TM112 Meeting2-Binary data representation.pptx

Meeting #2
Block 1 (Part 2 )
Binary data representation and computation
TM112: Introduction to Computing and
InformationTechnology
OU Materials, PPT prepared by Dr. Ahmad Mikati
1
14 October 2023 TM112-AOU

Contents
• Introduction
• 2.1 Representing integers and text in binary
• 2.2 Decimal numbers and some limitations of binary representations
• 2.3 Representing logic operations and logic circuits
• Summary
2

Introduction
• This part will provide you with a basic understanding of
how computers represent and process data.
• After studying it, you will be able to compare some of the
different binary representations of data and reason about
their efficiency.
• Also, you will appreciate that sometimes a particular
representation can lead to errors (with potentially far-
reaching consequences).
3

Binary logic
4
• Binary logic underpins most forms of digital technology. It was developed in
the mid-nineteenth century by George Boole.
• At the time, Boole was interested in exploring the fundamental nature of truth,
so he developed a system involving symbols – a symbolic logic – to help him
reason about truth and falsity.This system is often referred to as Boolean logic
or binary logic, and it focuses on the manipulation of 1s and 0s (1 to represent
TRUE and 0 to represent FALSE).
• When trying to understand the nature of the digital world, it is useful to know
something about the binary numbers, because binary numbers are the form in
which information is held in a computer.
• For example, 100011 can be considered as a binary number. When stored in a
computer, each digit of a binary number is referred to as a bit, which is a
contraction of the word’s binary digit.

1.1 Representing integers and text in binary
• The printed symbols shown in Figure 2.1 provide
convenient representations of short and long flashes of
light or short and long bleeps of sound, which is how
Morse code is normally transmitted.
5

1.1 Representing integers and text in
binary
• In any representation it is important that the symbols can
be distinguished from each other.
• Changes in electrical voltages or friction in mechanical
systems can cause random fluctuations, called noise,
which may distort how the symbol is perceived.
• In a binary system there are only two symbols, so it is
generally easier to make them different enough to be
distinguishable – for example, Morse code specifies that a
dash should be three times as long as a dot.
6

Binary representation systems in
computers
• Some very early computers, such as the ENIAC
(Electronic Numerical Integrator and Computer), tried to
represent data using our usual base-10 system. So, 0 volt
was used to represent the digit 0, 1 volt to represent the
digit 1, and so on, all the way up to 9 volts to represent
the digit 9.
• Hence, a lot of circuitry was needed just in order to
distinguish between the different voltages, which took up
a lot of space and generated a lot of heat.
• The advantage of representing data in binary is that only
two ranges of voltage need to be detected.
7

Number systems
We have different number systems:
• Decimal (Base 10)
• It uses 10 symbols (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9).
• Binary (Base 2)
• It uses 2 symbols (0 and1).
• Octal (Base 8)
• It uses 8 symbols (0, 1, 2, 3, 4, 5, 6 and 7).
• Hexadecimal (Base 16)
• It uses 16 symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F).
8

• Decimal (Base 10) System
• The system that is used worldwide.
• It uses ten digits (0 to 9) and each column counts groups
ten times bigger than those counted in the column to its
right.
• Examples:
• 37 = 7 + 3*101
• 345 = 5 + 4*101 + 3*102
• 4621 = 1 + 2*101 + 6*102 + 4*103
9
Number systems

• Octal (Base 8) System
• The system that is mostly used by computer scientist.
• It uses eight digits (0 to 7) and each column counts groups
eight times bigger than those counted in the column to its
right.
• Examples:
• 468 = 6 + 4*81 = 3810
• 1258 = 5 + 2*81 + 1*82 = 8510
10
Number systems

• Hexadecimal (Base 16) System
• The system that is mostly used by computer scientist.
• It uses eight digits (0 to F) and each column counts groups
sixteen times bigger than those counted in the column to
its right. (A is equivalent to 10 in decimal,….., F is
equivalent to 16 in decimal)
• Examples:
• 4616 = 6 + 4*161 = 7010
• A516 = 5 + 10*161 = 16510
11
Number systems

• Binary (Base 2) System
• The system that is used by the computer.
• It uses two digits (0 and 1) and each column counts
groups two times bigger than those counted in the
column to its right.
• Examples:
• 1102 = 0 + 1*21 + 1*22 = 610
• 11012 = 1 + 0*21 + 1*22 + 1*23 = 1310
12
Number systems

• A bit is short for binary digit and refers to a 1 or a 0 stored in the
computer, while a byte is a group of eight bits that can be used to
represent numbers between 0 and 255.
• A byte looks like this:
• The largest number we can store in a byte is 11111111 = (255)10.
• The smallest number we can store in a byte is 00000000 = (0)10.
• The word size for a computer is the number of bits that the CPU of a
particular computer can handle at one time. 13
Number systems

Binary to Decimal Conversion
14
To convert from binary to decimal a method known as Positional notation
can be used.The below figures illustrate the method with an example of
how to convert 100110112 to decimal.

Octal to Decimal Conversion
15
• To convert from octal to decimal the following formula can be used:
Decimal Form = Ʃ(ai x 8i)
• Where, 'a' is the individual digit being converted, while 'i' is the number of
the digit counting from the right-most digit in the number, starting with 0.
• Here are two examples:

Converting from Decimal to any Base
Note that the highest-order bit is the leftmost bit, and it is called the most significant bit (MSB).
On the other hand, the lowest-order bit is the rightmost bit, and it is called the least significant bit (LSB).
14 October 2023 TM112-AOU 16
Given the integer part of a decimal number, you can find the equivalent
value in any base, n, as follows:
1. Divide the number by n.
2. You will get Quotient and Remainder.The remainder will always be
less than the base n. Keep this remainder aside.
3. Use the quotient to repeat the above steps, until you get a
quotient equal to 0.
4. Now, group the remainders in order, such that the first remainder
will be the least significant bit, and the last one will be the most
significant bit.

Decimal to Binary Conversion
17

Decimal to Binary Conversion
18
• One of the easy methods used to convert from decimal to binary
is comparison with descending powers of two and subtraction.
• To convert a number from binary to decimal notation, we put the
number in the table and add up the values of each place value.
• So, to convert the binary number 1001 into decimal notation, we
can use the following table
Decimal number= 8+1=9

Decimal to Octal
19
The conversion of a decimal number to its octal equivalent is done by
the repeated division method.You simply divide the base 10 number by 8
and extract the remainders.
Here are two examples to illustrate this method:

Representing integers in binary
Unsigned integers
• In computer-speak, an unsigned integer is an integer that is greater
than or equal to zero.
• An unsigned integer is sufficient for any purpose where a value does
not become negative – for example, a counter counting upwards
from zero.
• The number of unsigned decimal values that we can represent in
binary depends on the number of bits we have available. If there are 3
bits available, we can represent 23 = 8 values. If we want to include 0,
this means that we can encode all of the unsigned integers from 0 to
7 in three bits.
20

Representing integers in binary
Unsigned integers
21
• The largest decimal value that can be represented in 3 bits is 7.
This is one less than 23 (or, in mathematical notation, 23 – 1)
because one of the 8 available codes has been used up to
represent 0.
• This scheme can be extended to systems with more bits.
• In general, if we have n bits, we can represent 2n unsigned
integers, and the largest integer that can be represented is 2n – 1.

Adding unsigned integers in binary
notation
• In decimal notation, adding two positive integers is very
straightforward. The two values are added, and the sign of the result
is automatically positive.
• Similarly, if the values are represented as two unsigned integers in
binary notation, their binary values can just be added.
• So, to add two binary numbers by hand, we use the same method as
adding two decimal numbers. The numbers are written one under the
other, so that each column has the same place value. Then the digits
in each column are added, column by column from the right, carrying
digits as necessary.
22

Adding unsigned integers in binary
notation
23
• To illustrate, here is the working to add the two unsigned
integers 110 and 101, where the ‘carry’ digit is shown in blue
below:

Sign-magnitude representation
• So for our 3 bits, there would be 23 = 8 possible binary codes, which
could be used to encode positive and negative integers as shown in
Table 2.7.
24
Signed magnitude is the most intuitive method for representing the
unsigned numbers.
The MSB (Most Significant Bit) of a binary number is kept as the
“sign” of the number
MSB = 1: negative number
MSB = 0: positive number
The remaining bits represent the magnitude (or absolute value) of
the numeric value.

Sign-magnitude representation
25
In an N bit word signed magnitude system
 1 bit is used for the sign of the number (MSB).
 N-1 bits are used for the magnitude of the number.
 The largest integer is 2N-1- 1
 The smallest integer is -(2N-1- 1)
Example: In an 8 bit word signed magnitude system give the decimal
representation of the following numbers: 00000001, 10000001
Answer:
•00000001:
-The MSB is 0:The number is positive
-The remaining 7 bits are: 00000012 = 110
-The decimal number is +1
•10000001:
-The MSB is 1:The number is negative
-The remaining 7 bits are: 00000012 = 110
-The decimal number is -1

Representing text in binary
• Most modern systems for encoding text derive in part from ASCII
(American Standard Code for Information Interchange, pronounced
‘askee’), which was developed in 1963.
• In the original ASCII system, upper-case and lower-case letters,
numbers, punctuation and other symbols and control codes (such as
a carriage return, backspace and tab) were encoded in 7 bits. As
computers based on multiples of 8 bits (or a byte) became more
common, the encoding system became an 8-bit system, and so could
be expanded to include more symbols.
• When binary numbers were assigned to each character in the original
ASCII system, careful thought was given to choosing sequences of
values for the characters of the alphabet and numerals that would
make it easy for a computer processor to perform common
operations on them. (These encodings were preserved in the 8-bit
system by simply padding out the leftmost bit with a 0.)
26

1.1.5 Representing text in binary
• Since 2007, the standard encoding system for characters has been
Unicode Transformation Format-8 (UTF-8) which uses a variable
number of bytes (up to 6) to encode characters in use across the
world. However, in order to maintain backward compatibility, the
original 127 ASCII codes are preserved in UTF-8.
27

Floating-point numbers and scientific
notation
• Consider the decimal number 2343.56.We could also write this as
23.4356 × 102 OR 0.234356 × 104 or 234356.0 × 10–2.
• The decimal point can ‘float’ to any position as long as the power of
10 is appropriate.
• Scientific notation is a special case of floating-point notation where
there is a single non-zero digit between 1 and 9 (inclusive) to the left
of the decimal point.
• To express a number greater than 10 in scientific notation, the first
stage is to divide it repeatedly by 10 until it is reduced to a number
that is less than 10.
28

notation
29
• So, the number 2343.56 can be represented in scientific notation
as 2.34356 × 103. Note that the exponent, 3, indicates that the
decimal point should be moved three places to the right to get
back the original decimal notation.
• Expressing a small number – between 0 and 1 – in scientific notation
involves a process very similar to that for large numbers.
• Take –0.000654 as an example. First you move the decimal point to the
right until it sits after the first non-zero digit: - 00006.54
• Then multiply this number by 10 raised to the power of minus the
number of places the decimal point has been moved:-6.54 × 10−4

notation
• The number –0.000654 in decimal notation can be written as -
6.54 × 10–4 in scientific notation. Here, the negative exponent (–4)
indicates that the decimal point should be moved 4 places to the left
to get back to the original decimal notation.
• Notice that scientific notation has three distinct parts, shown in
Figure 1.10:
• a sign
• an exponent (the power of 10)
• a mantissa (the decimal number part).
30

Representing logic operations and
logic circuits
• In the previous two sections, we have seen that we can use binary
encodings to represent numerical and textual data.We will now see
that operations, including arithmetical operations such as addition,
and comparison operations such as less than and equals, can be
encoded as one or more logic operations.These logic operations act
on the binary representations of the data.
• To move from the human to the computer view, the integers have to
be encoded as binary representations and the addition operator has
to be encoded as a sequence of logical operations that have what is
called the truth table.
31

logic circuits
• A truth table for a logic operation lists all the possible combinations
of input values, and for each possibility gives the output value for
that operation.As the operations we will consider will always be
applied to binary encodings, each input value must be either a 1 or a
0 and the result of the operation must also always be a 1 or a 0.
• We will start by looking at the truth tables for three of the
fundamental logical operations defined by Boole.We will then see
how these basic operations can be used as building blocks for the
logic circuits that perform more complex operations. By the end of
this subsection, you will see how these simple operations can be used
to build a logic circuit to add two binary numbers.
32

logic circuits
The NOT operation
• One of the most fundamental operations we might want to perform is
to ‘flip’ a single bit – let’s call the bit a. So, if 𝒙is 1, we want the result
to be 0, and if 𝒙is 0, we want the result to be 1.This operation is called
NOT 𝒙and is expressed as: 𝒙 or 𝒙’.
33
The behavior of NOT operator is characterized by the truth table
shown below:
The NOT truth table
To physically perform logic operations on binary
data in a computer, we need to use electrical
components. The components that represent the
most fundamental operations are called logic
gates, which can be combined in a logic circuit
in order to create more complex operations.
The NOT logic gate

1.3 Representing logic operations and
logic circuits
• The AND operation
• Most logical operations involve two input values.A truth table for
two binary inputs, x and y, has more rows because there are four
possible permutations (or ways of combining) the two input values,
as shown inTable 2.19.
AND logic gate
AND truth table
34

logic circuits
• The OR operation
• truth table for the logic operationOR (which Boole originally
designated by the symbol +) is shown below
OR logic gate
OR truth table
35

Building logic circuits
• Suppose we want to build a logic circuit with two inputs,A and B,
that tests if B is greater than A.
• The first step is to create a truth table showing the desired
outcomes: if B is greater than A, the result is 1 (True), otherwise the
result is 0 (False).
36

• To translate this into a logic expression – that is a combination of our
logic operations (NOT,AND and OR) – we follow this algorithm.
• Identify the row where the outcome (B > A) is 1.
• If inputA is 1, write A; otherwise write NOT A in the logic expression for
the selected row.
• If input B is 1, write B; otherwise write NOT B in the logic expression for
the selected row.
• Join these with anAND, and the final equation will be the sum of all the
deduced logic expressions.
37
Final equation: A’. B

• Here, the resulting logic expression NOTA AND B tells us that the
logic circuit that is equivalent to this truth table for each combination
of inputs can be constructed from two logic gates
• a NOT gate withA as an input, which gives an output of NOTA
• an AND gate that takes NOT A and B as inputs, which gives the
required result NOTA AND B as an output.
38

39
In the above circuit:
The output of gate 1 is: 𝒙 + 𝒚
The output of gate 2 is: 𝒚 + 𝒛
The output of gate 3 is: 𝒙 + 𝒚 ( 𝒚 + 𝒛)

What is inside a logic gate?
• How Logic gates are actually constructed , and what exactly is inside a
logic gate?
• A Logic gate is itself made up of a combination of more fundamental
components that act as on/off switches.
• In early computers, such devices were generally based on various
designs of vacuum tube (collectively called valves).
• In modern computers, they are based on transistors, which are
formed of layers of semiconducting material such as silicon.
40
A ‘pluggable’ unit made of valves from
an IBM computer of the mid-1950s
A chip containing six inverters

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