Historical, Psychological, Cultural,

and Logical Perspectives

Hossein
Arsham

University of Baltimore, Baltimore, Maryland, 21201, USA

harsham@ubmail.ubalt.edu

1.
Introduction

2.
Historical Perspective

3.
Psychological Perspective

4.
Cultural Perspective

5.
Logical Perspective

6.
Concluding Remarks

7.
Notes, Further Readings, and
References

The
introduction of zero into the decimal system in 13^{th} century was the
most significant achievement in the development of a number system, in which
calculation with large numbers became feasible. Without the notion of zero, the
descriptive and prescriptive modeling processes in commerce, astronomy,
physics, chemistry, and industry would have been unthinkable. The lack of such
a symbol is one of the serious drawbacks in the Roman numeral system. In
addition, the Roman numeral system is difficult to use in any arithmetic
operations, such as multiplication. The purpose of this article is to raise
students, teachers and the public awareness of issues in working with zero by
providing the foundation of zero form four different perspectives. Imprecise
mathematical thinking is by no means unknown; however, we need to think more
clearly if we are to keep out of confusions.

Our
discomfort with the concepts of zero (and infinite) is reflected in such humor
as 2 plus 0 still equals 2, even for large values, and popular retorts of
similar tone. A like uneasiness occurs in confronting infinity, whose proper
use first rests on a careful definition of what is finite. Are we mortals
hesitant to admit to our finite nature? Such lighthearted commentary reflects
an underlying awkwardness in the manipulation of mathematical expressions where
the notions of zero and infinity present themselves. A common fallacy is that, any number
divided by zero is infinity. It is not
simply a problem of ignorance by young novices who have often been mangled. The
same errors are commonly committed by seasoned practitioners, yea, and even
educators! These errors frequently can be found as well in prestigious texts
published by mainstream publishers.

Counting
is as old as prehistoric man is, after he learned to count, man invented words
for numbers and later still, symbolic numerals. The numeral system we use today
originated with the Hindus. They were devised to go with the 10-based, or
"decimal," method of counting, so named after the Latin word decima,
meaning tenth, or tithe. The first popularizer of this notation was a Muslim
mathematician, Al-Khwarizmi in the 9^{th} century, however it took the
new numbers about two centuries to reach Spain and then to England in a book
called Craft of Nombrynge.

The Two Notions of Zero:
The notion of zero was introduced to Europe in the Middle Ages by Leonardo
Fibonacci who translated from Arabic the work of the Persian (from Usbekestan
province) scholar Abu Ja'far Muhammad ibn (al)-Khwarizmi. The word
"algorithm," Medieval Latin 'algorismus', is a contamination of his
name and the Greek word arithmos, meaning "number, has come to represent
any iterative, step-by-step procedure. Khwarizmi in turn documented (in Arabic,
in the 7^{th} century) the original work of the Hindu mathematician
Ma-hávíral as a superior mathematical construction compared with the then
prevalent Roman numerals which do not contain the concept of zero. When these
scholarly treatises were being translated by European accountants, they
translated 1, 2, 3,.. upon reaching zero, they pronounced, "empty",
Nothing! The scribe asked what to write and was instructed to draw an empty
hole, thus introducing the present notation for zero.

Hindu
and early Muslim mathematicians were using a heavy dot to mark zero's place in
calculations. Perhaps we would not be tempted to divide by zero if we also
express the zero as a dot rather that the 0 character.

Babylonians
also used a zero, approximately at the same time as Egyptians, before 1500 BC.
Certainly, zero's application in our base 10 decimal system was a step forward,
as logarithms of Napier and others brought into use.

While
zero is a concept and a number, Infinity is not a number; it is the name for a
concept. Infinity cannot be considered as a number since it does not follow
numbers' properties. For example, (infinity + 2) is not more than infinity.
Since infinite is the opposite of finite, therefore whoever uses
"infinite" must first give an indication for what is finite. For
example, in the use of statistical tables, such as t-table, almost all
textbooks denote symbol of infinity ( ) for the parameter of any t-distribution
with values greater than 120. I share Cantor's view that "....in principle
only finite numbers ought to be admitted as actual."

Aristotle
considered the infinite, as something for which there is no exit in an attempt
to pass through it. In his Physics: Book III, he wrote "It is plain, too,
that the infinite cannot be an actual thing and a substance and
principle."

Many
writers have given much attention to clarifying the nature of the
"infinite": what is it, how can we know anything about it, etc. Many
constructively minded mathematicians such as David Hilbert choose to emphasize
that we can restrict ourselves to the finite and thereby avoid many of these
problems: this is the so-called "finitary standpoint".

Zero
as a concept, was derived, perhaps from the concept of a void. The concept of
void existed in Hindu philosophy and the Buddhist concept of Nirvana, that is:
attaining salvation by merging into the void of eternity. Ma-hávíral (born,
around 850 BC) was a Hindu mathematician, unfortunately, not much is known
about him. As pointed out by George Wilhelm Friedrich Hegel, "India, such
a vast country, has no documented history." In the West, the concept of
void and nothingness appeared first in the works of Arthur Schopenhauer during
the 19^{th} century, although zero as a number has been adapted much
earlier.

The
Arabic writing mathematicians not only developed decimal notation, they also
gave irrational numbers, such as square root of 2, equal rights in the realm of
Number. And they developed the language, though not yet the notation, of
algebra. One of the influential persons in both areas was Omar Khayyam, known
in the west more as a poet. I consider that an important point; too many people
still believe that mathematicians have to be dry and uninteresting.

Initially,
there was some resistance to accepting this significant modification to the
time-honored Roman notation. Among the trite objections to leaving Roman
numerals for the new notation was the difficulty in distinguishing between the
numeral 1 and 7. The solution, still employed in Europe, was to use a
cross-hatch to distinguish the numeral 7.

The
introduction of the new system indisputably marked the democratization of
mathematical computation by its simplicity and lack of mystery. Up to then the
"abacus" was the champion. Abacus was a favorite tool for a few and
praised by Socrates. The Greek's emphasis on geometry (i.e., measuring the land
for agricultural purposes, the earth, thus the world geography) so kept them
from perfecting number notation system. They simply had no use for zero.

Sacrilegious
as it may sound on first impression, the notation of zero is at heart nothing
more than a directional separator as in the case of a thermometer. It is, in
actuality, "not there." For example, in order to express the number
206, a symbol is needed to show that there are no tens. The digit 0 serves this
purpose. Zero became a part of the Natural Numbers System in the last century
when Giuseppe Peano puts it in his first of five axioms for his number theory.
One may think of an analogy. Zero is similar to the "color" black,
which is not a color at all. It is the absence of color, while the Sun Light
contains all the colors.

Zero
is the only digit, which cannot stand-alone. It is a lonely number, lonelier
than one. It requires some sort of companionship to give meaning to its life.
It can go on the left. On the right. Or both ways! Or in the middle as part of
a threesome. Witness "01", "10", or "102". Even
"1000". A relationship with other numbers gives it meaning (i.e. it
is a dependent number). By itself it is nothing!

When
we write 10, we mean 1 ten and 0 ones. In some number systems, it would be
redundant to mention the 0 ones, because zero means there are no objects there.
Place value uses relative positions. So an understanding of the role of 0 as
marking that a particular 'place' is empty is essential, as is its role of
maintaining the 'place' of the other digits. The usage of zero here is more of
a qualitative than quantitative. Therefore, it is called an operational zero.

Another
colleagues who is writing a paper for symbolic logic on zero stated that
"It seems to be it is 'nothing' in addition/subtraction, but if it is
nothing then how can it effect numbers in multiplication? Also as to your
comment on 2/0 being meaningless. I am wondering what the answer should be, if
it can be more clearly defined), and why."

Here,
my dear colleague has mixed the two distinct notions of zero: Zero as a number
being used in our numerical systems AND as a concept for 'nothing'. As a result
of this mixed-up, he is "wondering" at his own mental creature. We
used to think that if we know one, we know the other. We are finding out that
we must learn a great deal more about "AND".

Judging
from the treatment accorded to the concept of zero, we do practice a variety of
avoidance mechanisms rather than confront the imagery associated with this
seemingly difficult concept.

In
reciting one's telephone number, social security number, postal zip code or
post office box, room number, street number or any of a variety of other
numeric nominals, we carefully avoid pronouncing the digit "zero" and
instead substitute "oh." One may say "it is caused by our desire
to communicate quickly, if we can say the same thing in one syllable, why
not?" What about number seven, should we find a substitute for this too?

In
some parts of the world, the phrasing "naught" and "aught"
are used but it is quite uncommon to hear "zero." All the other
digits are correctly enunciated with this one curious exception.

Is
the presence of nothing (reflecting non-existence) different from the absence
of something (reflecting non-availability) or the absence of anything (reflecting
non-existence)? Zero is a symbol for "not there" which is different
from "nothing" "Not there" reflects that the number or
item(s) exists but they are not just available. "Nothing" reflects
nonexistence.

Zero
not only has the quality of being nothing, it is also a noun, verb, adverb, and
an adjective as in "zero possibility". "We zeroed in on the
cause," means we had isolated all the possibilities, and have discovered
the one remaining. In this use as a verb, zero equals one. However, "The
result was a big, fat, zero," uses the noun to express the idea of results
of "nothing". Here, zero has the quality of not being there. Zero as
an action appears in the Conservative Laws of physics.

Is zero a number?
Consider the following scene:

*Ernie*: I've put a number of cookies in that Jar. You can
have them if you give me your teddy.

*Bert*: Great While Ernie hands over the teddy and looks eagerly in the
jar, said:

Bert "Wait a Minute There's No Cookies Here. You Said You Put a Number of
Cookies in There"

*Ernie*: That's right, zero is a number.

Clearly some sort of
an avoidance mechanism is in operation. It is as though the name itself invokes
a kind of anxiety perhaps associated with "nothingness", a kind of
emptiness which humankind finds uncomfortable and prefers to avoid confronting.
As with all such anxiety- provoking ideas, some other imagery is substituted
which provides a veneer to mask the disquieting emotional undertones of the
discomforting idea. Zero represents the amount of nothing.

Today
zero has a meaning not just of a number, but as the bottom, or failure. He made
no baskets, or, he made zero baskets -- meaning he failed to score. Or he gave
zero assistance.

If
you are familiar with Numerology, you notice that there is no zero to work with
in the numbers that correlate with the alphabet, strange? Not at all. The
absence of zero may suggest that the Pythagorean who first developed the
duality between numbers and letters were not aware of the zero notion. The
notion of zero is much younger.

On
the telephone keypad, zero has the honor of representing the operator. There is
no zero in most games, such as plying cards (after all who wants to win zero!).
Zero is placed at the end of the keypad on the computer and at the bottom of
the keypad on the telephone. Is zero the beginning or the end? Notice that on a
calculator's keypad the numbers starts with the largest numbers on the top and
work their way down to zero. What about the o and 0 being right next to each
other on the PC keyboard? Numbers are located three places. First it is located
on the keyboard keys with the range 1, 2,...,0; this is the same order that
phone keypad. Second, on the right of the keyboard is a calculator-like pad
where zero is the last listed number. Finally, there is a list of functions
key, however there is no F0 because that could translate into no function and
what would be the point of having a key "without" function. There
will always be questions about the true meaning and function of zero. Is it the
end or the beginning? What does ground zero mean? Some use it as starting
point; the military uses it as an ending point.

The
resistance against zero can be noted even at the architectural level in
buildings where the ground level is rarely denoted as the zeroth-level as it
should be. However, for mathematicians it comes easily to label the floors of a
building to include zero, for example, the Department of Mathematics' building
at the University of Zagreb in Croatia has floors numbered as -1, 0, 1, 2, and
3. In fact, this is not a particularity of one building but a common practice
in modern buildings in Spain and in Spanish speaking world such as Argentina.
The feeling of comfort with zero in these countries could be due to the fact
that the Islamic culture had more influence in Spain than any other European
countries. Other countries do have a special word to say 'ground floor' in a
conversation, not using a "0 button" for the ground floor.

Other Apparent Cultural Difficulties with Zero: It may be considered frivolous hyperbole to
suggest that the demise of the Roman Empire was due to the absence of zero in
its number system, but one can only ponder the fate of our civilization given
the difficulty our culture seems to have with the presence of zero in our
number system.

The
notion of zero brings another wearying and yet intriguing questions: Is our
current century the 20^{th} century or the 21^{st} century?
According to the Holy Scriptures (see, Matthew chapter 2), King Herod was alive
when Jesus was born, and Herod died in 4 BC. Does that mean the millennium
actually started in 1996?

Ordinal
numbers, which the Gregorian calendar uses, indicate sequence. Thus "A.D.
1" (or the first year A.D.) refers to the year that begins at the zero
point and ends one year later. Think of a carpenter's ruler, if you will; the
first inch is the interval between the edge and the one-inch mark. Thus, e.g.,
the millennium ended with the passing of the two-thousandth year, not with its
inception. Cardinal numbers, which astronomers use in their calculations,
indicate quantity. Zero is a cardinal number and indicates a value; it does not
name an interval. Thus "zero" indicates the division between B.C. and
A.D., not the interval of the first year before or after this point. Continuing
with our example, put two rulers end to end: although there is a zero point,
there is no "zero'th" inch.

As
it stands now, we refer to years with ordinal numbers and to ages with cardinal
numbers. Thus a child less than a year old is usually said to be so many weeks or
months old, rather than "zero years old." If we changed over to this
system for our calendar (referring to the age of our era, rather than to the
order of the year), then there would be "zero years" for both A.D.
and B.C.! That is to say, the last twelve months before the birth of Christ and
the first twelve months after the birth of Christ would be the years 0 B.C. and
A.D. 0 respectively.

The
main confusion is between the notions of "time window length" and a
"point in time". There is an interval between 0 and 1. Considering
whether this century is 2000 or 2001, depends on whether you look at a number
as a points on time or a time interval. Years are intervals; numbers are
points. Therefore, it is always a mistake to treat years as points. For example,
consider the old arithmetic question: John was born in 1985 and Jane in 1986.

How
much older is John than Jane? The answer, of course, can be anywhere from a few
seconds to two years, depending on when in those intervals the two people were
born.

This
is quite revealing of the cultural predilections of the time when the calendar
was reorganized, first under the Julian scheme undertaken under the auspices of
the Roman Emperor, Julius Caesar, after whom the month of July was named, and
subsequently under the Gregorian calendar currently in use, which was devised
during the reign of Pope Gregory. What is quietly yet magnificently revealed by
this now-curious omission is the absence of the notion of zero in the numbering
systems then in use. When the notion of zero was subsequently introduced in the
west in the Middle Ages, it could hardly have been regarded as feasible to
rewrite the entire calendar, if the debate occurred in the first place. Clearly
then, our ideas about numbers permeate our culture.

The
Babylonians, and Chinese did not have a symbol for zero. The word zero comes
from the Arabic "al-sifer". Introduced to Europe during Italian
renaissance in the 12^{th} century by Leonardo Fibonacci (and by
Nemorarius a less well-known mathematician) as "cifra" from which we
have obtained our present cipher, meaning empty space. Sifer in turn is a
translation of Hindi word "sunya" meaning void or empty. In Hindi
"shunya" means zero. The terms aught, naught, and cipher are older
names in English for zero symbol. In French "chiffre" means zero. It
may also make you wonder that the word "cifra" in Russian means
"written numbers." Similarly, "Ziffer" in German means one
single written number; it is used in contrast to a single letter. Zero in
German is called "Null". The ancient Egyptians never used a zero
symbol in writing their numerals. Therefore there was no function for a zero in
writing their numerals. The two applications of the zero concept used by
ancient Egyptian scribes were:

1)
as a zero reference point for a system of integers used on construction
guidelines,

and

2)
as a value that resulted from subtracting a number from an equal number.

It
is quite extraordinary that neither the Egyptians nor the Greek were able to
create a symbol to represent zero, or nothingness. The conceptual difficulty
may have been that the zero is something that must be there in order to say
that nothing is there. The Hindu-Arabic numerals were used for written
calculations in the West not before the 12^{th} century, when Arabic
texts were translated into Latin.

Reading
the seventh edition of a book on Management Science (Taylor [64]), I found the author
dividing 2 by zero in the Simplex linear optimization tableau while performing
a column ratio test, with the stated conclusion, 2 ÷ 0 = infinity ( ). A typographical error? Confusion? Willful sin? A
telephone call bringing the obvious error to the attention of the publisher for
correction in future editions was met with an astonishing return call from the
editor of the text still insisting that 2 ÷ 0 = .

Although
both the author and editor insist on this computational outcome, they
nonetheless somehow decline to continue the Simplex calculation based on this
result, contrary to the logic of their conclusion.

Questions
I had were: How can you divide two by zero? Which number, when multiplied by
zero, gives you 2?

Dividing by Zero Can Get You into Trouble: If we persist in retaining such errata in our
educational texts, an unwitting or unscrupulous person could utilize the result
to show that 1 = 2 as follows:

(a).(a)
- a.a = a^{2} - a^{2}

for
any finite a. Now, factoring by a, and using the identity

(a^{2} - b^{2}) = (a - b)(a + b) for the other side, this can
be written as:

a(a-a)
= (a-a)(a+a)

dividing
both sides by (a-a) gives

a
= 2a

now,
dividing by a gives

1
= 2, Voila!

This
result follows directly from the assumption that it is a legal operation to
divide by zero because a - a = 0. If one divides 2 by zero even on a simple,
inexpensive calculator, the display will indicate an error condition.

Again,
I do emphasize, the question in this Section goes beyond the fallacy that 2/0
is infinity or not. It demonstrates that one should never divide by zero [here
(a-a)]. If one does allow oneself dividing by zero, then one ends up in the
Hell. That is all.

It
seems apparent that the zero paradox should be broken into to areas:
mathematical and physical. Not only is the need to define zero, but infinity as
well. For some it is not a question of whether it exists, but merely what the
definite result is."

One
must make a clear distinction between the abstract concepts and the concrete
concepts as well as their useful implications in modeling process of reality.
Therefore, one must engage in investigating mathematical knowledge, especially
the relation between conceptual and applied (procedural) knowledge. The
distinction between these knowledge types is possible at a theoretical,
epistemological and terminological level. One may classified them according to
their different approach to a given problem:

Applied knowledge: How
to get from where one is to where one wants to go in a finite number of steps.

Conceptual knowledge:
How to get from where one is to where one wants to go in a finite or an
infinite number of steps, or a leap without any steps at all.

An
example of conceptual knowledge would be

Where one is: natural numbers

Where one wants to go: the end of them

How: Infinite number of steps.

For
the applied knowledge it would be

Where one is: natural numbers

Where one wants to go: the end of them

How: In a finite number of steps depends on what calculator you are using.

As
you see, conceptuality is subjective while realization is objective. Most
conceptuality is metaphysical; while reality is mostly physical. One must
recall that: being definite has the property of being definable.

The
origin of the fallacy that any number divided by zero is equal to infinity goes
back to the work of Bháskara, an Hindu mathematician who wrote in the 12^{th}
century that "3/0 = , this fraction, of which
the denominator is cipher is termed an infinite quantity". He made this
false claim in connection with an attempt to correct the wrong assertion made
earlier by Brahmagupta of India that A / 0 = 0.

Notice
that by this fallacy one tries to define "infinity" in terms of zero.
Unfortunately, similar practices seem to prevail to the present day. A similar
fallacy exists for logarithms of zero which is believed by many to be
(negative) infinity.

Is Zero Either Positive or Negative? Natural numbers are positive integer numbers. One
horse, two trees, etc. However, the arrival of zero caused the inevitable rise
of the even more nefarious numbers: The negative numbers.

What
about negative numbers? The negative sign is an extension of the number system
used to indicate directionality. Zero must be distinguished from nothing. Zero belongs
to the integer set of numbers. Zero is neither positive nor negative but
psychologically it is negative. The concept of zero represents
"something" that is "not there," while zero as a number
represents the lowest of all non-negative numbers. For example, if a person has
no account in a bank, his/her account is nothing (not there). If he/she has an
account, he/she may have an account-balance of zero.

A
high school teacher told me that "...In High school Algebra books they
like to teach about numbers. You know whole numbers, natural numbers, rational
numbers, irrational numbers, and integers to name a few. The problem that I
often run across is where does the zero fit in. For instance 'a positive
integer', does this include zero? We know that whole numbers include 0, but it
is a positive whole number..."

She
is right, unfortunately some algebra books are confusing on categorizing zero
in our numerical systems. However, the accepted and widely use categories for
inclusion of zero as a positive number is "non-negative integers",
while for excluding it from positive integer the terminology "positive
integers" is used. Similarly, for the real numbers involving zero, the
following four categories: "positive", "negative",
"non-negative" and "non-positive" are being used. The last
two categories include zero, while the first two exclude zero, respectively.
Therefore, as you see, the first two sets are the subsets of the last two,
respectively.

Talking
to another high school teacher, he stated that ".. I always thought and
believed that zero is neither positive nor negative. It's only when we used the
book International Student (7th Ed., by Lial, Hornsby, and Miller, Addison
Wesley, 1999, page 6) that:

when
they presented inverse property of addition

a + (-a) = 0

they
wrote these:

__Number__ __Additive Inverse__

6 -6

-4 -(-4) or 4

2/3 -2/3

0 -0 or 0

This
is rather confusing to me and to my students because I told them that zero is
neither positive nor negative, then why did these authors attach a negative
sign on zero?

I
looked at other books and I found another one Modern Algebra and Trigonometry
(3rd Ed., by Elbridge Vance, 1995), that when he also presented Existence of
Additive inverses (axiom 6A), in one of his statements he wrote: 0 = -0.

All
these are confusing. It is also a difficult and uncomfortable situation when a
knowledgeable teacher want to correct the textbook, and the students taking the
textbook as the ultimate authority as if it's a Bible. One may like to remind
them by mentioning that the purpose of education is the critical thinking for
oneself.

The
additive inverse of any number is a unique number. Therefore, the additive
inverse of 0 cannot be " -0, or 0". (Thanks goodness! they did not
include, double zeroes -00, and 00, etc.)

Moreover,
the additive inverse of zero is itself. This property of zero also
characterizes the zero (i.e., no other number has such nice property).

Furthermore,
zero is the Null element for addition. Any operation has a unique Null. The
inverse of a Null element for any operation is itself. For example, the Null
element for both multiplication and division operations is 1.

Is Zero an Even or Odd Number? If one defines evenness or oddness on the integers (either positive
or all), then zero seems to be taken to be even; and if one only defines
evenness and oddness on the natural numbers, then zero seems to be neither.
This dilemma is caused by the fact that the concepts of even and oddness
predated zero and the negative integers. The problem posed by this question is
that zero is not to be really a number not that it is even or odd.

Most
modern textbooks apply concepts such as "even" only to "natural
numbers," in connection with primes and factoring. By "natural
numbers" they mean positive integers, not including zero. Those who work
in foundations of mathematics, though, consider zero a natural number, and for
them the integers are whole numbers. From that point of view, the question
whether zero is even just does not arise, except by extension. One may say that
zero is neither even nor odd. Because you can pick an even number and divide it
in groups, take, e.g., 2, which can be divided in two groups of "1",
and 4 can be divided in two groups of "2". But can you divide zero?
That's why there are so many "questions."

If
you feel that the question if zero is an even number is of no practical value
at all, let me quote the following news from the German television news program
(ZDF) "Heute" on Oct. 1, 1977:

Smog
alarm in Paris: Only cars with an odd terminating number on the license plate
are admitted for driving. Cars with an even digit terminating were not allowed
to be driven. There were problems: Is the terminating number 0 an even number?
Drivers with such numbers were not fined, because the police did not know the
answer.

"Is
zero odd or even? One of my students suggested a convention, i.e. a useful
unproved mechanism which makes her feel better, that zero is indeed Even! She
offered two arguments:

A1:
"Odd" numbers are spaced two apart. So are "even" numbers.
Proceeding downward, 8,6,4,2,0,-2,-4 .. should all be considered Even. While
odd numbers 9,7,5,3,1,-1,-3 ... skip over zero in a most stubborn manner.

A2:
Let two softball teams play a game, with each player betting one dollar a run
to the opposing team. Further presume that no runs are scored (due to beer
consumption) and no extra innings are allowed because it got dark.

The
final score is zero to zero. If a player is asked by his wife whether he won or
lost, he would probably indicate that he "broke even". As the old
math teacher said: " Proof? Why any fool can see that."

These
issues make themselves strongly felt in the classroom, textbook, in the
frequent mishandling of the notion of zero by the novice and professional alike
and therefore recommend themselves to our attention. These are among many
issues of how to teach these concepts to students at early age.

Continuous
data come in the forms of Interval or Ratio measurements. The zero point in an
Interval scale is arbitrary. The different scales for measuring temperature all
have a zero, yet each has a different value! For example, on a Celsius
thermometer, zero is set at the temperature at which pure water freezes at the
sea level altitude. While zero degrees Fahrenheit is 32 ° degrees below
freezing, and finally absolute zero is the theoretical point at which molecular
movement ceases. Therefore, since the absolute temperature can be created in
the laboratory, it is only a concept. So, here one must accept that the meaning
of zero is relative to its context. Now the question is: does 80 ° degrees
Fahrenheit temperature implies it's twice as hot as when it's 40 ° degrees? The
answer is a No. Why not?

Recently
one of my students asked me "I want to know what the opposite of zero
is." Well, not everything has an opposite. The concept of opposite is a
human invention in order to make the world manageable, there is no real
opposite in nature. Is day opposite of night? Is male opposite of female, or
they are complementary to each other? What is the opposite of color blue? Here
we must be cautious when we ask about apposite of zero. The difference is
between quality (which is a concept) versus quantity (which a number). For
example, what is "minus red?" or what is opposite of red? However, in
the context of the real line, you can say that the opposite of zero is itself,
while the opposite of +2 is -2 with respect to the origin point 0, as both have
the same distance from the origin while one in on its right-side and the other
on the left-side. This definition is acceptable if you accept the opposite of
left is the right. What is the opposite of 1/2? If you say, it's 2, then 0 has
no opposite.

Unfortunately
I find that the act of dividing by zero is not at all an uncommon practice.
Many references in applied mathematics can be found committing this and other errors.
And if educators profess division by zero as an appropriate mathematical
practice, they should not be surprised to see this error persist among their
students just as the teachers themselves learned this practice from their own
teachers. You might think, as one of my colleagues from Eastern Europe believed
that "... the Anglo-Saxons culture do not have a way with numbers."
While respecting this opinion, unfortunately, I found that this error is not
limited to a particular culture. In fact, it is the problem often initiated by
our educators worldwide. For example, in the textbook for Educacion Mathematica
by Gracia, et al. [1989, page 138], which is widely used in Spanish speaking
Schools of Education, you will find that the function y = 1/(X^{2} -
1), evaluated at X = -1 is 952380952. Where did this number come from? The
right question one might ask is who educates our educators?

Ball
[7] interviewed 10 elementary and 9 secondary teachers, asking, "Suppose
that a student asks you what 7 divided by 0 is. How would you respond? Why is
that what you'd say?" What she found was that 1 of the 10 elementary
teacher candidates could explain using the meaning of the terms, 2 gave the
correct rule, 5 gave an incorrect rule, and 2 didn't know. 4 of the secondary
candidates could explain using the meaning of the terms and 5 only gave the
correct rule, e.g.; "You can't divide by zero . . . It's just something to
remember," but gave no further justification when probed. Some of the
teacher who only gave the correct rule was math majors.

In
most Elementary Education programs for prospective teachers, such as the one at
the Towson State University in Maryland, it is required to take four math
courses, concepts of mathematics I and II, plus teaching mathematics in the elementary
school, together with a supervised math-teaching experience session. While the
standard is high, the main question is who educates our educators? Adding to
this, doubling the existing difficulties for the teachers, the school systems
hiring a teacher seems to be more concerned about "how he/she would handle
violence in the classroom?" Unfortunately, it is a miserable story to
tell.

There
must be a conviction that mathematics teacher and researchers in mathematics
education have much to learn from each other, especially at a time when the
school and adult curricula are converging. Based on my experience, I offer the
following three distinct headings:

· **Recruitment:** What can be done to encourage reluctant would-be mathematics teachers
to take the plunge?

· **Retention:**
What support do they need to enable them to become sufficiently competent,
confident and comfortable with mathematics so that they can teach it to others?

· **Re-training:** What is it like teaching mathematics without a strong background in
mathematics?

Unfortunately,
mathematics has been fundamentally depersonalized to "something machines
do" and that the meaningful response is that we need always to emphasis
that mathematics has little value divorced from imagination. Machines will
always do 'imaginationless' mathematics better than humans. But
"mathematics imagination meld" is needed by society and it can become
a fascinating subject for most children in the classroom.

Too
many pupils now think that mathematics is boring. Mathematics can and must be
made more fun, more relevant, and more challenging, for pupils and for
teachers. The use of Internet interactive technology in the classroom can add a
new and precious variety. This variety can help to engage and hold pupils'
attention, and can raise the chances that the lesson will have been judged a
success. The new interactive technology can help to attract and retain teachers
by making the whole process more business-like, more efficient and more
effective. However the provision of appropriate hardware, software and training
remain expensive and intractable hindrances to progress.

There
is a "math" video series [Harlan Meyer, Diamond Entertainment, 1996].
One is called Addition, then Subtraction, Multiplication and, of course,
Division. The division segment of the series starts by misspelling the word
quotient. Then the "star" of the video shows how to divide by using
repeated subtraction; however, she asks "If I have 12 doggy bones and I
take away 4 groups of 3 bones, how many will I have left?" She answers
herself, "Right, four." But it was the "trick" she claimed
for dividing by zero. Unfortunately, there are many instances like this, which
sent your blood pressure through the roof. Zero is nothing. So just remember
nothing INTO something is nothing. Teaching kids to count is fine, but teaching
them what counts is best.

One
may view "division" as a subtraction operation. When you write 20/5 =
4, what you really mean is that how many times you can subtract 5 from 20? And
the answer is 4 times. That is why division is the "inverse"
operation for multiplication, which is an addition. That is, 5 x 4 = 20, means,
adds to itself 5, 4 times, and you will get 20. So dividing by "0" has
no meaning, because the question: how many times you can subtract nothing from
something? The question itself makes no sense. The act of dividing by zero is
meaningless. Therefore, it does not make to ask further what is its result,
whether it is indeterminate or not?

Zero
is an important concept, so time should be spent establishing that from early
age one has some understanding of zero; zero, nought, nothing - as ever, the
language should be varied. In absence of a concept of zero there could have been
only positive numerals in computation, the inclusion of zero in mathematics
opened up a new dimension of negative numerals. Zero, when used as a counting
number (such as zero defects), means that no such objects are present. A
concept and symbol that connotes nullity represents a qualitative advancement
of the human capacity of abstraction. As always, concepts are only real in
their correct context.

Unfortunately,
there are teachers who continue misleading students as the following argument
illustrates: "When we multiply 4 times 3, what we're really doing is
adding 3 plus 3 plus 3 plus 3. So, in a sense, multiplication is just really
fast addition, right? Well, as it turns out, division is just really fast
subtraction. So, if you're diving 12 by 3, the answer is the number of times
you can subtract 3 from 12 before you get to zero (i.e., 12 - 3 - 3 - 3 = 0).
So, the answer is 4. Now that you know that, imagine what happens if you try to
divide 12 by 0. You start subtracting zeroes, you realize that you are doing it
infinite times. So, division by zero is infinity."

But
when you start subtracting zeroes, even infinite times, you never get down to
zero! One should never divide by zero. Our high school curriculum should put
more emphasis on mathematical modeling rather than maths which in most cases
are merely "puzzle solving" which has nothing to do with students'
lives. This will bring excitement in learning the language of mathematics and
its applications.

1. Abu Al-Hasan, *The
Arithmetic of Al-Uqlidisi*, translated by A. Saidan as *The Arithmetic*,
D. Reidel, Dordrecht, 1978. Al Uqlidisi (the Arabic for the Euclidean)
describes decimal notation, explains the algorithms for the four operations,
compares the notation to sexagesimal, and explains that the latter are more
suitable for scientific calculations and the former for business and everyday
use.

The use of comma's and points still remains a nuisance in understanding
numbers. In the English speaking world 1,000 means a thousand in many other
languages (such as Spanish) it means one, on the other hand 1.000 is a thousand
in some languages and only 1 in the English speaking world!

2. Aczel A., The Mystery
of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity, Four
Walls Eight Windows, 2000. Contains some engaging historical accounts of mathematical
mysteries, and paradoxes, and its theological dimension!

3. Alperin R., A
mathematical theory of origami construction and numbers, *New York Journal of
Mathematics*, 16(1), 119-134, 2000.

4. Anglin W., *Mathematics:
A Concise History and Philosophy*, Springer-Verlag, 1994.

5. Anglin W., *The
Philosophy of Mathematics: The Invisible Art*, Edwin Mellen Press, 1997.

6. Azzouni J., *Metaphysical
Myths, Mathematical Practice: The Ontology and Epistemology of the Exact Sciences*,
Cambridge Univ Pr., 1994. This is a book about the Philosophy of Mathematics,
written for scientific philosophers.

7. Ball D., Prospective
elementary and secondary teachers' understanding of division, *Journal for
Research in Mathematics Education*, 21(2), 132-144, 1990.

8. Bashmakova I., and G.
Smirnova, *The Beginnings and Evolution of Algebra*, Mathematical Assn. of
Amer., 2000. It gives a good description of the evolution of algebra from the
ancients to the end of the 19th century.

9. Bell E.,* Men of
Mathematics*, Touchstone Books, 1986, also Econo-Clad Books, 1999. It
contains some women of mathematics too. It is a kind of inspirational
literature containing a certain amount of fiction.

10. Berka K., *Measurement: Its concepts,
Theories and Problems*, Boston Studies in the Philosophy, Vol. 72, Boston,
Kluwer, 1983.

11. Berggren J. L., *Episodes in the
Mathematics of Medieval Islam*, Springer-Verlag New York, 1986. In contains
(p. 102) a good discussion on origin of the world Algebra. The word
"algebra" is derived from the first word of the Arabic "al-jabr
wa-l'muqabala". Al-jabr and al-muqabala are the names of basic algebraic
manipulations. al-jabr means "restoring", that is, e.g., taking a subtracted
quantity from one side of the equation and placing it on the other side, where
it is made positive. al-muqabala is "balancing", that is,
"replacing two terms of the same type, but on different sides of an
equation, by their difference on the side of the larger. What makes the solution
of a problem an algebraic solution is the method, not necessarily the use of
notation.

12. Boyer C., and U. Merzbach, *A History
of Mathematics*, John Wiley & Sons, 1991. Among other discoveries, it
claims that "It is quite possible that zero originated in the Greek world,
perhaps at Alexandria, and that it was transmitted to India after the decimal
position system had been established in India."

13. Brann E., *The Ways of Naysaying: No,
Not, Nothing, and Nonbeing*, Roman & Littlefield Pub., 2001. The author
mounts an inquiry into what it means to say something is not what it claims to
be or is not there or is nonexistent or is affected by nonbeing.

14. Brann E., *Plato's Sophist: The
Professor of Wisdom*, Focus Pub., 1996. A very good reading for
understanding the concept of "nothingness" in the Sophist world view.

15. Butterworth B., *The Mathematical
Brain*, Macmillan, London, UK., 1999. It contains some helpful materials
relevant to the so-called "dyslexia" when some children approach mathematical
concepts.

16. Butterworth B., A head for figures, *Science*,
284, 1999, 928-929.

17. Cajori F., *A History of Mathematical
Notations*, Chicago, Open Court, 1974, 2 vols. Also in Dover Publications,
1993. A good source for mathematical notations' history.

18. Cajori F., *A History of Mathematics*,
Chelsea Pub Co., 1999. Covers the period from antiquity to the close of World
War I.

19. Calinger R. J. Brown, and T. West, *A
Contextual History of Mathematics*, Prentice Hall, 1999. It provide a good
argument on the distinction between the words "abbacus" and
"abacus", the latter referring to the counting board. The 'abbacus'
is not counting board but the decimal numerals system, while mentioning that
Italian teachers of the new commercial mathematics were called "Maestri
d'Abbaco". pp. 367-368.

20. Cohen I. B., *Revolution in Science*,
Harvard Univ Pr., 1987. Contains his well accepted essential criteria for
scientific investigations, including mathematics and its revolution.

21. Conant L., *The Number Concept: Its
Origin and Development*, New York, MacMillan and Co., 1896. It has a short
note (page 80) on the Hottentots' a group of Khoisan-speaking pastoral peoples
of southern Africa, legend that their language had no words for numbers greater
than three.

22. Crowe M., *A History of Vector
Analysis: The Evolution of the Idea of a Vectorial System*, Dover, 1994.
States that the first attempt to represent complex numbers geometrically was
made in the 18th century.

23. Crump T., *The Anthropology of Numbers*,
Cambridge Univ Press, 1992.

24. Dauben J., *Georg Cantor: His
Mathematics and Philosophy of the Infinite*, Princeton Univ Press, 1990.

25. Dauben J., *et al.*, (Eds.), *History
of Mathematics: States of the Art*, Academic Press, 1996. It is cited in
Klaus Barner's preprint "Diophant und die negativen Zahlen", where he
tries to credit Diophantus with the invention of negative numbers.

26. Davis Ph., R. Hersh, and E. Marchisotto,
(eds.), *The Mathematical Experience*, Springer Verlag, 1995. The chapter
titled Dialectical vs Algorithmic has a good discussion on Conceptual vs
Procedural Knowledge.

27. Detlefsen M., *et al.*, Computation
with Roman Numbers, *Archive for History of Exact Science*, 15(2),
141-148, 1976.

28. Dilke O., *Reading the Past:
Mathematics and Measurement*, University of California Press, 1987. This
small book (only 61 pages long) provides interesting information covering the
Ancient Near East including Egyptian, Babylonian, Greek and Roman mathematics.

29. Driver R., J. Ewing (Editor), and F.
Gehring, (Eds.), *Why Math?*, Springer Verlag, 1995. A very relevant book
for a general education mathematics course.

30. Foucault M., *Aesthetics, Method, and
Epistemology*, New Press, 1998. His Discourse on Language, has a good
analysis with discussion on Greek's interest on geometry rather than
arithmetic.

31. Fowler D., *The Mathematics of Plato's
Academy: A New Reconstruction*, Oxford University Press, 1999. Plato in his
work POLITEIA, Book Z, 524E, makes reference to the number one (1) and 956;
951; 948; 949; 957; (zero) or better the not-one. It seems that the Greeks were
influenced by Indian culture much earlier than we thought it did. The culture
as is often assumed, did not move in one direction namely from west to the
east. It traveled in both directions.

32. Franci R., and L. Rigatelli, Towards a
history of algebra from Leonardo of Pisa to Luca Pacioli, *JANUS*,
72(1-3), 17-82, 1985.

33. Gillies D., (Ed.), *Revolutions in
Mathematics*, Oxford Univ Press, 1996. It points out that revolutions in
mathematical notation, mathematical pedagogy, standards of mathematical rigor
add up to revolutions in mathematics.

34. Gillies D., *Philosophy of Science in
the Twentieth Century: Four Central Themes*, Blackwell Pub, 1993. It traces
the development during the 20thcentury of four central themes: subjective,
conventionalism, the nature of observation, and the demarcation between science
and philosophy.

35. Grabiner J., *The Origins of Cauchy's
Rigorous Calculus*, MIT Press, 1981. Contains a good discussion on the
genesis of Cauchy's ideas including the convergence. The original meaning of
"calculus" is as a "pebble", small stones or clays (kept in
a sack used in the ancient time by shepherds containing one calculi for each,
e.g., sheep, as a counting tool in finding out if there were is any missing
sheep at the end of each day). This word persists in modern medical English
where a kidney stone, is technically known as a "urinary calculus".

36. Gracia L., A. Martinez, and R. Minano, *Nuevas
Tecnologias y Ensenanza De Las Matematicas*, Editorial Síntesis, Madrid,
1989.

37. Grattan-Guinness, *Fontana History of
the Mathematical Sciences*, Fontana Press, 1997. It mentioned the used of
Arabic numeral system starting with Fibonacci and gradual began to take firm
place, especially in Italy, whose practitioners are called
"abacists". The choice of this name is unfortunate, for it did not
use any kind of abacus, p. 139.

38. Haylock D., *Mathematics Explained for
Primary Teachers*, Sage Publications Ltd, London, 2001. Contains curriculum
on numeracy strategy, and the basic skills test in numeracy for schools in UK.

39. Houben G., *5000 Years of Weights*,
Zwolle, Netherlands, 1990. Among others, it mentions systems of weights of
power of 2. The oldest known set of weights dates the year 1229 and the
longest, still existing set has weights 1/8, 1/4, 1/2, 1, 2, 4, 8 ounces.

40. Ifrah G., *From One to Zero: A
Universal History of Numbers*, Viking Penguin Inc., New York, 2000, a
translation of *Histoire Universelle des Chiffres*, Seghers, Paris, 1981.
Ifrah drew attention to number four, claiming that "Early in this century
there were still peoples in Africa, Oceania, and America who could not clearly
perceive or precisely express numbers greater than 4." p.6. He also
provides a discussion and cites some Arabic texts as the evidence that
"early Islamic mathematics relied substantially on earlier Hindu
mathematics." p.361. In addition to the Menninger book, this book is also
an excellent source of information on the origin and development of number
symbols in ancient and medieval societies.

41. Ifrah G. , *The Universal History of
Numbers: From Prehistory to the Invention of the Computer*, Wiley, 1999,
(Translated from the French by D. Bellos,* et al.*). It is a complete
account of the invention and evolution of numbers the world over. A marvelous
journey through humankind's grand intellectual epic including how did many
cultures manage to calculate for all those centuries without a zero?

42. Jaouiche K., *La Theorie Des
Paralleles En Pays D'islam: Contribution a La Prehistoire Des Geometries
Non-euclidiennes*, Paris, Vrin, 1986. It includes texts by al-Nayrizi,
al-Jawhari, Thabit ibn Qurra, ibn al-Haytham, al-Khayyam, and Nasir al-Din
al-Tusi among others.

43. Katz V., (Ed.), *Using History to
Teach Mathematics: An International Perspective*, Mathematical Assn of
Amer., 2000. Contains 26 essays from around the world on how and why an
understanding of the history of mathematics is necessary for the informed
teachers.

44. Klein J., *Greek Mathematical Thought
and the Origin of Algebra*, Dover Pub., 1992. It points out the fact that
the difference between arithmetic and logic is viewed concerning relationships
or not. However, they distinguished between practical and theoretical logic.
Also a good discussion about the fact that to the Greeks, 1 was never a number.
A number was a multitude of units and 1 is a unit, not a multitude.

45. Kline M., *Why the Professor Can't
Teach: Mathematics and the Dilemma of University Education*, St. Martin's
Press, New York, 1977.

46. Kline M., *Mathematics in Western
culture*, Oxford University Press, 1964. Mostly, the book deals with the
cultural history of mathematics.

47. Knorr W., *Textual Studies in Ancient
and Medieval Geometry*, Springer Verlag, 1989. Contains a good discussion
and argument on whether the Greeks have any notion for fractions and what
really they meant by a "ratio?"

48. Lancy D., (Editor), *Cross-Cultural
Studies in Cognition and Mathematics*, Academic Press, 1985. Deals mostly on
the anthropology aspects of counting number systems.

49. Laugwitz D., *Bernhard Riemann,
1826-1866: Turning Points in the Conception of Mathematics*, trans. Abe
Shenitzer, Birkhaeuser, 1999. It concerns with the mathematics from both the
operational style of Euler and the conceptual style initiated by Riemann later.

50. Lesh R., and H. Doerr, *Symbolizing,
Communicating, and Mathematizing: Key Concepts of Models and Modeling*, in
P. Cobb, E. Yackel, and K. McClain (Eds.), *Symbolizing and Communicating in
Mathematics Classrooms: Perspectives on Discourse, Tools, and Instructional
Design*, Lawrence Erlbaum Associates, N.J., 361-383, 2000. By definition the
mathematical modeling process of reality is the mathematization of reality as
we perceive it. Mathematizing could be in the forms of quantifying, graphical
visualizing, tabular coordinating and/or symbols notation systems to develop
mathematical descriptions and explanations that make heavy demands on modelers'
representational capabilities.

51. Livio M., *The Accelerating Universe:
Infinite Expansion, the Cosmological Constant, and the Beauty of the Cosmos*,
Wiley, John & Sons, 2000. This book helps the reader to think, understand,
draw, and evaluate mathematical patterns of order and chaos that is a part of
this universe with its physical laws.

52. Mankiewicz R., *The Story of
Mathematics*, Casell &Co., London, 2000. The author points out the fact
that the Babylonians, and Chinese did not have a symbol for zero.

53. Mankiewicz R., and Ian Stewart, *The
Story of Mathematics*, Princeton Univ Press, 2001. A popular illustrated
cultural history of mathematics.

54. Marshak A., *The Roots of
Civilization: The Cognitive Beginnings of Man's First Art, Symbol and Notation*,
Moyer Bell, 1991. The author claims to find numerical writing and calenders on
prehistoric carved bones tens of thousands of years before the usually dated
advent of writing with civilization.

55. Netz R., *The Shaping of Deduction in
Greek Mathematics: A study in cognitive history*, by Reviel (Ideas in
Context, 51), Cambridge University Press, 1999. The main consideration concerning
the relative unpopularity of mathematics is quite simple, the author states:
"Mathematics is difficult."

56. Neugebauer O.., *The Exact Sciences in
Antiquity*, Dover, 1969. Provides some justifications faced by the Babylonian
place value notation which are due to the lack of a symbol for zero.

57. Neugebauer O., (editor), *Astronomical
Cuneiform Texts : Babylonian Ephemerides of the Seleucid Period for the Motion
of the Sun, the Moon, and the Planets*, Springer Verlag, 1983.

An interesting hypothesis is the connection between partitioning a circle into
360 degrees and number of days in a year. There are two main theses about the
origin of the 360º system:

The first underlines the mathematical suitability of 360 (its factors are 2, 3,
4, 5, 6, 8, 9 ,10, 12, etc) in problems related to the division of a whole in
equal parts, the second points out the connection with come astronomical
constants (as 365).

The second thesis is the fact that the Babylonian had a sexagesimal system,
which was used in Greek astronomy. The fact, that a year consists of little
more than 360 days, seems to be secondary. The Babylonians did have a calendar
with 360 days per year, plus suitable "additional days". Actually, it
is supported by a clear 'semantic' link (day=degree) and by some historical
facts: for example Chinese astronomy had 365 and 1/4 degrees, the Babylonian
ephemerides were based on mean synodic months divided in 30 parts and the year
was divided in 12 parts, etc.

The sexagesimal system seems to have been a basis of ancient thinking. Their
day measurement was the development of a 24 hour system (spherically, each hour
being one half of 30 degree segments relative to 360 degrees)... hours also
divided into 60 minutes, minutes into 60 seconds. Attempts to develop
measurable systems of "time" added their own bit of complexity to
what was already a complex and culturally variant attempt to juxtapose
precision in calendar and time systems congruent with a celestial system which
seemed to defy precision at the time.

Our desire for a mathematical modeling of the universe and its processing
difficulties is apparent here too. Some interesting analogous ones existed also
in music, architecture, etc. These models required the fitting between small
integer numbers, easy to be represented and dealt with, and complex phenomena
whose numerical parameters did not exactly fit in the integer-based scheme. It
is credible that the 360-system, and the 6-8-9-12 scheme in music, were the
results of this conflict, being mathematically suitable and semantically
justified.

58. Paulos J., *Once Upon a Number: The
Hidden Mathematical Logic of Stories*, Basic Books, 1999. A bridge between
science and culture.

59. Pears I., *An Instance of Fingerpost*,
Penguin, 1999. (A fingerpost is a directional sign, shaped like a finger,
pointing the direction to go). This book is a mathematical criminal novel about
a cryptanalyst trying to solve a "code," though this word was not
used that way until the early 1800's. The 17th century term was
"cipher."

60. Regiomontanus, Johann, *De Triangulis
Omnimodis*, 1464. It contains a systematic account of methods for solving
triangles with applications to Astronomy mostly for Calenders. An English
translation by Barnabas Hughes published by the University of Wisconsin Press,
1967. The original book contributed to the dissemination of Trigonometry in
Europe in the 15th Century.

61. Scriba C., and P. Schreiber, *5000
Jahre Geometrie: Geschichte, Kulturen, Menschen* (5000 Years of Geometry:
History, Cultures, People), Springer, 2001. Provides an overview of the
historical developments of geometrical conceptions and its realizations. Its
Chapter 3 deals with oriental view of geometry in the contexts of cultural
environments such as Japan, China, India, and the Islamic world.

62. Seife Ch., and M. Zimet , *Zero: The
Biography of a Dangerous Idea*, Viking Press, 2000. Good answers to
questions such as Why did the Church reject the use of zero? How did mystics of
all stripes get bent out of shape over it? Is it true that science as we know
it depends on this mysterious round digit?, can be found in this recent book.

63. Snape Ch., and H. Scott, *Puzzles,
Mazes and Numbers*, Cambridge Univ Pr., 1995. It contains the historical
development of the topics in its title.

64. Taylor III, B., *Introduction to
Management Science*, Prentice Hall, 2002. Module A: The Simplex Solution
Method, pp. 26-27.

65. Van Der Waerden, B., *Geometry and
Algebra in Ancient Civilizations*, Springer Verlag, 1983. Points out that
unlike Greeks, the Babylonians were engage in some algebraic concepts (not
algorithmic methods) such as solving systems of equations: determine x and y
when the product xy, and the sum x+y (or the difference x-y) is known. However,
by geometric means as application of areas, not by any algebraic methods.

66. Vilenkin N., In Search of Infinity,
Provides a good discussion on the paradoxes generated by the theory of infinite
sets, Springer Verlag, 1995.

67. Urton G., *The Social Life of Numbers*,
University of Texas Press, Austin, 1997. The author points out the fact that
the inability to count beyond three in some tribes around the world, they are
able to perceive the difference in numbers, by some "gestalt" form of
perception.

68. Zaslavsky C., *Africa Counts*,
Lawrence Hill, 1999. Zaslavsky, when dealing with the early counting, has
pointed out that "questions of number recognition are different from
questions of counting (and from telling anthropologists about it); using a
small set of number words as basis for a number system is different again , pp.
32-33.

Note also that in classic languages the first few numbers were adjective (i.e.
inflected for gender, number, case): 1, 2, 3, 4 in Greek, 1, 2, 3 in Latin. In
the old Russian language when following 2, 3, 4, and all their compounds the
noun is in the Genitive Singular however, when following 5, 6, 7, 8, 9, and all
their compounds as well as 10 and 11 the noun is in the Genitive Plural. Also
when following 100 and its multiples.