Transcript Number Systems Decimal Binary Denary Octal Hexadecimal Click the mouse or Press the space bar to Continue.

Number Systems
Decimal
Binary
Denary
Octal
Hexadecimal
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Exponents
20 = 1
 21 = 2
 22 = 2 x 2 =4
 23 = 2 x 2 x 2 = 8
 x5 + x10 = x15
 1 / x2 = x -2

Click to Continue
Decimal Numbering systems



Base: 10
Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Representation
5234
Thousands
Hundreds Tens Units
5
2
3
4
Click to Continue
Decimal Numbering systems

Example:
103 = 1000
5
523410
102 = 100
2
101 = 10 100 = 1
3
4
5,234 = 5 x 1000 + 2 x 100 + 3 x 10 + 4 x 1
Click to Continue
Binary Numbering systems
Base: 2
 Digits: 0, 1
 binary number:
1101012
positional powers of 2: 25 24 23 22 21 20
decimal positional value: 32 16 8 4 2 1
binary number:
1 1 0 1 0 1

Click to Continue
Binary to Decimal Conversion
To convert to base 10, add all the values where
a one digit occurs.
Ex:
1101012
positional powers of 2: 25 24 23 22 21 20
decimal positional value: 32 16 8 4 2 1
binary number:
1 1 0 1 0 1
32 + 16 + 4 + 1 = 5310

Click to Continue
Binary to Decimal Conversion
Ex:
1010112
positional powers of 2: 25
decimal positional value:
binary number:
24
23 2 2 21 20
Click to Continue
Decimal to Binary Conversion
The Division Method. Divide by 2 until you reach zero,
and then collect the remainders in reverse.
Ex 1:
5610
=
1110002
2 ) 56 Rem:
2 ) 28 0
2 ) 14 0
2) 7
0
2) 3
1
2) 1
1
0 1
Click to Continue
Decimal to Binary Conversion
Ex 2: 3510 =
2)
2)
2)
2)
2)
2)
Rem:
Answer:
3510 =
2
Click to Continue
Decimal to Binary Conversion
The Subtraction Method:
Subtract out largest power of 2 possible (without
going below zero) each time until you reach 0.
Place a one in each position where you were
able to subtract the value, and a 0 in each
position that you could not subtract out the
value without going below zero.
Click to Continue
Decimal to Binary Conversion
Ex 1:
56
- 32
24
- 16
8
- 8
0
5610
26 | 25 24 23 22 21 20
64 | 32 16 8 4 2 1
| 1 1 1 0 0 0
Answer: 5610 = 1110002
Click to Continue
Decimal to Binary Conversion
Ex 2:
3810
38
26 | 25 24 23 22 21 20
|
|
Answer: 3810 =
2
Click to Continue
Character Representation
ASCII Table
Rightmost
Four Bits
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
000
NUL
SOH
STX
ETX
EOT
ENQ
ACK
BEL
BS
HT
LF
VT
FF
CR
SO
SI
001
DLE
DC1
DC2
DC3
DC4
NAK
SYN
ETB
CAN
EM
SUB
ESC
FS
GS
RS
US
Leftmost Three Bits
010
011
100
Space
0
@
!
1
A
"
2
B
#
3
C
$
4
D
%
5
E
&
6
F
'
7
G
(
8
H
)
9
I
*
:
J
+
;
K
,
<
L
=
M
.
>
N
/
?
O
101
P
Q
R
S
T
U
V
W
X
Y
Z
[
\
]
^
_
110
`
a
b
c
d
e
f
g
h
I
j
k
l
m
n
o
111
p
q
r
s
t
u
v
w
x
y
z
{
|
}
~
DEL
Click to Continue
Character Representation
Ex: Find the binary ASCII and decimal ASCII values for the ‘&’ character.
Rightmost
Four Bits
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
000
NUL
SOH
STX
ETX
EOT
ENQ
ACK
BEL
BS
HT
LF
VT
FF
CR
SO
SI
001
DLE
DC1
DC2
DC3
DC4
NAK
SYN
ETB
CAN
EM
SUB
ESC
FS
GS
RS
US
Leftmost Three Bits
010
011
100
Space
0
@
!
1
A
"
2
B
#
3
C
$
4
D
%
5
E
&
6
F
'
7
G
(
8
H
)
9
I
*
:
J
+
;
K
,
<
L
=
M
.
>
N
/
?
O
101
P
Q
R
S
T
U
V
W
X
Y
Z
[
\
]
^
_
110
`
a
b
c
d
e
f
g
h
I
j
k
l
m
n
o
111
p
q
r
s
t
u
v
w
x
y
z
{
|
}
~
DEL
Click to Continue
Character Representation
ASCII Table
From the chart:
‘&’ = 0100110
(binary ASCII value)
Convert the binary value to decimal:
01001102 = 32 + 4 + 2 = 3810
Therefore:
‘&’ = 38
(decimal ASCII value)
Click to Continue
Octal Numbering systems
Base: 8
 Digits: 0, 1, 2, 3, 4, 5, 6, 7
 Octal number:
12468
powers of :
84
83 82 81 80
decimal value: 4096 512 64 8 1
Octal number:
1 2 4 6

Click to Continue
Octal to Decimal Conversion
To convert to base 10, beginning with the
rightmost digit multiply each nth digit by 8(n-1),
and add all of the results together.
Ex:
12468
positional powers of 8:
8 3 82 81 80
decimal positional value: 512 64 8 1
Octal number:
1
2 4 6
512 + 128 + 32 + 6 = 67810

Click to Continue
Octal to Decimal Conversion
Ex:
103528
positional powers of 8:
decimal positional value:
Octal number:
84
83
82
Click to Continue
81 80
Decimal to Octal Conversion
The Division Method. Divide by 8 until you reach zero,
and then collect the remainders in reverse.
Ex 1:
433010
= 103528
8 ) 4330
Rem:
8 ) 541
2
8 ) 67
5
8) 8
3
8) 1
0
0
1
Click to Continue
Decimal to Octal Conversion
Ex 2: 81010 =
8 ) 810
8)
8)
8)
Rem:
Answer:
81010 =
8
Click to Continue
Decimal to Octal Conversion
The Subtraction Method:
Subtract out multiples of the largest power of 8
possible (without going below zero) each time
until you reach 0. Place the multiple value in
each position where you were able to subtract
the value, and a 0 in each position that you
could not subtract out the value without going
below zero.
Click to Continue
Decimal to Octal Conversion
Ex 1:
2018
- 1536
482
- 448
34
- 32
2
- 2
0
201810
84 | 83 82 81 80
4096 | 512 64 8
1
| 3
7 4
2
Answer: 201810 = 37428
Decimal to Octal Conversion
Ex 2:
76510
765
84
| 83
|
|
82
Answer: 76510 =
81
80
8
Hexadecimal Numbering systems
Base: 16
 Digits: 0, 1, 2, 3, 4, 5, 6, 7,8,9,A,B,C,D,E,F
 Hexadecimal number:
1F416
powers of :
164
163 162 161 160
decimal value: 65536 4096 256 16 1
Hexadecimal number:
1 F 4

Hexadecimal Numbering systems
Four-bit Group
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
Decimal Digit
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Hexadecimal Digit
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
Hexa to Decimal Conversion
To convert to base 10, beginning with the
rightmost digit multiply each nth digit by 16(n-1),
and add all of the results together.
Ex:
1F416
positional powers of 16:
163 162 161 160
decimal positional value: 4096 256 16 1
Hexadecimal number:
1
F 4
256 + 240 + 4 = 50010

Hexa to Decimal Conversion
Ex:
7E16
positional powers of 16:
decimal positional value:
Hexa number:
163
162
161 160
Decimal to Hexa Conversion
The Division Method. Divide by 16 until you
reach zero, and then collect the remainders in
reverse.
Ex 1:
12610
= 7E16
16) 126 Rem:
16) 7
14=E
0
7
Decimal to Hexa Conversion
Ex 2: 81010 =
16 ) 810
16 )
16 )
Rem:
Answer:
81010 =
16
Decimal to Hexa Conversion
The Subtraction Method:
Subtract out multiples of the largest power of 16
possible (without going below zero) each time
until you reach 0. Place the multiple value in
each position where you were able to subtract
the value, and a 0 in each position that you
could not subtract out the value without going
below zero.
Decimal to Hexa Conversion
Ex 1:
810
- 768
42
- 32
10
- 10
0
81010
163 | 162
4096 | 256
| 3
161 160
16 1
2 A
Answer: 81010 = 32A16
Decimal to Hexa Conversion
Ex 2:
156
15610
162 | 161
|
|
160
Answer: 15610 =
16
Binary to Octal Conversion
Since the maximum value represented in 3 bit is equal
to:
23 – 1 = 7
i.e. using 3 bits we can represent values from 0 –7
which are the digits of the Octal numbering system.
Thus, three binary digits can be converted to one octal
digit.
Binary to Octal Conversion
Three-bit Group
000
001
010
011
100
101
110
111
Decimal Digit
0
1
2
3
4
5
6
7
Octal Digit
0
1
2
3
4
5
6
7
Octal to Binary Conversion
Ex :
Convert 7428 =
2
7 = 111
4 = 100
2 = 010
7428 = 111 100 0102
Binary to Octal Conversion
Ex :
Convert 101001102 =
110 = 6
100 = 4
010 = 2
8
( pad empty digits with 0)
101001102 = 2468
Binary to Hexa Conversion
Since the maximum value represented in 4 bit is equal
to:
24 – 1 = 15
i.e. using 4 bits we can represent values from 0 –15
which are the digits of the Hexadecimal numbering
system.
Thus, Four binary digits can be converted to one
Hexadecimal digit.
Binary to Hexa conversion
Four-bit Group
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
Decimal Digit
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Hexadecimal Digit
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
Hexa to Binary Conversion
Ex :
Convert 3D98 =
2
3 = 0011
D = 1101
9 = 1001
3D98 = 0011 1101 10012
Binary to Hexa Conversion
Ex :
Convert 101001102 =
0110 = 6
1010 = A
101001102 = A616
8
Octal to Hexa Conversion
To convert between Octal to Hexadecimal
numbering systems and visa versa convert from one
system to binary first then convert from binary to
the new numbering system
Hexa to Octal Conversion
Ex :
Convert E8A16 =
8
1110
1000 10102
111
010
001
010
7
2
1
2
E8A16 = 72178
(group by 3 bits)
Octal to Hexa Conversion
Ex :
Convert 7528 =
16
111
101
0102
0001
1110
1010
1
E
A
7528 = 1EA16
(group by 4 bits)