2. 2.1 Introduction
A bit is the most basic unit of information in a
computer.
It is a state of “on” or “off” in a digital circuit.
Sometimes these states are “high” or “low”
voltage instead of “on” or “off..”
A byte is a group of eight bits.
A byte is the smallest possible addressable unit
of computer storage.
The term, “addressable,” means that a
particular byte can be retrieved according to
its location in memory.
2
3. 2.1 Introduction
A word is a contiguous group of bytes.
Words can be any number of bits or bytes.
Word sizes of 16, 32, or 64 bits are most
common.
In a word-addressable system, a word is the
smallest addressable unit of storage.
A group of four bits is called a nibble.
Bytes, therefore, consist of two nibbles: a “high-
order nibble,” and a “low-order” nibble.
3
4. 2.2 Positional Numbering Systems
Bytes store numbers using the position of each
bit to represent a power of 2.
The binary system is also called the base-2
system.
Our decimal system is the base-10 system. It
uses powers of 10 for each position in a
number.
Any integer quantity can be represented
exactly using any base (or radix).
4
5. 2.2 Positional Numbering Systems
The decimal number 947 in powers of 10 is:
The decimal number 5836.47 in powers of 10 is:
5
5 103
+ 8 102
+ 3 101
+ 6 10
0
+ 4 10-1
+ 7 10-2
9 102
+ 4 101
+ 7 100
6. 2.2 Positional Numbering Systems
The binary number 11001 in powers of 2 is:
When the radix of a number is something other
than 10, the base is denoted by a subscript.
Sometimes, the subscript 10 is added for
emphasis:
110012 = 2510
6
1 24
+ 1 23
+ 0 22
+ 0 21
+ 1 20
= 16 + 8 + 0 + 0 + 1 = 25
8. 2.3 Converting Between Bases
Converting 0.8125 to binary . . .
You are finished when the product is
zero, or until you have reached the
desired number of binary places.
Our result, reading from top to bottom
is:
0.812510 = 0.11012
This method also works with any base.
Just use the target radix as the
multiplier.
8
9. 2.3 Converting Between Bases
The binary numbering system is the most
important radix system for digital computers.
However, it is difficult to read long strings of binary
numbers -- and even a modestly-sized decimal
number becomes a very long binary number.
For example: 110101000110112 = 1359510
For compactness and ease of reading, binary
values are usually expressed using the
hexadecimal, or base-16, numbering system.
9
11. 2.3 Converting Between Bases
Using groups of four bits, the binary number
110101000110112 (= 1359510) in hexadecimal is:
Octal (base 8) values are derived from binary by
using groups of three bits (8 = 23
):
11
Octal was very useful when computers used six-bit
words.
If the number of bits is not a
multiple of 4, pad on the left
with zeros.
12. 2.4 Signed Integer Representation
The conversions we have so far presented have
involved only unsigned numbers.
To represent signed integers, computer systems
allocate the high-order bit to indicate the sign of a
number.
The high-order bit is the leftmost bit. It is also
called the most significant bit.
0 is used to indicate a positive number; 1
indicates a negative number.
The remaining bits contain the value of the number
(but this can be interpreted different ways)
12
13. 2.4 Signed Integer Representation
There are three ways in which signed binary
integers may be expressed:
Signed magnitude
One’s complement
Two’s complement
In an 8-bit word, signed magnitude
representation places the absolute value of
the number in the 7 bits to the right of the
sign bit.
13
14. 2.4 Signed Integer Representation
For example, in 8-bit signed magnitude
representation:
+3 is: 00000011
- 3 is: 10000011
Computers perform arithmetic operations on
signed magnitude numbers in much the same
way as humans carry out pencil and paper
arithmetic.
Humans often ignore the signs of the operands
while performing a calculation, applying the
appropriate sign after the calculation is
complete.
14
15. 2.4 Signed Integer Representation
Example:
Using signed magnitude
binary arithmetic, find
the sum of 75 and 46.
First, convert 75 and 46 to
binary, and arrange as a sum,
but separate the (positive)
sign bits from the magnitude
bits.
15
16. 2.4 Signed Integer Representation
Signed magnitude representation is easy for
people to understand, but it requires
complicated computer hardware.
Another disadvantage of signed magnitude is
that it allows two different representations for
zero: positive zero and negative zero.
For these reasons (among others) computers
systems employ complement systems for
numeric value representation.
16
17. Complementary Number Representation
Two types of complements for base R number system:
- R's complement and (R-1)'s complement
The (R-1)'s Complement
Subtract each digit of a number from (R-1)
Example
- 9's complement of 83510 is 16410
- 1's complement of 10102 is 01012(bit by bit complement operation)
The R's Complement
Add 1 to the low-order digit of its (R-1)'s complement
Example
- 10's complement of 83510 is 16410 + 1 = 16510
- 2's complement of 10102 is 01012 + 1 = 01102
Complements
18. Complementary Number Representation
In the r’s complement representation, r represents the
radix. Depending upon the radix or base the
complimentary representation is as follows:
Binary number system: 1’s complement and 2’s
complement.
Decimal number system: 9’s complement and 10’s
complement.
Octal number system: 7’s complement and 8’s
complement.
Hexadecimal number system: F’s complement and
16’s complement.
To determine the (r-1)’s we need to subtract the given number
from the maximum number of the given base. Similarly to
determine the r’s complement first determine the (r-1)’s
complement of the given number and finally add 1 to the
LSB(Least Significant Bit).
19. Determine the 1’s and 2’s complement of
the binary number 1011
First find 1’s complement of a number by flipping
numbers e.g 1’s complement of 1011 is 0100
Second, Determine the 2’s complement of number
by adding 1 to the LSB of the 1’s complement result.
0100
+ 0001
———–
0101→ This is the 2’s complement of 1011
20. Subtraction of Binary numbers
using 1’s and 2’s complement
Consider the two binary numbers X = 1010100 and Y = 1000011, we perform
the subtraction X – Y and Y - X using 2's complemenfs:
21. Determine the 9’s and 10’s
complement of the decimal
number
a) first find 9’s complement of decimal number by
subtracting each digit from 9
example for 6298, 9’s complement is
9999
– 6298
———–
3701 → This is the 9’s complement of 6298
b) To find Find 10’s Complement of a decimal number
add 1 to its 9’s Complement
example 10s complement of 6298 is
=9s complement of 6298+1=3701+1
=3702
23. 2.5 Floating-Point Representation
The signed magnitude, one’s complement,
and two’s complement representation that we
have just presented deal with signed integer
values only.
Without modification, these formats are not
useful in scientific or business applications
that deal with real number values.
Floating-point representation solves this
problem.
23
24. 2.5 Floating-Point Representation
If we are clever programmers, we can perform
floating-point calculations using any integer format.
This is called floating-point emulation, because
floating point values aren’t stored as such; we just
create programs that make it seem as if floating-
point values are being used.
Most of today’s computers are equipped with
specialized hardware that performs floating-point
arithmetic with no special programming required.
24
25. 2.5 Floating-Point Representation
Floating-point numbers allow an arbitrary
number of decimal places to the right of the
decimal point.
For example: 0.5 0.25 = 0.125
They are often expressed in scientific notation.
For example:
0.125 = 1.25 10-1
5,000,000 = 5.0 106
25
26. 2.5 Floating-Point Representation
Computers use a form of scientific notation for
floating-point representation
Numbers written in scientific notation have three
components:
26
27. 2.5 Floating-Point Representation
Computer representation of a floating-point
number consists of three fixed-size fields:
This is the standard arrangement of these fields.
27
Note: Although “significand” and “mantissa” do not technically mean the same
thing, many people use these terms interchangeably. We use the term “significand”
to refer to the fractional part of a floating point number.
28. 2.5 Floating-Point Representation
We introduce a hypothetical “Simple Model” to
explain the concepts
In this model:
A floating-point number is 14 bits in length
The exponent field is 5 bits
The significand field is 8 bits
28
29. 2.5 Floating-Point Representation
Example:
Express 3210 in the simplified 14-bit floating-point
model.
We know that 32 is 25
. So in (binary) scientific
notation 32 = 1.0 x 25
= 0.1 x 26
.
In a moment, we’ll explain why we prefer the
second notation versus the first.
Using this information, we put 110 (= 610) in the
exponent field and 1 in the significand as shown.
29
30. 2.5 Floating-Point Representation
The IEEE has established a standard for
floating-point numbers
The IEEE-754 single precision floating point
standard used to represent 32 bit data in which
1 bit for sign, 8-bit exponent (with a bias of 127)
and a 23-bit significand.
The IEEE-754 double precision standard used
to represent 64 bit data in which 1 bit for sign,
11-bit exponent (with a bias of 1023) and a 52-
bit significand.
30
31. 2.5 Floating-Point Representation
In both the IEEE single-precision and double-
precision floating-point standard, the significant has
an implied 1 to the LEFT of the radix point.
The format for a significand using the IEEE format
is: 1.xxx…
For example, 4.5 = .1001 x 23
in IEEE format is 4.5 =
1.001 x 22
. The 1 is implied, which means is does
not need to be listed in the significand (the
significand would include only 001).
31
32. 2.5 Floating-Point Representation
Example: Express -3.75 as a floating point number using
IEEE single precision.
First, let’s normalize according to IEEE rules:
3.75 = -11.112 = -1.111 x 21
The bias is 127, so we add 127 + 1 = 128 (this is our exponent)
The first 1 in the significand is implied, so we have:
Since we have an implied 1 in the significand, this equates to
-(1).1112 x 2 (128 – 127)
= -1.1112 x 21
= -11.112 = -3.75.
32
(implied)
33. 2.6 Character
Codes
Calculations aren’t useful until their results can
be displayed in a manner that is meaningful to
people.
We also need to store the results of calculations,
and provide a means for data input.
Thus, human-understandable characters must be
converted to computer-understandable bit
patterns using some sort of character encoding
scheme.
33
34. 2.6 Character
Codes
Binary-coded decimal
(BCD) was one of these
early codes. It was used
by IBM mainframes in
the 1950s and 1960s.
34
Decimal
Digit
BCD
8 4 2 1
0 0 0 0 0
1 0 0 0 1
2 0 0 1 0
3 0 0 1 1
4 0 1 0 0
5 0 1 0 1
6 0 1 1 0
7 0 1 1 1
8 1 0 0 0
9 1 0 0 1
35. 2.6 Character
Codes
In 1964, BCD was extended to an 8-bit code,
Extended Binary-Coded Decimal Interchange
Code (EBCDIC). See textbook Page 89
EBCDIC was one of the first widely-used
computer codes that supported upper and
lowercase alphabetic characters, in addition to
special characters, such as punctuation and
control characters.
EBCDIC and BCD are still in use by IBM
mainframes today.
35
36. 2.6 Character
Codes
Other computer manufacturers chose the 7-bit
ASCII (American Standard Code for Information
Interchange) as a replacement for 6-bit codes.
see textbook page 90.
While BCD and EBCDIC were based upon
punched card codes, ASCII was based upon
telecommunications (Telex) codes.
Until recently, ASCII was the dominant
character code outside the IBM mainframe
world.
36
37. 2.6 Character
Codes
Many of today’s systems embrace Unicode, a 16-
bit system that can encode the characters of
every language in the world.
The Java programming language, and some
operating systems now use Unicode as their
default character code.
The Unicode codespace is divided into six parts.
The first part is for Western alphabet codes,
including English, Greek, and Russian.
37
38. 2.6 Character
Codes
The Unicode codes-
pace allocation is
shown at the right.
The lowest-numbered
Unicode characters
comprise the ASCII
code.
The highest provide for
user-defined codes.
38
