Numbersystem 130913004355-phpapp02-converted

NumberSystem
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Number system
 In earlier days, people used to exchange their things for
other things. The requirement for numbers primarily
originated from the need to count.
 They used the numbers 1,2,3,.that served the people for
many years because all they needed to count was their
crops, and animals.
 Later on numbers such as zero, integers, rational
numbers, irrational numbers were introduced.
 There is evidence that as early as 30,000 BC our ancient
ancestors were tallying or counting things. That is where
the concept of number systems began.
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Numbers
 Natural Numbers:
A natural number is a number that comes naturally,
Natural numbers are greater than zero we can use this
numbers as counting numbers: {1, 2, 3, 4, 5, 6 ….…, }.
 Whole numbers:
Whole numbers are just all the natural numbers plus a
zero: {0, 1, 2, 3, 4, 5, ……………… , }.
 If our system of numbers was limited to the Natural
Numbers then a number such as –2 would have no
meaning. The next number system is the Integers.

numbers
 Integers:
Integers include the Natural numbers, zero, and the
negative Natural numbers.
Numbers in the form of negative and positive numbers {
….-4, -3, -2, -1, 0, 1, 2, 3,4, …. }.
 Rational number:
Which can be written in the form of
Where p and q are integers and q ≠ 0 is called a rational
number, so all the integers are rational number .
.p
q

numbers
 Irrational numbers : p
The number can not be written in the form of . q
Pythagorean in Greece were first to discover irrational
number .
 2, 3, are irrational number .

numbers
 Real numbers:
 All the numbers including rational and irrational numbers
are called real number
 The official symbol for real numbers is a bold R.
 Prime numbers:
 The real number which is divisible by 1 and itself is called
prime number Ex- 1,2,3,5,7,11,13,17, …..

The Real Number System
Real Numbers
(all numbers are real)
Rational Numbers Irrational Numbers
…-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5
Integers
Whole Numbers
Natural Numbers
…any number that is
not rational
Example:
= 3.14159……
e= 2.71828…..
Which can be written in the form of
.
p
q

Number system
 A number system defines how a number can be
represented using distinct symbols.
 A number can be represented differently in different
systems.
 For example, the two numbers (2A)16 and (52)8 both refer to
the same quantity, (42)10, but their representations are
different.

Common Number Systems
Number system can be categorized as
Decimal number system
Binary number system
Octal number system
Hexadecimal Number System

Common Number Systems
 Each number system is associated with a base or radix
The decimal number system is said to be of base or radix
10
 A number in base r contains r digits 0,1,2,...,r-1
Decimal (Base 10): 0,1,2,3,4,5,6,7,8,9
System Base Symbols
Used by
humans?
Used in
computers?
Decimal 10 0, 1, … 9 Yes No
Binary 2 0, 1 No Yes
Octal 8 0, 1, … 7 No No
Hexa-
decimal
16 0, 1, … 9,
A, B, … F
No No

The decimal system (base 10)
 The word decimal is derived from the Latin root decem
(ten). In this system the base b = 10 and we use ten
symbols.
S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.
Binary system (base 2)
 The word binary is derived from the Latin root bini (or
two by two).
 In this system the base b = 2 and we use only two
symbols,
S = {0, 1}
 The symbols in this system are often referred to as
binary digits or bits.

The hexadecimal system
(base 1)
 The word hexadecimal is derived from the Greek root
hex (six) and the Latin root decem (ten).
 In this system the base b = 16 and we use sixteen
symbols to represent a number.
 The set of symbols is
S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E,F}
 The symbols A, B, C, D, E, F are equivalent to
10, 11, 12, 13, 14, and 15 respectively.
 The symbols in this system are often referred to as
hexadecimal digits.

The o c t a l system (base 8)
 The word octal is derived from the Latin root octo (eight).
 In this system the base b = 8 and we use eight symbols
to represent a number.
 The set of symbols is:
S = {0, 1, 2, 3, 4, 5, 6, 7}

Converting Decimal t o Binary
162 / 2 = 81 rem 0
81 / 2 = 40 rem 1
40 / 2 = 20 rem 0
20 / 2 = 10 rem 0
10 / 2 = 5 rem 0
5 / 2 = 2 rem 1
2 / 2 = 1 rem 0
1 / 2 = 0 rem 1
 To convert a decimal integer into binary, keep dividing by 2
until the quotient is 0. Collect the remainders in reverse
order
 To convert a fraction, keep multiplying the fractional part by
2 until it becomes 0. Collect the integer parts in forward
order
 Example: 162.375: So, (162.375)10 = (10100010.011)2
0.375 x 2 = 0.750
0.750 x 2 = 1.500
0.500 x 2 = 1.000
14

Octal and Hexadecimal
Numbers
The octal number system: Base-8
Eight digits: 0,1,2,3,4,5,6,7
(127 .4) 1 82
2 81
7 80
4 8 1
(87 .5)
8 10
 The hexadecimal number system: Base-16
 Sixteen digits: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
 For our purposes, base-8 and base-16 are most useful as a
“shorthand” notation for binary numbers
(B65 F ) 11 16 3
6 16 2
5 161
15 16 0
( 46687 )
16 10
15

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