““The appropriate numbering systemThe appropriate numbering system
to represent a value depends on theto represent a value depends on the
application. There are advantagesapplication. There are advantages
and disadvantages to each numberingand disadvantages to each numbering
Numeration SystemNumeration System
Past and Present ..Past and Present ..
The symbols for writing numbers are called
numerals and the methods for calculating are
called algorithms. Taken together, any
particular system of numerals and algorithms is
called a numeration system.
The earliest means of recording numbers
consisted of a set of tallies – marks on stone,
stones in a bag, notches in a stick – one-for-one
each item being counted.
Example: Often such a stick was split in half with
one half going to the debtor and the other half to
It is still common practice today to keep count
by making tallies or marks with the minor but
useful refinement of marking off tallies in group
of five. Thus,
IIII IIII IIII IIII III
is much easier to read as twenty-three than
But such system for recording numbers were
much too simplistic for large numbers and for
Numbers actually originated in the history of
the Hindus of India. They have changed greatly
over the centuries, passing first to the Arabs of
the Middle East and finally to Europe in the
As early as 3400 B.C., the Egyptians developed
a system for recording numbers on stone
tablets using hieroglyphics.
Used number systems other than base 10, like
the duodecimal (base 12) in which they counted
fingers-joints instead of fingers.
Divided their days into twenty-four periods,
which is, in turn, why we have twenty-four
hours in a day.
The Egyptians had symbols for one and the first
few powers of ten, and then combined symbols
to represent other numbers.
The Ancient Egyptians ..
Famous for their astrological observations and
Used a sexagesimal (base 60) numbering system
Made use of six and ten as sub-bases.
First to develop the concept of cipher position
or place value, in representing larger numbers
Even the Chinese ..
One 一 (Y )ī
Two 二 (Èr)
Four 四 (Sì)
Six 六 (Liù)
Seven 七 (Q )ī
Eight 八 (B )ā
Nine 九 (Ji )ǔ
Ten 十 (Shí)
Hundred 百 (B iǎ )
Thousand 千 (Qi n)ā
10 Thousands 万
100 Millions 亿 (Yì)
The Aztecs, Eskimos
and Indian Merchants ..
Aztecs - Developed vigesimal (base 20) systems
because they counted using both fingers and
The Ainu of Japan and the Eskimos of
Greenland are the two people who make use of
vigesimal systems to the present day.
Relatively easy to understand is quinary (base 5)
which uses five digits: 0,1,2,3,4. This system is
particularly interesting, in that a quinary finger
counting scheme is still in use today by Indian
merchants near Bombay. This allows them to
perform calculations on one hand while serving
their customers with the other.
the Hebrews ..
Used letters for numbers.
Every letter in the Greek or Hebrew alphabet
corresponded to a different number.
For example, in Greek, the letter Alpha
corresponded to 1 and the letter Theta
corresponded to 9.
The Romans ..
Devised a system that was an important
improvement over hash marks
Roman Numeral System
Also used letters to represent numbers
DISADVANTAGES: Very difficult to apply in
large numbers; there is no provision for
representing the number zero or negative
numbers which are very important concepts in
Transmitted to us the Hindu Numeral System
The use of Arabic Numerals in Europe is attributed
to the Italian mathematician Fibonacci.
In 1202, he published a book called Liber Acci,
which taught Arabic Numerals and Algebra and
strongly advocated the use of Arabic numerals in
The use of the Hindu-Arabic numerals is now the
prevalent number system throughout the world.
With the development of the printing press in the
century, the numerals have become
standardized, and this only increased with the
development of computers.
Used on daily basis and has 10 digits
Latin decem “ten”
Symbols used to represent these digits arrived
in Europe around the 13th
century from the
Said to be base 10 or radix 10 where the term
radix comes from the Latin word meaning
Decimals also refer to decimal fractions, either
separately or in contrast to vulgar fractions
It has been also adopted almost universally =
due to the fact that we happen to have 10
A place value system – the value of a particular
digit depends both on the digit itself and on its
position within the number
Almost any other base would be as good as or
better than base 10. This is because, for many
arithmetic operations, the use of a base that is
wholly divisible by many numbers especially the
smaller values , conveys certain advantages.
Some mathematician would ideally prefer a
system with a prime number as a base: for
example, seven or eleven
Nature of ten fingers, place value system, good
choice for interfacing with non-engineers
Mathematical terminology and concept use is
identical to the base that is use for counting
Values for currency, demography, economics,
for quantitative backup in science, etc.
The decimal numeration system
uses ten ciphers, and place-
weights that are multiples of ten.
What if we made a numerationWhat if we made a numeration
system with the same strategysystem with the same strategy
of weighted places, except withof weighted places, except with
fewer or more ciphers?fewer or more ciphers?
The binary numeration system
is such a system ..
Instead of ten different cipher symbols, with
each weight constant being ten times the one
before it, we only have two cipher symbols, and
each weight constant is twice as much as the
one before it.
The two allowable cipher symbols for the
binary system numeration are “1” and “0”, and
these ciphers are arranged right=to=left in
doubling values of weight.
Binary is referred to as “base 2” numeration.
Each cipher position in binary is called bit
The primary reason that the binary numeration
system is used in modern electronic computers
is because of the ease of representing two
cipher states electronically.
Binary numeration also lends itself well to the
storage and retrieval of numerical information:
on magnetic tape, optical disks, or a variety of
other media types.
The real number system evolved over
time by expanding the notion of
what we mean by the word
At first, “number” meant something you could
count, like how many sheep a farmer owns.
These are called the natural numbers, or
sometimes the counting numbers.
The use of three dots at the end of the list is a
common mathematical notation to indicate
that the list keeps going forever.
At some point, the idea of “zero” came to be
considered as a number. If the farmer does not
have any sheep, then the number of sheep that
the farmer owns is zero. We call the set of
natural numbers plus the number zero the
After more than a, 500 years of potentially
inaccurate calculations, the Babylonians finally
began the special sign for zero.
Many historians believe that this sign, which
first around 300 BC, was one of the most
significant inventions in the history of
However, the Babylonians only used their
symbol as place holder and they didn’t have the
concept of zero as an actual value.
The concept of zero first appeared in India
around 600 AD.
Even more abstract than zero is the idea of
negative numbers. If, in addition to not having
any sheep, the farmer owes someone 3 sheep,
you could say that the number of sheep that
the farmer owns is negative 3. It took longer
for the idea of negative numbers to be
accepted, but eventually they came to be seen
as something we could call “numbers.” The
expanded set of numbers that we get by
including negative versions of the counting
numbers is called the integers.
Whole numbers plus negative numbers
First appeared in India around 600 AD
century, the great Swiss mathematician
Leonhard Euler believed that negative numbers
were greater than infinity, and it was common
practice to ignore any negative results
returned by equations on the assumptions that
they were meaningless
While it is unlikely that a farmer owns a
fractional number of sheep, many other things
in real life are measured in fractions, like a half-
cup of sugar. If we add fractions to the set of
integers, we get the set of rational numbers.
All numbers of the form , where a and b are
integers (but b cannot be zero)
Rational numbers include what we usually call
Notice that the word “rational” contains the
word “ratio,” which should remind you of
Now it might seem as though the set of
rational numbers would cover every possible
case, but that is not so. There are numbers
that cannot be expressed as a fraction, and
these numbers are called irrational because they
are not rational.
Cannot be expressed as a ratio of integers.
As decimals they never repeat or terminate
(rationals always do one or the other)
Rationals + Irrationals
All points on the number line
Or all possible distances on the number
An Ordered Set ..
The real numbers have the property that they
are ordered, which means that given any two
different numbers we can always say that one
is greater or less than the other. A more
formal way of saying this is:
For any two real numbers a and b, one and
only one of the following three statements is
1. a is less than b, (expressed as a < b)
2. a is equal to b, (expressed as a = b)
3. a is greater than b, (expressed as a > b)
Every real number corresponds to a distance
on the number line, starting at the center
Negative numbers represent distances to the
left of zero, and positive numbers are distances
to the right.
The arrows on the end indicate that it keeps
going forever in both directions.
Absolute Value ..
When we want to talk about how “large” a
number is without regard as to whether it is
positive or negative, we use the absolute value
function. The absolute value of a number is the
distance from that number to the origin (zero)
on the number line. That distance is always
given as a non-negative number.
If a number is positive (or zero), the absolute
value function does nothing to it
If a number is negative, the absolute value
function makes it positive
Chara Nina Marie
Leslie Bernolo /emh