                                                                                                  (12b)

Unlike vectors, complex numbers have a bit more active properties. They act as rotation operators, which is demonstrated below.

First, define a unit complex number that makes an angle of φ with the real axis,

u = cos(φ) + i sin(φ)                                                                                       (13a)

u z = [cos(φ) + i sin(φ)] |z| [cos(θ) + i sin(θ)]                                              (13b)

u z = |z|[ cos(φ)cos(θ) - sin(φ)sin(θ) + i(sin(φ)cos(θ)+cos(φ)sin(θ))].         (13c)

Applying two the following two trigonometric identities

sin(A+B)= sin(A) cos(B) + cos(A) sin(B)                                                         (14a)


cos(A+B)=cos(A) cos(B)- sin(A) sin(B),                                                           (14b)

reduces (13c) to

u z = |z|[cos(φ+θ)+ i(sin(φ+θ)],                                                                   (15)

which is a new vector having the same length as the original z, but rotated counter clockwise through an angle φ.

One more detail brings us a full circle in our journey through the mathematical maze. Using (10a) and (12a) and (12b), we write z in a polar representation

z = |z| [cos(θ) + i sin(θ)],                                                                                (16a)

and supposing that the angle is a function of t, write

dz(t)/dt = |z| [-sin(θ) + i cos(θ)] dθ/dt

dz(t)/dt =i |z|[cos(θ) + i sin(θ)] dθ/dt

dz(t)/dt =i z dθ/dt,

the solution of which is as follows:

z=|z|e.                 (16b)

Euler was able to use the exponential representation of complex numbers to link up three important numbers in mathematics (e, i and π) in a simple relation called, Euler's identity, which has been called by some mathematicians as the most beautiful equation

e+1=0.                 (17)

phthon code

ipython --pylab
from pylab import *

plot([0, 4.99],[1,1],'r')


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