Lecture 4: Number Systems (Chapter 3)

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Lecture 4 Number Systems (Chapter 3)
(1) Data Types Section 3-1 (2)
Complements Section 3-2 (3) Fixed Point
Representations Section 3-3 (4) Floating Point
Representations Section 3-4 (5) Other Binary
Codes Section 3-5 (6) Error Detection
Codes Section 3-6
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Data Types
Information that a Computer is dealing
with Data Numeric Data Numbers (Integer,
real) Non-numeric Data Letters,
Symbols Relationship between data
elements Data Structures Linear Lists,
Trees, Rings, etc Program (Instructions)
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Data Types Numeric Data Representation
Nonpositional number system Roman number
system Positional number system Each digit
position has a value called a weight associated
with it Examples Decimal, Octal,
Hexadecimal, Binary Base (or radix) R
number Uses R distinct symbols for each
digit Example A R a n-1 a n-2 ... a 1 a 0 .a
-1 a -m V(A R) SUM (a k R k) for k -m to
n-1 R 10 Decimal number system R
2 Binary R 8 Octal R 16 Hexadecimal
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Data Types Numeric Data Representation
Why a Positional Number System for Digital
Computers??? Major Consideration is the COST and
TIME Cost of building hardware Arithmetic and
Logic Unit, CPU,Communications Time to
processing Arithmetic - Addition of Numbers -
Table for Addition Non-positional Number
System Table for addition is infinite --gt
Impossible to build, very expensive even if it
can be built Positional Number
System Table for Addition is finite --gt
Physically realizable, but cost wise the smaller
the table size, the less expensive --gt
Binary is favorable to Decimal
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Positive (Unsigned) Binary Numbers
Unsigned binary numbers are typically used to
represent computer addresses or other values
that are guaranteed not to be negative. An n-bit
unsigned binary integer A an-1 an-2... a1
a0 has a value of For example, 1011
1 x 23 0 x 22 1 x 21 1 x 20 8
2 1 11 An n-bit unsigned binary integer has a
range from 0 to 2n - 1.
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Octal and Hexadecimal Numbers
Octal, base-8, numbers were used in the early
days of computing to represent binary
numbers Octal numbers are made by grouping
binary numbers together three bits at a
time Hexadecimal, base-16, numbers are the
representation of choice today Hex numbers
are made by grouping binary numbers together
four bits at a time For example Octal 7
2 5 1 7 5 2 2
. Binary 1 1 1 0 1 0 1 0 1 0 0 1 1 1 1 1 0
1 0 1 0 0 1 0 Hex E A 9
F 5 2
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Negative (Signed) Binary Numbers
Positional representation using n bits X X n X
n-1 X n-2 X 1 X 0 . X -1 X -2 ... X
-m Sign-magnitude format Left most bit position
(X n) is the sign bit -- only bit that is
complemented 0 for positive number 1 for
negative number Remaining n-1 bits represent
the magnitude Min - (2 n - 2 -m) 1111 1111
. 1111 1111 Max (2 n - 2 -m) 0111 1111 .
1111 1111 Zero - 0 1000 0000 . 0000
0000 Zero 0 0000 0000 . 0000 0000
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Complements of Numbers
Two types of complements for base R number
system Rs complement (R-1)s complement The
(R-1)s Complement Subtract each digit of a
number from (R-1) Examples 9s complement of
835 10 is 164 10 1s complement of 1010 2 is
0101 2 (bit by bit complement operation) The
Rs Complement Add 1 to the low-order digit of
its (R-1)s complement Examples 10s
complement of 835 10 is 164 10 1 165 10 2s
complement of 1010 2 is 0101 2 1 0110 2
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Negative (Signed) Binary Numbers
Ones complement format Negative numbers are
represented by a bit-by-bit complementation
of the (positive) magnitude (the process of
negation) Sign bit interpreted as in
sign-magnitude format Examples (8-bit
words) 42 0 00101010 -42 1
11010101 Min - (2 n - 2 -m) 1111 1111 .
1111 1111 Max (2 n - 2 -m) 0111 1111 . 1111
1111 Zero - 0 1111 1111 . 1111
1111 Zero 0 0000 0000 . 0000 0000
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Negative (Signed) Binary Numbers
Twos complement format Negative numbers, -X, are
represented by the pseudo- positive number 2n
- X An n-bit unsigned binary integer A an-1
an-2... a1 a0 has a value of For
example 1011 -1 x 23 0 x 22 1 x 21 1 x
20 -8 2 1 -5 With 2n
digits 2 n-1 -1 positive numbers 2 n -1
negative numbers Given the representation for
X, the representation for -X is found by
taking the 1s complement of X and adding 1
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Negative (Signed) Binary Numbers
Twos complement format Most significant bit is
the sign bit. Number representation is not
symmetric. Only one representation for
zero. Easy to negate, add, and subtract
numbers. A little bit trickier for multiply and
divide. Min - (2 n) 1000 0000 . 0000
0000 Max (2 n - 2 -m) 0111 1111 . 1111
1111 Zero 0000 0000 . 0000 0000
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Signed 2s Complement Addition
Add the two numbers, including their sign bit,
and discard any carry out of left-most(sign)
bit Examples 6 0 0110 -6 1 1010
9 0 1001 9 0 1001 15 0 1111 3 0
0011 6 0 0110 -9 1 0111 -9 1 0111
-9 1 0111 -3 1 1101 -18 (1) 0 1110
9 0 1001 9 0 1001 18 1 0010
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Detecting 2s Complement Overflow
When adding two's complement numbers, overflow
will only occur if the numbers being added have
the same sign the sign of the result is
different If we perform the addition an-1
an-2 ... a1 a0 bn-1bn-2 b1
b0 ----------------------------------
sn-1sn-2 s1 s0 Overflow can be detected
as where cn-1and cn are the carry in and carry
out of the most significant bit.
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Signed 2s Complement Subtraction
To subtract two's complement numbers we first
negate the second number and then add the
corresponding bits of both numbers.
Examples 3 0011 -3 1101 -3 1101
3 0011 - 2 0010 - -2 1110 - 2 0010 -
-2 1110 become 3 0011 -3 1101
-3 1101 3 0011 -2 1110 2 0010
-2 1110 2 0010 1 0001 -1 1111
-5 1011 5 0101
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Sign-Extension / Zero-Extension
Sign-extension is used for signed immediates and
signed values from memory To sign-extend an n bit
number to nm bits, copy the sign-bit m times.
For example, with n 4 and m 4,
1011 -4 0101 5 11111011 -4
00000101 5 Zero-extension is used
for logical operations and unsigned values from
memory To zero-extend an n bit number to nm
bits, copy zero m times. For example, with n 4
and m 4, 1011 11 0101
5 00001011 11 00000101 5
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Floating Point Number Representation
The location of the fractional point is not fixed
to a certain location --gt The range of the
representable numbers is wide --gt high
precision F EM m n e k e k-1 ... e 0 m n-1 m
n-2 ... m 0 . m -1 ... m -m sign exponent
mantissa Mantissa Signed fixed point number,
either an integer or a fractional
number Exponent Designates the position of the
radix point
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Floating Point Number Representation
Decimal Value V M R E Where M
Mantissa E Exponent R Radix
(10) Example (decimal) 1234.5678 Exponent
Mantissa Sign Value Sign Value 0 4 0 0.12345
678 gt 0.12345678 x 10 4
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Floating Point Number Representation
Example (binary) 1001.11 ( 9.75) Make a
fractional number, counting the number of
shifts .100111 gt 4 shifts Exponent Man
tissa Sign Value Sign Value 0 100 0 1001111
Or for a 16-bit number with a sign, 5-bit
exponent, 10-bit mantissa 0 00100 1001111000
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Other Representations- Gray Codes
Characterized by having their representations of
the binary integers different in only one digit
between consecutive integers Useful in
analog-digital conversion. Decimal Gray
Binary Decimal Gray Binary 0 0 0 0 0 0 0 0
0 8 1 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 9 1
1 0 1 1 0 0 1 2 0 0 1 1 0 0 1 0 10 1 1 1 1 1
0 1 0 3 0 0 1 0 0 0 1 1 11 1 1 1 0 1 0 1
1 4 0 1 1 0 0 1 0 0 12 1 0 1 0 1 1 0 0 5 0
1 1 1 0 1 0 1 13 1 0 1 1 1 1 0 1 6 0 1 0 1 0
1 1 0 14 1 0 0 1 1 1 1 0 7 0 1 0 0 0 1 1 1
15 1 0 0 0 1 1 1 1
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Other Representations- ASCII Characters
4MSBs 3LSBs 0 1 1 3 4 5 6 7 0 (hex) NUL DLE SP
0 _at_ P p 1 SOH DC1 ! 1 A Q a q 2 STX
DC2 " 2 B R b r 3 ETX DC3 3 C
S c s 4 EOT DC4 4 D T d t 5 ENQ NAK
5 E U e u 6 ACK SYN 6 F V f v 7 BEL ETB
7 G W g w 8 BS CAN ( 8 H X h x 9 HT EM )
9 I Y i y A LF SUB J Z j z B VT ESC
K k C FF FS , lt L \ l D CR GS -
M m E SO RS . gt N n F SI US / ?
O _ o DEL
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Error Detecting Codes- Parity
Parity System Simplest method for error
detection One parity bit attached to the
information Even Parity and Odd Parity Even
Parity One bit is attached to the information so
that the total number of 1 bits is an even
number 1011001 0 1010010 1 gt B even B n-1
() B n-2 () B 0 Odd Parity One bit is
attached to the information so that the total
number of 1 bits is an odd number 1011001
1 1010010 0 gt B odd B n-1 () B n-2 () B
0 () 1
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Error Detecting Codes- Parity
Even Parity Generator Circuit
B 0
B 1
B 2
B 3
B 4
B even
B 5
B 6
Even Parity Checker Circuit
B 0
B 1
B 2
B 3
B 4
B 5
ERROR
B 6
B even