This ppt slide is all about number system. here we learn- To represent numbers To know about different system How number system works 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.

1. ‫يم‬ِ‫ح‬‫ه‬‫الر‬ ِ‫ن‬َ‫م‬ْ‫ح‬‫ه‬‫الر‬ ِ ‫ه‬
‫اَّلل‬ ِ‫م‬ ْ
‫س‬ِ‫ب‬
In the name of Allah, the Beneficent, the Merciful

2. Welcome
To
The Presentation

3. Acknowledgment
We would like to thank Mahabubur Rahman Sir for giving me an
opportunity to express such a important topic via mathematical
presentation. I am also thankful to our friends for their ideas and co-
operation they provided to me. I am grateful to all of them.

4. Who I am...
S.M. Fazla Rabbi
Roll No. 37
Batch E-53
Department of CSE

5. Special Thanks to...
Chairman
Department of
Computer science and Engineering
For
Giving
Me
opportunity

6. Number Systems in
Mathematics
Presentation
on

7. Why need to learn?
▪ To represent numbers
▪ To know about different system
▪ How number system works

8. Content
Introduction
Classification
Discussion
History
Question
1
2
3
4
5

9. Introduction
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.

10. Brief Introduction About Numbers
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.

11. The history of number system
Modern Number System came
from different civilizations

12. The history of number system
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…

13. The history of number system
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.

14. The history of number system
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

15. The history of number system
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

16. 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.
The history of number system

17. The history of number system
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 history of number system
Finally, In the sections following this one, we will give a chronological survey of number
systems throughout our mathematical history. We begin with prehistoric number systems,
which—in a way—predate even counting. We then travel through the proverbial wormhole,
if you will, and arrive at about ~3000 B.C.E. where the Egyptians are ruling supreme. Not
alone though, as the Summerians are more or less side by side and show signs of a much
higher mathematical understanding (~3000 B.C.E.). As the Summerians more or less
become the Babylonians around 2000 B.C.E. The Chinese, working largely in isolation, are
next: ~2500 B.C.E. Greek mathematicians become a reckoned—in more than one sense of
the word—force around 500 B.C.E. The mathematically impaired Romans provide no real
mathematical interest, but rotationally they are of large interest (~0 C.E.). Four hundred
years later (~400 C.E.) and across the globe, the astrologists of the Mayan civilization are
making incredible computations using, what seems to be, a number system developed in
isolation. The Hindus make achievements during a long period of time. Perhaps the most
important ones to us around 300 B.C.E. to 400 C.E. Then again, we make a rather big jump in
history and arrive in modern times, where we discuss computers and number systems.

19. Classification of Number System
Number Systems
Complex
Real
Rational
Imaginary
Irrational
Positive
Integer
Negative Zero Whole/ Non-negative
Natural
Composite
Fraction
Odd
Even
Prime
Terminating and
Repeating
NoTerminating and No
Repeating

20. Discussion
On various types of number

21. Prime & Composite
A prime number (or a prime) is a natural number greater than 1 that has no positive divisors
other than 1 and itself.
There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime
numbers: the Egyptian fraction expansions in the Rhind papyrus, for instance, have quite different forms for
primes and for composites. However, the earliest surviving records of the explicit study of prime numbers come
from the Ancient Greeks. Euclid's Elements (circa 300 BC) contain important theorems about primes
Example: 5 can only be divided evenly by 1 or 5, so it is a prime number.
When a number can be divided up evenly it is a Composite Number. Ex- 4,6,8,9….
Composite
History
Definition
Definition

22. Integer
An integer is a number that can be written without a fractional component . For example, 21, 4,
0, and −2048 are integers, while 9.75, 5 1⁄2, and √2 are not. Integer is denoted as Z.
Z= {……,-3,-2,-1,0,1,2,3,…..}
Whole/ Non-
negative
Odd
Even
Positive {1,2,3,…….}
Negative {….,-3,-2,-1,}
Zero { 0 }
{0,1,2,3,…….}
{2,4,6,8……}
Any integer that can be divided exactly by 2 is an even number.The last digit is 0, 2, 4, 6 or 8
Example: −24, 0, 6 and 38 are all even numbers.
{1,3,5,7,……}
Any integer that can be divided exactly by 2 is an even number.The last digit is 0, 2, 4, 6 or 8 .
Example: −3, 1, 7 and 35 are all odd numbers.
Zero’s origin>>4000 yrs Mesopotamia-Sumerian ”space”>>> third
century B.C. in ancient Babylon zero symbol >> 350 A.D.Mayans
calender.>>>seventh century A.D. in India mathematician
Brahmagupta >>>Middle East 773 A.D Mohammed ibn-Musa al-
Khowarizmi It was al-Khowarizmi who first synthesized Indian
arithmetic and showed how the zero could function in algebraic
equations.

23. Fraction
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.

24. Real
In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this
context was introduced in the 17th century by René Descartes, who distinguished between real
and imaginary roots of polynomials. Julius Wilhelm Richard Dedekind was a German mathematician who
made important contributions to abstract algebra (particularly ring theory), algebraic number theory and
the foundations of the real numbers. The Real Numbers had no name before Imaginary Numbers were
thought of. Mathematician got called "Real" because they were not Imaginary.
Rules
* Real Numbers are measurable.The sets of real number, are those numbers
That can be mapped on a number line.
* Real Numbers have a concrete value
* Real Numbers can be manipulated, all can rewritten as a decimal
In mathematics, a real number is a value that represents a
quantity along a continuous line. The real numbers include
all the rational numbers, such as the integer −5 and
the fraction 4/3, and all the irrational numbers such as
√2 (1.41421356… the square root of two, an
irrational algebraic number) and π (3.14159265…, a
transcendental number).
History
Definition

25. Rational & Irrational
In mathematics, a rational number is any number that can be expressed
as the quotient or fraction p/q of two integers, a numerator p and a non-
zero denominator q. Since q may be equal to 1, every integer is a rational
number. Rational numbers are usually denoted by a boldface Q, it was
thus denoted in 1895 by Giuseppe Peano. Example: ⅛, ⅔.
History
&
Definition
Characteristics
Definition
Characteristics
Terminating>>> ½ = 0.5
Repeating>>> 10/3 =3.333333
No terminating & No repeating >>> √2= 1.4142135624 ………
The numbers cannot be written as a ratio of two integers are
called Irrational Numbers. Example: π (Pi) is a famous irrational
number. π = 3.1415926535897932384626433832795... (and more)

26. Imaginary
An imaginary number is a complex number that can be written as a real
number multiplied by the imaginary unit i, which is defined by its property i2 =
−1. The square of an imaginary number bi is −b2. For example, 5i is an imaginary number,
and its square is −25. Zero is considered to be both real and imaginary.
Greek mathematician and engineer Heron of Alexandria is noted as the first to have
conceived these numbers,Rafael Bombelli first set down the rules for multiplication of
complex numbers in 1572. The concept had appeared in print earlier, for instance in work
by Gerolamo Cardano.
Imaginary numbers can be very useful for solving engineering problems. On example is
if you have a pendulum swinging, it starts to slow down and eventually stop. If you want
to work out the motion of the pendulum over a certain time (ie derive a formula) then
the best way to do it is to use complex numbers.
History
Definition
Uses

27. Complex
A complex number is a number that can be expressed in the form a + bi,
where a and b are real numbers, and i is the imaginary unit (which satisfies
the equation i2 = −1). In this expression, a is called the real part of the
complex number, and b is called the imaginary part. If z=a+bi, then we
write {Re} (z)=a and {Im} (z)=b .
For example, −3.5 + 2i is a complex number.
The Italian mathematician Gerolamo Cardano is the first person known to have introduced complex
numbers. He called them "fictitious" during his attempts to find solutions to cubic equations in the 16th
century.
Complex numbers are the building blocks of more intricate math, such as
algebra. They can be applied to many aspects of real life, especially in
electronics and electromagnetism.
History
Definition
Uses

28. 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.

29. In Short
Natural, N={1,2,3,4,5……..}
Integer, I={-3,-2,-1,0,1,2,3}
Real, R=[-∞, +∞]
Rational, Q= {
𝑝
𝑞
, 𝑝, 𝑞 ∈ 𝐼, 𝑞 ≠ 0} ex.
1
2
=0.5 (Terminating),
10
3
=3.333333 (repeating)
Irrational, Qi={√2, √3, √5, √7} ex. 2 = 1.4142135624 (no terminating & no repeating)
Complex, Z= x+iy, x,y ∈ 𝑅 where, x= real part and iy= Imaginary part…we can say
All real number are Subset of complex number.
Imaginary, i = Square root of negative number. Ex. −1, −2, −3….Where i exist.

30. 1) Binary Number System
A Binary number system has only two digits that are 0 and 1. Every number (value) represents
with 0 and 1 in this number system. The base of binary number system is 2, because it has only
two digits.
2) Octal number system
Octal number system has only eight (8) digits from 0 to 7. Every number (value) represents with
0,1,2,3,4,5,6 and 7 in this number system. The base of octal number system is 8, because it has
only 8 digits.
3) Decimal number system
Decimal number system has only ten (10) digits from 0 to 9. Every number (value) represents with
0,1,2,3,4,5,6, 7,8 and 9 in this number system. The base of decimal number system is 10, because it
has only 10 digits.
4) Hexadecimal number system
A Hexadecimal number system has sixteen (16) alphanumeric values from 0 to 9 and A to F. Every
number (value) represents with 0,1,2,3,4,5,6, 7,8,9,A,B,C,D,E and F in this number system. The
base of hexadecimal number system is 16, because it has 16 alphanumeric values. Here A is 10, B
is 11, C is 12, D is 13, E is 14 and F is 15.
Most popular number
systems
Used
In
Computer
System
And
Mathematics
Also

31. S o m e
R e f e r e n c e s
Work behind
en.wikipedia.org
www.archimedes-lab.org
www.math.chalmers.se www.slideshare.net
www.google.com
translate.google.com
www.mathsisfun.com www.youtube.com
www.purple-math.com
www.whatis.techtarget.com www.basic-mathematics.com

32. Thank you !
Anything else?