REPRESENTATION OF NUMBERS
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International Journal of Scientific Engineering and Technology
(ISSN : 22771581)
www.ijset.com, Volume No.1, Issue No.4, pg : 3941
01 Oct. 2012
REPRESENTATION OF NUMBERS
Sundaram Ramchandran
SUNY Binghamton, Binghamton, NY, Aug 2001
Email: [email protected], [email protected]
Abstract

Transcendental numbers such as pi , e that cannot be
For modulating signals, the compact (see note 1)
so represented (Of course they can be represented
representation of the signals assumes importance to
through
transcendental
functions
such
as
simplify the design of circuits and in the context of design
trigonometric functions and / or differential equations
of systems that are robust This article examines the nature
but that is another story)
of numbers and goes on to survey the methods of
representing algebraic irrational numbers. It also briefly
There have also been mathematical definitions of rational
examines the methods of physically realizing the same,
numbers as limits of sequences both irrational algebraic and
especially in the context of frequency modulation
nonalgebraic or transcendental numbers and of integers as
techniques , speech processing and other applications as
limits of rational numbers. Of course integers form a subset of
also to issues of sampling in DSP.
the group of rational numbers with Prime Integers forming a
Keywords
subset of the integers.
Irrational, Transcendental, Polynomial, Roots, Phi,
imaginary, complex, recursive, dynamical system, chaos,
Real numbers again form a subset of the set of complex
hysteresis, base, fields, rings, algebraic structure.
numbers. Interestingly, according to the Fundamental
Theorem of Algebra, the set of complex numbers is complete
INTRODUCTION
in the sense that any nth degree polynomial with complex
At the risk of stating the obvious, real numbers can be divided
coefficients will have n roots (including multiple roots)
into the following categories:
(which, as we have seen, is not the case with polynomials with
integral / rational or real coefficients). Those who want to

Natural numbers / Integers
know more about complex numbers can go through the

Rational Numbers
Appendix. The more mathematically inclined may like to

Irrational Algebraic numbers
investigate Cantor's theories relating to the cardinality of the

Irrational Nonalgebraic Transcendental Numbers
set of real numbers, the set of rational numbers (including
Natural numbers / Integers are of course easily represented. Of
integers as a subset including the primes as a subsubset) and
course, there is always the issue of compact representation as
the group of complex numbers.
well as the largest integer that can be represented in a given
system. Some methods that have been adopted for handling
Prime polynomials over a field are those polynomials that are
this are: (Other than as a string of characters)
irreducible and do not have roots in that field. For example, if
a polynomial has only imaginary roots, it is considered prime

The floating point / exponential representation
over the real field. Interestingly, there seems to be a dual
(exponent mantissa)
relationship between transcendental numbers which cannot be

As a polynomial to the base of a large number
expressed as roots of polynomials with real coefficients and

In terms of its prime factors
prime polynomials over the real field though their cardinalities
(sizes) are at opposite ends of the spectrum. In this section,
Rational numbers can be represented as an ordered pair of
we will cover the representation of algebraic irrational
numbers with the basic arithmetical operations represented
numbers.
appropriately (The more mathematically inclined may want to
investigate and prove / disprove the nature of rational numbers
METHODS
as a field). One interesting idea could be to map the above to
The compact representation of irrational numbers assumes
the complex plane after embedding it in the complex number
importance especially in signal processing applications such as
system (defined later)
signal modulation by carrier signals (for example generation
of sinusoidal carrier modulation, especially those with
Irrational numbers are basically of two types:
frequencies that are pairwise mutually incommensurate (not

Algebraic numbers (Those that can be represented as
rationally related to one another)). Accurate representations of
the roots of polynomials with rational coefficients).
such numbers beyond those provided by current methods may
This is typically of the form of roots of integers and
be necessary to prevent instabilities especially in applications
their sums, products etc.
and systems that are chaotic or extremely sensitive to initial
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International Journal of Scientific Engineering and Technology
(ISSN : 22771581)
www.ijset.com, Volume No.1, Issue No.4, pg : 3941
01 Oct. 2012
conditions. One of the simplest methods of representing
and possibly involve feedback circuits in general. This kind of
irrational square roots would be to represent them as the
representation has the advantage that it can be extended to any
distance / magnitude of a vector in the plane (links to
kind of data including signals / images / 2ds and 3d shapes.
pythagoras theorem ) with possible conditions on the phase
(This is also linked to what is known as Waveform
(angle between the coordinates) (Also known as the ladder
computing). This may also be related to / extended to realizing
method). But this would only be valid for square roots and not
a matrix of functions. It may be possible to solve constrained
for nth roots in general.
optimization problems by implementing inequalities through
bounds / limits set through stabilizing mechanisms.
One of the methods that have been adopted to approximate
irrational numbers is that of a dynamical system / process /
One could sample a real interval fairly densely within a certain
algorithm (linked to continued fractions and related areas). For
interval by choosing a fairly high degree polynomial and
example, a recursive / iterative algorithm for computing
increasing / decreasing the value of the coefficients (with
square roots is as follows.Suppose the square root of alpha is
bounds on the height/norm of the polynomial). The ordering
to be computed. We choose the initial value x(1) which is
could be based on the powers of the coefficients.
greater than alpha and then define x(n+1) as :
x(n+1) = (x(n) + (alpha /x(n)) / 2
Interestingly, this is also connected to the dense covering of
the unit circle by the superposition of 2 oscillators moving on
The more mathematically inclined may like to prove that the
mutually orthogonal axes with mutually incommensurate
sequence decreases monotonically and find the rate of
frequencies (lissajous figures). If the orthogonality of these
convergence and find better methods of accelerating
oscillators (linked to vector sum) could be represented by a
convergence such as shanks methods. (This is taken from the
pair of signals in quadrature like sine and cosine), these may
book "Principles of mathematical analysis" by Walter Rudin 
provide an alternative (possibly simpler) way of representing
> Third Edition. Page 81 Chapter 3 > Numerical Sequences
and generating frequency modulation signals as compared to
and Series ex 16.) Similar methods (linked to continued
those involving imaginary phases which may involve damped
fractions) have also been used to approximate transcendental
exponentials. This could also be connected to the modeling of
numbers such as pi and e but these will be covered at a later
voice signals that are said to be quasiperiodic. In fact, for
point of time. A more general and more mathematical
many mutually incommensurate values of the frequencies and
method is to represent them as the zero(s) of a polynomial
appropriate weights, the resultant signal could possibly be one
with integer coefficients (extended to rational coefficients).
of a family of space filling curves. It would be interesting to
(Again, the more mathematically inclined may like to
consider extensions to quaternions. Again, the more
investigate extensions to the above to fractional exponents
mathematically inclined may like to investigate the embedding
(links to elliptic curves where y^2 is equated to a cubic in x on
of rational numbers within the irrational numbers and vice
the right)) as also to application to problems involving nested
versa with links to approximation of integrals.
roots / horner's polynomials.
Those interested in physics may like to investigate
One method of realizing the above electronically seems to be
connections of the above to circular polarization.
representing the polynomial through an array of delay
elements with integral or rational weights (links to z transform
SOME INTERESTING RESULTS
or weighted powers of the time variable. Of course the system
The above method could also be used to represent i (the square
can be implemented in parallel) and have a system with the
root of 1), the basis of extension of the real number system to
resultant equated / clamped to zero and with taps for
the complex plane as the root(s) of the polynomial:
extracting the different roots to avoid cycling between the
different roots or being sensitive to initial conditions. To solve
X^2 + 1 = 0
for repeated roots, one could possibly have phase locked loops
which synchronize the various tap outputs and equalize them.
Interestingly, the golden mean or phi (1+/ sqrt (5))/ 2 which
Essentially, a general extension of the idea would be to have a
occur(s) in various areas of aesthetics, geometry, architecture
kind of network which could possibly be reconfigured (linked
is / are defined as the roots of the polynomial:
to what is known as neural networks) where the processing
units could realize various functions (which could be linked to
X^2 = X + 1 or X^2  X 1 = 0.
other functions) and where the links could denote both input
as well as operations such as addition / component wise
In fact, this simple relationship between scaling and addition /
multiplication, general polynomial multiplication (Linked to
multiplication (In fact any power of phi can be represented as
convolution of signals), correlation and others. This would
a multiple of phi + a constant (which would be a member of
allow us to represent variables and symbolic computation by
the Fibonacci series which starts with 1 1and 1 where every
linking them to the input. Function evaluation would proceed
term is the sum of two preceding terms for which there is a
in forward while solving equations would start with the output
simple formula for computing the nth term) which is valid for
40
International Journal of Scientific Engineering and Technology
(ISSN : 22771581)
www.ijset.com, Volume No.1, Issue No.4, pg : 3941
01 Oct. 2012
all powers is the key to simplification of computation,
One could also combine the two themes) such as control
compact representations of numbers and their powers as well
systems, regulating the volume of fluid in dispensing systems,
as implementing the shift register schemes as mentioned
controlling dosage of medicine which may be important not
above. Especially since the smaller value of phi , i.e. (1sqrt
only because a higher dosage may be more toxic but also
(5))/2, is smaller than 1, it may be possible in certain cases to
because there may be a nonlinear relationship between dosage
realize computations of infinite series as sums of geometric
and effect. The above may also be important in bounding a
progressions.
system (as mentioned earlier in the example of optimization
problems) and slowing the process of saturation as also in
Geometrically, it occurs in a variety of packing, dissection and
better utilization of physical memory.
tiling problems and has the property that if a line (say of unit
length) is divided in the golden mean say a/b where a is the
It would be interesting to investigate the applications /
smaller part, then by definition, b / 1 (or a + b) = a/b
consequences of extending / complementing the natural /
integral number system (and its extension the rationals) by
The slow growth of powers of this number (This could be the
including the golden mean , i.e., (1sqrt(5))/2 and its relevance
simplest such irrational number since it is the root of a
to multivalued / fuzzy logic. (For example, 1 = hi^2  phi , 0
polynomial which has a combination of a low degree and low
= phi  phi 1 = phi phi ^2 and so on) with possible links to
height in the sense of absolute value of coefficients. Similar
alternative bases for computing architectures (instead of the
roots of higher degree polynomials may create problems in the
usual 0/1.
sense that not all roots may be pairwise incommensurate or
small) may be important in a number of applications where
At a more theoretical level, one interesting theme is the link
fine tuning and regulation is important (as compared to speed
between multiple roots of a polynomial and non invertible
of computation which may not be an issue nowadays) to
systems with links to multistability and possibly chaos (as
prevent instabilities, overshooting etc (Of course Chebyshev
well as the link between Fourier transform and mappings from
polynomials are also applicable in this context but these may
coefficient to root space and the reverse). The above could
not be so easily computed.
also be linked to chebyshev polynomials of irrational
argument / order.
CONCLUSION
the length of the representation in memory and secondly, the
ease of understanding and the accuracy and the length of the
In this article we examined some methods of representing
representation for display purposes.
irrational numbers (especially algebraic irrational numbers)
and some interesting results (especially those relating to Phi ,
REFERENCES
the golden mean, the imaginary number i ) and some possible
applications to dynamical (possibly chaotic) systems etc (the
1. Fractals, Chaos, Power Laws: Minutes from an
latter will be investigated in subsequent articles).
Infinite Paradise Manfred Schroeder
2. Principles of Mathematical Analysis  Walter
Another interesting direction would be that of generalization
Rudin  3rd Edition
of the above to arbitrary algebraic structures (for example
3. Abstract Algebra  Herstein
polynomial rings over abstract fields) , such as defining a
4. S .Ramchandran and M.Chatterjee, "Nonlinear
continuous group (Lie group ??) through discrete groups with
Dynamics of a Bragg cell under intensity feedback
possible links to Galois theory etc.
in the nearBragg, fourorder regime".Applied
Optics / Vol 41, No 29, / October 2002
Also,. another interesting aspect that has been examined is the
5. M.Chatterjee and S.Ramchandran, "Feedback
idea of the best physical representation of the essence of a
correction of angular error in grating readout",
symbolic and abstract algebraic structure.
SPIE
Conference,
San
Diego,
July
2001,
vol. 4470, pp. 127137
Notes:1. In this document, compact representation is meant to
mean the simplest representation in 2 senses, one in terms of
APPENDIX
(i*theta) where r is the magnitude which is defined as sqrt
(a^2+b^2) and theta is the phase which is defined by tan
Complex Numbers
(theta) = b/a e is related to the exponential function and is
Complex numbers are of the form a+bi where both a and b are
also called the natural logarithm.
real (and relate to x and y coordinates in the plane) and i=sqrt
(1) as shown earlier in the document. An alternative
representation of complex numbers is of the form r*e^
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