Transcript ICS312 Lecture1

ICS312 Set 2
Representation of Numbers and
Characters
Number Systems - Decimal
System (base 10)
Uses positional representation
Each digit in a number is associated with a power of 10
based on its position in the number. The one's digit
(zero power of 10) is the rightmost digit and the powers
of 10 increase as you move to the left.
5,325 = 5 * 103 + 3 * 102 + 2 * 101 + 5 * 100
For the decimal system:
10 = base
10 digits are: 0, 1, 2, 3, 4, 5, 6, 7, 8 , 9
The concept of positional representation applies to the
binary and hexadecimal numbering systems as well.
Number Systems - Binary
Number System
Base = 2
2 Digits: 0, 1
Examples:
1001b = 1 * 23 + 0 * 22 + 0 * 21 + 1 * 20
=8+1
=9
1010 1101b = 1 * 27 + 1 * 25 + 1 * 23 + 1 * 22 + 1
= 128 + 32 + 8 + 4 + 1
= 173
Note: it is common to put binary digits in groups of
4 to make it easier to read them.
Number Systems Hexadecimal Number System
Used as a shorter, easier to read notation than binary.
Base = 16
16 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E,F
Examples:
10h = 1 * 161 + 0 * 160 = 16 (decimal)
200h = 2 * 16^2 = 2 * 256 = 512
45h = 4 * 16^ 1 + 5 = 69
3Fh = 3 * 16 + 15 = 48 + 15 = 63
Converting from binary or hex to
decimal:
Use positional representation to expand the
number and convert to decimal equivalent.
Examples:
0011 1010b
015Eh
Converting from decimal to
binary:
Use repeated division by 2. The remainders give
the binary digits, starting from the lowest
power of 2.
Examples:
Convert
14
104
Division Step
Quotient/Remainder
Binary Digits
Converting from hex to binary
Rewrite each hex digit as its (4-digit)
binary equivalent.
Examples:
Hex Value
4Bh
32FFh
F38Bh
Binary equivalent
Converting from binary to hex
Rewrite each group of 4 binary digits as its
hex equivalent. Pad zeroes on the left as
needed.
Examples:
Binary
1101 1011 0100 0111 b
0110 1111 1010 1001 b
Hex Equivalent
Converting from decimal to
hexadecimal:
It’s simplest to convert the decimal number to binary and then
convert that binary number to hexadecimal.
Arithmetic
Apply the same concepts as for decimal
arithmetic to binary and hexadecimal
arithmetic.
Binary Addition:
Underlying examples: how carries work
1 + 1 b = 10b
1 + 1 + 1 b = 10b + 1 = 11b
Examples:
0011b
0010 1101b
+ 1001b
+ 0101 1110b
=
Hexadecimal Addition:
Underlying
F + 1 h =
E + 2 h =
A + B h =
Examples:
4Ah
+
5Bh
=
4B7Fh
+
2A93h
=
examples: how carries work
10h
10h
10d + 11d = 21d = 15h
Representing Integers in the
Computer (1)
Signed Numbers and Unsigned Numbers
Most Significant Bit (MSB)
leftmost or high-order bit (bit 15 or 7)
Least Significant Bit (LSB)
rightmost or low-order bit (bit 0)
Representing Integers in the
Computer (2)
Signed Integers
•Allows positive and negative integers to be represented
•MSB used to indicate sign:
0 = positive, 1 = negative
•Since 1 bit is used to represent the sign, only 7 or 15 bits can be
used for the magnitude, with positive and negative values for each +
one extra negative value:
1 byte = 8-1 bits = 27 => range = -128 to 127
2 bytes = 16-1 bits = 215 => range = -32,768 to +32,767
Representing Integers in the
Computer (3)
1's Complement
•Formed by reversing (complementing) each bit of value.
2's Complement form
•Formed by adding 1 to 1's complement of a number.
Examples:
•Express -9 in 2's complement form (8-bit)
•Add 9 and its 2's complement value, ignoring carries:
•What is 2's complement of –5?
•Calculate 9 - 5 using 2's complement form for –5, ignoring carries.
Intel's peculiar method of storing
multibyte numbers in the computer
Note: each hex digit is encoded in a nibble (4 bits).
e.g., 1011 1010 = BAh
Example of a multibyte number: 3A 4D is a 2 byte hex number
The memory of the computer is numbered byte 0, byte 1, byte 2, .
For this example, if the number is stored at bytes 20 and 21, it
is stored as follows: byte 20
4D
3A
byte 21
Intel's peculiar method of storing multibyte
numbers in the computer (Cont. 1)
The address of the 2 byte word (a word is 2 bytes, a byte is
8 bits) is given as the lowest address used, which is byte 20.
Note again that the bytes are stored in reverse order with the
least significant byte at the lowest memory address and the
most significant byte stored at the highest memory address.
Intel's peculiar method of storing multibyte
numbers in the computer (Cont. 2)
Example with a double word (= 2 words or 4 bytes).
E.g., 32 4A 3F 22h
This would be stored in memory as:
memory contents:
22
3F
byte # in memory:
50
51
4A
52
32
53
The address for the (entire) number stored in memory is 50,
which is the lowest memory address used.
Character Representation
ASCII
•American Standard Code for Information Interchange
•Standard encoding scheme used to represent characters
in binary format on personal computers
•7-bit encoding => 128 characters can be represented
•codes 0-31 and 127 are control characters
(nonprinting characters)
•control characters used on PC's are:
LF, CR, BS, Bell, HT, FF
Character Representation
(Cont.)
•ASCII characters codes are used for input
and output.
E.g., when the number 5, is
entered on the keyboard, the value read
into the processor is the ASCII character
code 35h. In order to do numeric processing,
the ASCII code must be converted to the true
numeric value by subtracting 30h, which gives
the value 5. Then, before the result can be
displayed the numeric value must be
reconverted to an ASCII code which can be
displayed
Character Representation
(Cont.)
•UNIDEC is a system (supported e.g by HTML) for
representing characters employing 2 bytes per
character.
• With 216 possible characters, it is possible
to include characters of languages other that
English, such as Chinese,Korean, Russian, etc.