Number System

Number System for class 10th students

1. NUMBER NUMBER
SYSTEMSYSTEM
       The mysterious world of numbers…
22

2. Contents
Acknowledgment
Introduction
Brief Introduction about
numbers
History of Number System
Number System according to
different civilizations
Types of Numbers
Decimal Expansion Of
Number System
Scientists related to Number
System
 What is a number line?
What is the difference
between numeral and number?
Word Alternatives

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 e­mail 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 according to Number System according to
different civilizations… different civilizations…

12. THE DECIMAL NUMBER
SYSTEM
The number system  we use on day­to­day 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 
radix­10. Outside  of specialized requirement  
such as computing , base­10 numbering system 
have been adopted almost universally. The 
decimal system with which we are fated is a 
place­value system, which means that the value 
of a particular digit depends both on the itself 
and on its position within the number.

13. MAYANNUMBERSYSTEM
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
binarynumeralsystem
, or -2
base numbersystem
,
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 19
th

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
Non­terminating, 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. Project made and Compiled by ~
Samarth Agrawal
Yogesh Surve
Arnab Das
Arijit Sharma
Ankita Sinha
Ayushi Sur
Nimisha Singh