based on ncert books

1. NUMBERNUMBER
SYSTEMSYSTEM
The mysterious world of numbers…
22

2. Contents
Acknowledgment
I nt r oduct ion
Brief I nt roduct ion about
numbers
Hist ory of Number Syst em
Number Syst em according t o
dif f erent civilizat ions
Types of Numbers
Decimal Expansion Of
Number Syst em
Scient ist s relat ed t o Number
Syst em
 What is a number line?
What is t he dif f erence
bet ween numeral and number?
Word Alt ernat ives

3. ACKNOWLEDGEMENT
We would like to thank Teema madam for giving us an
opportunity to express ourselves via mathematical
projects. We are also thankful to Bharti madam, our
computer teacher for letting us use the school computers
for presentation and providing us with an e-mail ID.
Also, we thank our friends for their ideas and co-
operation they provided to us. We are grateful to all of
them.

4. IntroductionIntroduction
A number system defines a set of values used to
represent a quantity. We talk about the number of
people attending school, number of modules taken
per student etc.
Quantifying items and values in relation to each
other is helpful for us to make sense of our
environment.
The study of numbers is not only related to
computers. We apply numbers everyday, and
knowing how numbers work, will give us an
insight of how computers manipulate and store
numbers.

5. A number is a mathematical object used in
counting and measuring. It is used in counting
and measuring. Numerals are often used for
labels, for ordering serial numbers, and for codes
like ISBNs. In mathematics, the definition of
number has been extended over the years to
include such numbers as zero, negative numbers,
rational numbers, irrational numbers, and
complex numbers.

6. Numbers were probably first used many thousands
of years ago in commerce, and initially only whole
numbers and perhaps rational numbers were
needed. But already in Babylonian times, practical
problems of geometry began to require square
roots.
Certain procedures which take one or more
numbers as input and produce a number as output
are called numerical operation.

7. The History Of Number
System
The number system with which we are
most familiar is the decimal (base-10)
system , but over time our ancestor have
experimented with a wide range of
alternatives, including duo-decimal (base-
12), vigesimal (base-20), and sexagesimal
(base-60)…

8. The Ancient Egyptians
The Ancient Egyptians experimented with duo-decimal (base-
12) system in which they counted finger-joints instead of finger .
Each of our finger has three joints. In addition to their base-
twelve system, the Egyptians also experimented with a sort –of-
base-ten system. In this system , the number 1 through 9 were
drawn using the appropriate number of vertical lines.
A human hand palm was the way
of counting used by the
Egyptians…

9. The Ancient Babylonians
Babylonians, were famous for their astrological observations
and calculations, and used a sexagesimal (base-60) numbering
system. In addition to using base sixty, the babylonians also
made use of six and ten as sub-bases. The babylonians
sexagesimal system which first appeared around 1900 to 1800
BC, is also credited with being the first known place-value of
a particular digit depends on both the digit itself and its
position within the number . This as an extremely important
development, because – prior to place-value system – people
were obliged to use different symbol to represent different
power of a base.

10. Aztecs, Eskimos, And Indian
Merchants.
Other cultures such as the Aztecs, developed vigesimal (base-
20) systems because they counted using both finger and toes.
The Ainu of Japan and the Eskimos of Greenland are two of
the peoples who make use of vigesimal systems of present
day . Another system that is relatively easy to understand is
quinary (base-5), which uses five digit : 0, 1, 2, 3, and 4. The
system is particularly interesting , in that a quinary finger-
counting scheme is still in use today by Indian merchant near
Bombay . This allow them to perform calculations on one
hand while serving their customers with the other.
Aztecs were the ethnic
group of Mexico

11. Number System accordingNumber System according
to different civilizations…to different civilizations…

12. THE DECIMAL NUMBER
SYSTEM
The number system we use on day-to-day basis
in the decimal system , which is based on ten
digits: zero through nine. As the decimal system
is based on ten digits, it is said to be base -10 or
radix-10. Outside of specialized requirement
such as computing , base-10 numbering system
have been adopted almost universally. The
decimal system with which we are fated is a
place-value system, which means that the value
of a particular digit depends both on the itself
and on its position within the number.

13. MAYAN NUMBER
SYSTEM
This system is unique to our current decimal
system, as our current decimal system uses base
-10 whereas, the Mayan Number System uses
base- 20.
The Mayan system used a combination of two
symbols. A dot (.) was used to represent the units
and a dash (-) was used to represent five. The
Mayan's wrote their numbers vertically as
opposed to horizontally with the lowest
denomination on the bottom.
Several numbers according to Mayan
Number System

14. BINARY NUMBER
SYSTEM
The binary numeral system, or base-2 number system,
represents numeric values using two symbols, 0 and 1. More specifically,
the usual base-2 system is a positional notation with a radix of 2.
Owing to its straight forward implementation in digital electronic
circuitry using logic gates, the binary system is used internally by all
modern computers. Counting in binary is similar to counting in any
other number system. Beginning with a single digit, counting proceeds
through each symbol, in increasing order. Decimal counting uses the
symbols 0 through 9, while binary only uses the symbols 0 and 1.

15. FRACTIONS AND ANCIENT
EGYPT
Ancient Egyptians had an understanding of fractions, however they
did not write simple fractions as 3/5 or 4/9 because of
restrictions in notation.The Egyptian scribe wrote fractions with
the numerator of 1.They used the hieroglyph “an open mouth"
above the number to indicate its reciprocal.The number 5,
written , as a fraction 1/5 would be written as . . .There are some
exceptions.There was a special hieroglyph for 2/3, , and some
evidence that 3/4 also had a special hieroglyph. All other fractions
were written as the sum of unit fractions. For example 3/8 was
written as 1/4 + 1/8.

18. The real numbers include all of the measuring numbers .
Real numbers are usually written using decimal numerals ,
in which a decimal point is placed to the right of the digit
with place value one.
 It includes all types of numbers such as Integers,Whole
numbers, Natural numbers, Rational number, Irrational
numbers and etc… Let us see them in detail…

19. A rational number is a number that can
be expressed as a fraction with an
integer numerator and a non-zero
natural number denominator.The
symbol of the rational number is ‘Q’. It
includes all types of numbers other
than irrational numbers, i.e. it includes
integers, whole number, natural
numbers etc…

20. This is a type of a rational number. Fractions are written
as two numbers, the numerator and the denominator
,with a dividing bar between them.
 In the fraction m/n ‘m’ represents equal parts, where ‘n’
equal parts of that size make up one whole.
 If the absolute value of m is greater than n ,then the
absolute value of the fraction is greater than 1.Fractions
can be greater than ,less than ,or equal to1 and can also
be positive ,negative , or zero.

21. If a real number cannot be written as a fraction of two
integers, i.e. it is not rational, it is called irrational
numbers . A decimal that can be written as a fraction
either ends(terminates)or forever repeats about which
we will see in detail further.
Real number pi (π) is an example of irrational.
π=3.14159365358979……the number neither start
repeating themselves or come in a specific pattern.

22.  Integers are the number which includes positive and
negative numbers.
 Negative numbers are numbers that are less than
zero. They are opposite of positive numbers .
Negative numbers are usually written with a negative
sign(also called a minus sign)in front of the number
they are opposite of .When the set of negative
numbers is combined with the natural numbers zero,
the result is the set of integer numbers , also called
‘Z’.

23.  The most familiar numbers are the natural
numbers or counting numbers: One, Two, Three
and so on….
 Traditionally, the sequence of natural numbers
started with 1.However in the 19th
century,
mathematicians started including 0 in the set of
natural numbers.
 The mathematical symbol for the set of all natural
numbers is ‘N’.

24. Moving to a greater level of abstraction, the real numbers
can be extended to the complex numbers. This set of
number arose historically, from trying to find closed formulas
for the roots of cubic and quadratic polynomials. This led to
expressions involving the square roots of negative numbers,
eventually to the definition of a new number: the square root
of negative one denoted by “I”. The complex numbers consist
of all numbers of the form (a+bi) ; Where a and b are real
numbers.

25. Other Types
There are different kind of other numbers too. It includes
 hyper-real numbers,
 hyper-complex numbers,
 p-adic numbers,
 surreal numbers etc.
These numbers are rarely used in our day-to-day life.
Therefore, we need not know about them in detail.

26. Decimal Expansion of
Numbers
A decimal expansion of a number can be either,
 Terminating
 Non-terminating, non recurring
 Non terminating, recurring
Let us see each of the following
briefly…

27. Terminating decimal
A decimal expansion in which the remainder becomes
zero. For example, 54 9 =
Terminating decimal is always a rational number. It can
be written in p/q form.
549
6
54
0
As the remainder is zero, this
is a terminating decimal

28. Non terminating non
recurring
“Recurring” means “repeating”. In this form, when we
divide a number by another, remainder never becomes
zero, and also the number does not repeat themselves in
any specific pattern. If a number is non terminating and
non repeating, they are always classified as irrational
number. For example,
0.10100100010000100000100.... does have a pattern,
but it is not a fixed-length recurring pattern, so the
number is irrational.

29. Non terminating, recurring
In this form, when a number is divided by the other, the
remainder never becomes zero, instead the numbers of
the quotient start repeating themselves. Such numbers are
classified as rational numbers. For example,
3.7250725072507250…
In this example, “7250” have started repeating itself.
Hence, it is a rational number. It can be expressed in p/q
form.

30. Mathematicians related to
Number System
Euclid :
Euclid was an ancient mathematician from
Alexandria, who is best known for his major work,
Elements. He told about the division lemma,
according to which,
A prime number that divides a product of two
integers must divide one of the two integer.
Euclid – The father of geometry

31. Mathematicians related to Number
System
R. Dedekind And G. Cantor :
In 1870s two German mathematicians; Cantor and
Dedekind, showed that :
Corresponding to every real number, there is a point
on the number line, and corresponding to every point
on the number line, there exists a unique real
number.
R. DedekindG. Cantor

32. Archimedes :
He was a Greek mathematician. He was the first to
compute the digits in the decimal expansion of π (pi). He
showed that -
3.140845 < π < 3.142857
Mathematicians related to Number
System
Archimedes

33. A number line is a line with marks on it that are
placed at equal distance apart. One mark on the
number line is usually labeled zero and then each
successive mark to the left or to the write of the zero
represents a particular unit such as 1, or 0.5. It is a
picture of a straight line.
A number line

34. No, number and numerals are not same. Numerals
are used to make numbers. It is a symbol used to
represent a number.
For example, the NUMERAL 4 is the name of
NUMBER four.
Numeral “7”
Number 7

35. Word AlternativesWord Alternatives
Some numbers traditionally have words to express them,
including the following:
 Pair, couple, brace: 2
 Dozen: 12
 Bakers dozen: 13
 Score: 20
 Gross: 144
 Ream(new measure): 500
 Great gross: 1728

37. Pro je ct m ade and Co m pile d
by: -
AKASH DIXIT
X A