**Introduction**

The number zero is a very powerful tool in mathematics that has many different

applications and rules. An interesting fact about the number zero is that
according to our calendar

(the Gregorian calendar), there is no “year zero” in our history. There is also
no “zeroth” century

as time is recorded from centuries B.C. to the 1st century A.D. However, certain
calendars do

have a year zero. In the astronomical year numbering system year zero is defined
as year 1 BC.

Buddhist and Hindu lunar calendars also have a year zero. In this paper I am
going to discuss

many different uses of the number zero in mathematics and in the world. I am
also going to

discuss the origins of the number zero.

**Zero as a Numeral**

I learned from Wikipedia, the word zero comes from the Arabic translation of the

Sanskrit sunya, around 400 B.C. in India. Sunya means void or empty. The Romans
translated

the word between the 5^{th} and 1^{st} century B.C. using Latin
to become zephyrus which meant “west

wind”. The word zephyr is used today to mean a light breeze. Fibonnaci, a famous
Italian

mathematician, who lived from 1170 to 1250, introduced this concept to Europe
and the word

became zefiro in Italian, and was then changed to zero in the Venetian dialect.
The actual

symbol 0 that we use was developed in India in the 9^{th} century A.D.

The number zero is an important number because it serves
as a place holder to help

distinguish the difference between numbers like 230 and 23. The ancient
Babylonians, who used

a base 60 system, did not have zero as a placeholder, but instead used a
separator mark to

separate places. The Babylonians had 59 different numbers that fit into a single
position. Each

number consisted of a different combination of symbols. The first position from
the right is the

60^{0} or units place, the next place to the left is the 60^{1}
place, and so on through 60^{n}. The number

8 32 would be 8 × 60^{1} + 32 × 60^{0} which would be 480 + 32
which equals 512. The base 60

system is still used in our society today through the way that we measure time
(sixty seconds in a

minute, sixty minutes in a hour) and the way that we measure the degrees in a
circle.

Numbers with Many Zeros

A googol is defined as10^{100}, which can be written as a 1 followed by
100 zeros. It can

also be obtained by taking 1 million to the 16^{th} power and then
multiplying that result by ten

thousand. On Wikipedia, I found the term googol was coined in 1920 by a nine
year old named

Milton Sirotta, the nephew of a famous mathematician named Edward Kasner. A
googol has

only two prime factors, they are 2 and 5 and there are 100 of each of them.

There are many names for very large numbers that I found while looking through
at the

online encyclopedia called Wikipedia. I learned that a quadrillion is 10^{15},
an octillion is 10^{27} , a

googolplex is and there are many more names
for very large numbers that are not listed

here. An example of an amount that is in the quadrillions is the number of BTU’s
that were used

in the United States in 2002 according to the United States Department of
Energy. Because it is

so difficult for most people to grasp numbers this large, the names for numbers
larger than

quadrillion are not commonly used. Instead, large numbers are more commonly
expressed in

scientific notation. Large numbers which must be written in scientific notation
can be found in

RSA codes, distances in outer space, and genome structure.

Length and Dimension Zero

In geometry a point is defined to have a dimension of zero while a line or line
segment

has dimension 1. A plane is 2-dimensional by definition.

** Division by Zero : Undefined, indeterminate, or infinity**

Division by zero is undefined. The reason for this is made clear by examining
the inverse

of division, namely, multiplication. Every division statement can be written as
a corresponding

multiplication statement. For example 6 ÷ 3 = 2 corresponds to 2× 3 = 6 .
Suppose we state that

5 ÷ 0 = 0 , then the corresponding multiplication statement gives 0× 0 = 5 ,
which is clearly a

false statement. Thus, 5 ÷ 0 ≠ 0 and it follows that x ÷ 0 ≠ 0 for any nonzero
x. Now suppose

that 5 ÷ 0 = n for some n. Then n must satisfy 0× n = 5 . However, by the
multiplicative property

of zero, the product of any number times zero is equal to zero. In general, if x
÷ 0 = n then this

implies that 0× n = x . For nonzero x there is no value of n which makes this
true, so that x ÷ 0 is

undefined. If x = 0 we see that 0 ÷ 0 is indeterminate, because 0× n = 0 for any
n.

It is possible to divide zero by a non-zero number. For example, 0 ÷ 8 = 0
corresponds to

0 = 8× 0 which is not a contradiction.

Sometimes students in upper level mathematics might be taught that a number
divided by

zero is infinity. This is a reasonable statement if their teacher explains that
the zero in the

denominator represents numbers that approach zero ( and are positive ). Dividing a
number by

(positive) numbers which get smaller and smaller, produces values which get
larger and larger

and approach infinity.

**Different Algebraic Expressions with Zero**

It is very interesting to examine the role of zero in different algebraic
expressions. The

expression is a valid expression because 0^{2}
= 0× 0 = 0 . Similarly the expression is

valid because 0^{3} = 0× 0× 0 = 0 . The expression x^{0} is
valid because any non-zero real number

raised to the zero power is defined as 1. 0^{0} is an interesting
special case of x^{0} because a lot of

people will look at 0^{0} and think that the answer is 0. As I learned
from a website named Ask Dr.

Math, 0^{0} is said to be an indeterminate form. This means in some
cases we think about it as

having one value , and in other cases we say it has a different value. Setting 0^{0}
equal to 1 lets

mathematicians use definitions in other areas of mathematics without having to
make special

cases for 0. A concept where having 0^{0} = 1 is useful is when using
the binomial theorem . 0^{x }is

a valid algebraic expression because the exponent x tells you how many times to
write the base 0.

No matter how many times you multiply zero by itself the product is always zero.
Recall that ,

or 0 ÷ 0 , is an indeterminate form (see previous discussion). A good example of
why is an

indeterminate form can be found on the Ask Dr. Math website. If you evaluate the
limit of the

expressions and
as x approaches zero, although substituting
zero in for x demonstrates

that both expressions approach the form , the
expression will approach 1 in the limit and
the

expression will have a limiting value of 7.
Thus, limits which approach the form must be

evaluated by alternative means since this expression does not determine a unique
limit.

**The Zero Discriminant and the Zero Determinant**

The discriminant is a very useful tool in Algebra. The descriminant of the
quadratic

polynomial, ax^{2} + bx + c , is the expression b^{2} − 4ac,
where values a, b, and c come from the

polynomial. The descriminant allows the students to determine how many solutions
there are to

a quadratic polynomial before plugging the coefficients into the quadratic
formula. The graph of

a quadratic polynomial is a parabola, and the discriminant can be used to
determine the number

of x-intercepts of the parabola. Specifically, if the discriminant is equal to a
negative number,

the parabola does not cross the x-axis; if the discriminant is equal to 0, then
the parabola has one

x-intercept (i.e. the vertex of the parabola is on the x-axis); if the
discriminant is positive, then

the parabola has two, distinct x-intercepts.

It is interesting to show that shifting a parabola horizontally does not change
the

discriminant. To do this, we can compare the equation y = x^{2} + 0 (a
parabola that opens

upwards and has its vertex at the origin) with the equation y = (x − 3)^{2}
+ 0 (a parabola that opens

upwards with vertex (3,0); a horizontal shift of the first equation ).
Evaluating the descrimiant of

the first parabola results in 0^{2} − 4(1)(0) , which is equal to zero.
The second equation can be

written as y = x^{2} − 6x + 9, which has determinant (−6)^{2} −
4(1)(9)=36−36 = 0. Similarly, any

horizontal shift of the equation y = x^{2} + 0 results in a zero
discriminant, thus the discriminant in

this example is invariant under horizontal translation. However, the
discriminant will vary under

vertical translations because the number of x-intercepts can change as the
parabola moves up or

down

The quadratic formula indicates that solutions to the quadratic equation ax^{2}
+ bx + c = 0

are given by. Notice that the discriminant
appears in this formula. Again,

the discriminant plays an important role in evaluating the quadratic formula; if
the discriminant

is equal to zero, the quadratic formula reduces to x=− b ± 0 , so that x=− b is
the only solution.

A way to obtain information about two lines in the x-y coordinate plane in a
system of

equations is to set up a 2 by 2 matrix where
the entries are the parameters in the

equations ax+ by = j and cx+ dy = k.

Solving ax + by = j and cx+ dy = k for y gives:

and

If , then the graphs of the lines determined
by these two equations have the same slope,

and so are either parallel lines or the same exact line. If we consider the
cross products of the

proportion , we obtain ad = bc, or ad - bc =
0. The expression ad – bc also happens to be

the determinant of the matrix
. Based on the
argument above, if the determinant of a

matrix whose entries are obtained from two linear equations is equal to zero,
then the two lines

have the same slope. This means that the two lines are
parallel (if j and k are distinct) or they

are the same line (if j = k).