Transcript Number and Coding Systems

Number Systems
 Denary
Base 10
 Binary Base 2
 Hexadecimal Base 16
 Conversion between number systems.
 Signed numbers, 2’s complements binary and binary
fractions.
 Floating point formats.
 Simple arithmetic operations using binary number.
 Bits, bytes and words
1-1
Denary Base 10


The base-10 system (decimal) is the most
commonly used today ( 0 1 2 3 4 5 6 7 8 9)
To represent the number four thousand three
hundred and twenty one (4321)
Thousand
Hundreds
Tens
Units
103
102
101
100
1000
100
10
1
4×1000
3×100
2×10
1×1
4000
+300
+20
1
Total
4321
1-2
Binary Base 2

The base-2 system (Binary) the binary system is used
internally by virtually all modern computers. ( ‘0’ or ‘1’ )
0
1
0
1
1
1
0
1
27
26
25
24
23
22
21
20
127
64
32
16
8
4
2
1
0×127
1×64
0×32
1×16
1×8
1×4
0×2
1×1
0
+64
+0
+16
+8
+4
+0
+1
Total
93
1-3
Hexadecimal Base 16


The base-16 system (Hexadecimal) uses the
numerals “0 1 2 3 4 5 6 7 8 9 B C D E F”
Easy to represent binary code (form of
shorthand) in which one hexadecimal digit
stands in place of four binary bits.
F
A
161
160
16
1
15×16
10×1
240
+10
Total
240


Hex
f
a
Binary ,1111 1010
1-4
Converting from denary to binary
 1) Start with the number you want (44)



2) Divide by the base you want to convert to (2)
3) Calculate the remainder from the sum
4) Place result of the sum on the next row
Number

Base required
Remainder
44
÷
2
0
22
÷
2
0
11
÷
2
1
5
÷
2
1
2
÷
2
0
1
÷
2
1
44 denary = 101100 binary
1-5
Converting from denary to hex




1) Start with the number you want (4321)
2) Divide by the base you want to convert to (16)
3) Calculate the remainder from the sum
4) Place result of the sum on the next row
Number

Base required
Remainder
4321
÷
16
1
270
÷
16
E (14)
16
÷
16
0
1
÷
16
1
4321 Denary = 10E1 Hexadecimal
1-6
Denary, Hex & Binary
Denary
Hex
Binary
0
0
0
0
0
0
1
1
0
0
0
1
2
2
0
0
1
0
3
3
0
0
1
1
4
4
0
1
0
0
5
5
0
1
0
1
6
6
0
1
1
0
7
7
0
1
1
1
8
8
1
0
0
0
9
9
1
0
0
1
10
A
1
0
1
0
11
B
1
0
1
1
12
C
1
1
0
0
13
D
1
1
0
1
14
E
1
1
1
0
15
F
1
1
1
1
1-7
Signed numbers




In mathematics negative numbers can be prefixed by the
'-' symbol.
On a computer system several methods cane be used
Use of a sign bit
Often Most Significant bit is used for the sign
Number
Binary
7
0
1
1
1
......1
0
0
0
1
0
0
0
0
0
-0
1
0
0
0
-1.........
1
0
0
1
-7
1
1
1
1


4 Bits gives a range -7 to
+7
Two representations of '0'
1-8
1's complement



To obtain the negative number the positive
binary value is complemented
'0' replaced with '1' and '1' replaced with '0'
For example binary 00001010 (decimal 10)
complemented becomes 11110101
1-9
2's complement





To obtain the 2's complement of a number
First the 1's complement is obtained
Then 1 is added
i.e. decimal 10 = 00001010, complemented =
11110101 add 1 = 11110110.
Advantage of 2's complement is that no
separate addition and subtraction circuits are
required
1-10
Binary fractions

So far only integer numbers have been
considered
Whole Number
15
Fraction
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
sign 26
25
24
23
22
21
20
2-1
2-2
2-3
2-4
2-5
2-6
2-7
2-8



Decimal 10.75 would be 1010.11
Limited by the number of bits assigned to whole
number
Accuracy is lost smallest value is 0.00390625
1-11
Floating point numbers





In scientific applications we often need very small or
large numbers
In the floating point system number is represented by a
signed fractional component called the mantissa and a
signed exponent.
mantissa × 10exponent
The value is normalised i.e. 1230000.0 would be
become 0.123×107
In a typical system where a float is 32 bits 24 bits hold
the signed mantissa and 8bits for the exponent
Floating point calculations often carried out in a separate
co processor
1-12
Binary arithmetic

Binary addition
A
op
B
Sum
Carry
0
+
0
0
0
0
+
1
1
0
1
+
0
1
0
1
+
1
0
1
Dec
10
1
+9
19
1
0
1
Binary
0
1
0
0
0
1
0
1
1
1-13
Binary arithmetic

Binary subtraction
A
0
op
-
B
0
Sum
0
borrow
0
0
1
1
-
1
0
1
1
1
0
1
1
0
* * **
1101110
−10111
---------------=1010111
(starred columns are borrowed from)
1-14
Binary arithmetic

Multiplication - Two numbers A and B can be multiplied by partial
products: for each digit in B, the product of that digit in A is
calculated and written on a new line, shifted leftward so that its
rightmost digit lines up with the digit in B that was used. The sum of
all these partial products gives the final result.
1 0 1 1 (A)
(decimal 11)
× 1 0 1 0 (B)
(decimal 10)
--------0 0 0 0 ← Corresponds to a zero in B
+ 1 0 1 1 ← Corresponds to a one in B
+0000
+1011
--------------= 1101110
(decimal 110)
1-15
Binary arithmetic

Division i.e. divide 0110101 (53) by 0101 (5).
___ 1010
(10 answer)
0101| 0110101
0101
110
0101
11 (3 remainder)
1-16
Bits Bytes & Words

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

Bit either a '1' or a '0'
Byte comprises 8 bits ' 10110011'
Nibble - Half a byte (4 bits) '1011'
Word Normally the bus width of the
microprocessor – in this case 32 bits
Hex numbers in the keil compiler are prefixed by
'0x' (0xa7)
Floating point numbers – exponent is
represented by the letter 'e' (1.25e-3)
1-17
ASCII character codes



Integer numbers are easily represented in
binary. This is not the case for characters and
other punctuation marks.
The standard ASCII code defines 128 character
codes (from 0 to 127), of which, the first 32 are
control codes (non-printable), and the other 96
are represent printable characters.
The ASCII code requires a 7 bit binary code.
Memory is usually a 8-bit wide which leaves 1
unused bit – use to provide additional characters
called Extended ASCII. Not standardised!
The Standard ASCII Table
*
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
0
NUL
SOH
STX
ETX
EOT
ENQ
ACK
BEL
BS
TAB
LF
VT
FF
CR
SO
SI
1
DLE
DC1
DC2
DC3
DC4
NAK
SYN
ETB
CAN
EM
SUB
ESC
FS
GS
RS
US
!
"
#
$
%
&
'
(
)
*
+
,
-
.
/
2
3
0
1
2
3
4
5
6
7
8
9
:
;
<
=
>
?
4
@
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
5
P
Q
R
S
T
U
V
W
X
Y
Z
[
\
]
^
_
6
`
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
7
p
q
r
s
t
u
v
w
x
y
z
{
|
}
~
•
* This table is organized to be easily read in hexadecimal: row numbers represent the first
digit and the column numbers represent the second one. For example, the A character is
located at the 4throw and the 1st column, for that it would be represented in hexadecimal
as 0x41 (6510).
Unicode





ASCII only supports 128 characters
ASCII was extended to 255 characters using the
iso 8859 method, where 15 regions/language
groups were defined.
Unicode method contains all the worlds
characters (well almost) in one coding list.
Uses multibyte encoding of 1,2 or 4 bytes
Common coding is UTF-8
 first
128 codes same as ASCII so provides backwards
compatibility
See http://www.unicode.org