# 1. Introduction

• Computers are classified according to functionality, physical size and purpose.
• Functionality, Computers could be analog, digital or hybrid. Digital computers process data that is in discrete form whereas analog computers process data that is continuous in nature. Hybrid computers on the other hand can process data that is both discrete and continuous.
• In digital computers, the user input is first converted and transmitted as electrical pulses that can be represented by two unique states ON and OFF. The ON state may be represented by a “1” and the off state by a “0”.The sequence of ON’S and OFF’S forms the electrical signals that the computer can understand.
• A digital signal rises suddenly to a peak voltage of +1 for some time then suddenly drops -1 level on the other hand an analog signal rises to +1 and then drops to -1 in a continuous version.

• Although the two graphs look different in their appearance, notice that they repeat themselves at equal time intervals. Electrical signals or waveforms of this nature are said to be periodic.Generally,a periodic wave representing a signal can be described using the following parameters
• Amplitude(A)
• Frequency(f)
• periodic time(T)
• Amplitude (A): this is the maximum displacement that the waveform of an electric signal can attain.
• Frequency (f): is the number of cycles made by a signal in one second. It is measured in hertz.1hert is equivalent to 1 cycle/second.
• Periodic time (T): the time taken by a signal to complete one cycle is called periodic time. Periodic time is given by the formula T=1/f, where f is the frequency of the wave.
• When a digital signal is to be sent over analog telephone lines e.g. e-mail, it has to be converted to analog signal. This is done by connecting a device called a modem to the digital computer. This process of converting a digital signal to an analog signal is known as modulation. On the receiving end, the incoming analog signal is converted back to digital form in a process known as demodulation.

# 2. concepts of data representation in digital computers

• Data and instructions cannot be entered and processed directly into computers using human language. Any type of data be it numbers, letters, special symbols, sound or pictures must first be converted into machine-readable form i.e. binary form. Due to this reason, it is important to understand how a computer together with its peripheral devices handles data in its electronic circuits, on magnetic media and in optical devices.

Data representation in digital circuits

• Electronic components, such as microprocessor, are made up of millions of electronic circuits. The availability of high voltage(on) in these circuits is interpreted as ‘1’ while a low voltage (off) is interpreted as ‘0’.This concept can be compared to switching on and off an electric circuit. When the switch is closed the high voltage in the circuit causes the bulb to light (‘1’ state).on the other hand when the switch is open, the bulb goes off (‘0’ state). This forms a basis for describing data representation in digital computers using the binary number system.

Data representation on magnetic media

• The laser beam reflected from the land is interpreted, as 1.The laser entering the pot is not reflected. This is interpreted as 0.The reflected pattern of light from the rotating disk falls on a receiving photoelectric detector that transforms the patterns into digital form.The presence of a magnetic field in one direction on magnetic media is interpreted as 1; while the field in the opposite direction is interpreted as “0”.Magnetic technology is mostly used on storage devices that are coated with special magnetic materials such as iron oxide. Data is written on the media by arranging the magnetic dipoles of some iron oxide particles to face in the same direction and some others in the opposite direction

Data representation on optical media

In optical devices, the presence of light is interpreted as ‘1’ while its absence is interpreted as ‘0’.Optical devices use this technology to read or store data. Take example of a CD-ROM, if the shiny surface is placed under a powerful microscope, the surface is observed to have very tiny holes called pits. The areas that do not have pits are called land.

# Reason for use of binary system in computers

• It has proved difficult to develop devices that can understand natural language directly due to the complexity of natural languages. However, it is easier to construct electric circuits based on the binary or ON and OFF logic. All forms of data can be represented in binary system format. Other reasons for the use of binary are that digital devices are more reliable, small and use less energy as compared to analog devices.

Bits, bytes, nibble and word

• The terms bits, bytes, nibble and word are used widely in reference to computer memory and data size.
• Bits: can be defined as either a binary, which can be 0, or 1.It is the basic unit of data or information in digital computers.
• Byte: a group of bits (8 bits) used to represent a character. A byte is considered as the basic unit of measuring memory size in computer.
• A nibble: is half a byte, which is usually a grouping of 4 bytes.
• Word: two or more bits make a word. The term word length is used as the measure of the number of bits in each word. For example, a word can have a length of 16 bits, 32 bits, 64 bits etc.

# Types of data representation

• Computers not only process numbers, letters and special symbols but also complex types of data such as sound and pictures. However, these complex types of data take a lot of memory and processor time when coded in binary form.
• This limitation necessitates the need to develop better ways of handling long streams of binary digits.
• Higher number systems are used in computing to reduce these streams of binary digits into manageable form. This helps to improve the processing speed and optimize memory usage.

Number systems and their representation

• A number system is a set of symbols used to represent values derived from a common base or radix.
• As far as computers are concerned, number systems can be classified into two major categories:
• decimal number system
• binary number system
• octal number system

Decimal number system

• The term decimal is derived from a Latin prefix deci, which means ten. Decimal number system has ten digits ranging from 0-9. Because this system has ten digits; it is also called a base ten number system or denary number system.
• A decimal number should always be written with a subscript 10 e.g. X10
• But since this is the most widely used number system in the world, the subscript is usually understood and ignored in written work. However ,when many number systems are considered together, the subscript must always be put so as to differentiate the number systems.
• The magnitude of a number can be considered using these parameters.
• Absolute value
• Place value or positional value
• Base value
• The absolute value is the magnitude of a digit in a number. for example the digit 5 in 7458 has an absolute value of 5 according to its value in the number line.
• The place value of a digit in a number refers to the position of the digit in that number i.e. whether; tens, hundreds, thousands etc.
• The total value of a number is the sum of the place value of each digit making the number.
• The base value of a number also k known as the radix, depends on the type of the number systems that is being used .The value of any number depends on the radix. for example the number 10010 is not equivalent to 1002.

# Binary number system

It uses two digits namely, 1 and 0 to represent numbers. unlike in decimal numbers where the place value goes up in factors of ten, in binary system, the place values increase by the factor of 2.binary numbers are written as X2.consider a binary number such as 10112.The right most digit has a place value of 1×20 while the left most has a place value of 1×23.

Octal number system

Consists of eight digits ranging from 0-7.the place value of octal numbers goes up in factors of eight from right to left.

This is a base 16 number system that consists of sixteen digits ranging from 0-9 and letters A-F where A is equivalent to 10,B to 11 up to F which is equivalent to 15 in base ten system. The place value of hexadecimal numbers goes up in factors of sixteen.

• A hexadecimal number can be denoted using 16 as a subscript or capital letter H to the right of the number .For example, 94B can be written as 94B16 or 94BH.

Further conversion of numbers from one number system to another

• To convert numbers from one system to another. the following conversions will be considered.
• Converting between binary and decimal numbers.
• Converting octal numbers to decimal and binary form.
• Converting hexadecimal numbers to decimal and binary form.
1. a) Conversion between binary and decimal number
2. Converting binary numbers to decimal numbers
• To convert a binary number to a decimal number, we proceed as follows:
• First, write the place values starting from the right hand side.
• Write each digit under its place value.
• Multiply each digit by its corresponding place value.
• Add up the products. The answer will be the decimal number in base ten.

EXAMPLE

Convert 1011012 to base 10(or decimal) number

 Place value 25 24 23 22 21 20 Binary digits 1 0 1 1 0 1

Multiply each digit by its place value

 N10=(1*25) +(0*24)+(1*23)+(1*22)+(0*21)+(1*20) N10=32+0+8+4+0+1 =4510

32*1=32

16*0=0

8*1=8 4*1=4

2*0=0

1*1=1

=4510

NB: remember to indicate the base subscript since it is the value that distinguishes the different systems.

# Teksti

• The binary equivalent of the fractional part is extracted from the products by reading the respective integral digits from the top downwards as shown by the arrow next page.
• Combine the two parts together to set the binary equivalent.

Convert 0.37510 into binary form

0.375×2=0.750

0.750×2=1.500

0.500×2=1.000 (fraction becomes zero)

Therefore 0.37510=0.0112

NB: When converting a real number from binary to decimal, work out the integral part and the fractional parts separately then combine them.

Example

Convert 11.0112 to a decimal number.

Solution

Convert the integral and the fractional parts separately then add them up.

2×1= 2.000

1×1= +1.000

3.00010

 Weight 21 20 . 2-1 2-2 2-3 Binary digit 1 1 . 0 1 1 Values in base 10 2 1 . 0 0.25 0.125

0.50×0 =0.000

0.25×1 =0.250

0.125×1=+0.125

0.37510

3.00010+0.37510= 3.37510

Thus 11.0112=3.37510

1. iv) Converting a decimal fraction to binary

Divide the integral part continuously by 2.For the fractional part, proceed as follows:

Multiply the fractional part by 2 and note down the product

• Take the fractional part of the immediate product and multiply it by 2 again.
• Continue this process until the fractional part of the subsequent product is 0 or starts to repeat itself.

# Teksti

• The following examples illustrate how to convert hexadecimal number to a decimal numberExample

Convert octal number 3218 to its binary equivalent

Solution

Working from left to the right, each octal number is represented using three digits and then combined we get the final binary equivalent. Therefore:

3=0112

2=0102

1=0012

Combining the three from left to right

 3 2 1 011 010 001

3218 =0110100012

Converting binary numbers to hexadecimal numbers

• To convert binary numbers to their binary equivalents, simply group the digits of the binary number into groups of four from right to left e.g. 11010001.The next step is to write the hexadecimal equivalent of each group e.g.

1101- D

0001- 1

The equivalent of 11010001 is D1H or D116

Converting hexadecimal numbers to decimal and binary numbers.

Converting hexadecimal numbers to decimal number

• To convert hexadecimal number to base 10 equivalent we proceed as follows:
1. First, write the place values starting from the right hand side.
2. If a digit is a letter such as ‘A’ write its decimal equivalent
• Multiply each hexadecimal digit with its corresponding place value and then add the products

The binary equivalent of the fractional part is extracted from the products by reading the respective integral digits from the top downwards as shown by the arrow next pag

• Combine the two parts together to set the binary equivalent.

Convert 0.37510 into binary form

0.375×2=0.750

0.750×2=1.500

0.500×2=1.000 (fraction becomes zero)

Therefore 0.37510=0.0112

Converting octal numbers to decimal and binary numbers

Converting octal numbers to decimal numbers

• To convert a base 8 number to its decimal equivalent we use the same method as we did with binary numbers. However, it is important to note that the maximum absolute value of a octal digit is 7.For example 982 Is not a valid octal number because digit 9 is not an octal digit, but 7368 is valid because all the digits are in the range 0-7.Example shows how to convert an octal number to a decimal number.

Example 1.13

Convert 5128 to its base 10 equivalent

Solution

 Place value 82 81 80 64 8 1 Octal digit 5 1 2

Write each number under its place value as shown below

Multiply each number by its place value.

 N10=(5 x 82)+(1 x 81 )+(2 x 80 ) =(5 x 64)+8+2 =320+8+2 N10=33010

64 x 5=320

8 x 1= 8

1 x 2=+ 2

330

Therefore5128 =33010

Converting octal numbers to binary numbers

• To convert an octal number to binary, each digit is represented by three binary digits because the maximum octal digit i.e. 7 can be represented with a maximum of seven digits. See table:

 Octal digit Binary equivalents 0 000 1 001 2 010 3 011 4 100 5 101 6 110 7 111

# Teksti

Example

Convert the hexadecimal number 11116 to its binary equivalent.

Solution

Place each number under its place value.

 162 161 160 1 1 1

256 x1= 256

16 x 1 = 16

1 x 1= + 1

273

Therefore 11116 =27310

Example

Convert octal number 3218 to its binary equivalent

Solution

Working from left to the right, each octal number is represented using three digits and then combined we get the final binary equivalent. Therefore:

3=0112

2=0102

1=0012

Combining the three from left to right

 3 2 1 011 010 001

3218 =0110100012

Converting binary numbers to hexadecimal numbers

To convert binary numbers to their binary equivalents, simply group the digits of the binary number into groups of four from right to left e.g. 11010001.The next step is to write the hexadecimal equivalent of each group e.g.

1101- D

0001- 1

The equivalent of 11010001 is D1H or D116

Converting hexadecimal numbers to decimal and binary numbers.

Converting hexadecimal numbers to decimal number

To convert hexadecimal number to base 10 equivalent we proceed as follows:

First, write the place values starting from the right hand side.

1. If a digit is a letter such as ‘A’ write its decimal equivalent
• Multiply each hexadecimal digit with its corresponding place value and then add the products
• The following examples illustrate how to convert hexadecimal number to a decimal number

Example

Convert the hexadecimal number 11116 to its binary equivalent

Solution

Place each number under its place value.

 162 161 160 1 1 1

256 x1= 256

16 x 1 = 16

1 x 1= + 1

273

Therefore 11116 =27310

# Converting hexadecimal numbers to binary numbers

• Since F is equivalent to a binary number11112 the hexadecimal number are therefore represented using4 digits as shown in the table below
 Hexadecimal digit Decimal equivalent Binary equivalent 00 00 0000 01 01 0001 02 02 0010 03 03 0011 04 04 0100 05 05 0101 06 06 0110 07 07 0111 08 08 1000 09 09 1001 A 10 1010 B 11 1011 C 12 1100 D 13 1101 E 14 1110 F 15 1111

The simplest method of converting a hexadecimal number to binary is to express each hexadecimal digit as a four bit binary digit number and then arranging the group according to their corresponding positions as shown in example

Example 1

Convert 32116

 Hexadecimal digit 3 2 1 Binary equivalent 0011 0010 0001

Combining the three sets of bits, we get 0011001000012

32116 = 0011001000012

Example 2

Convert 5E616 into binary

 5 E 6 Binary equivalent 0101 1110 0110

5E616 = 0101111001102

# Symbolic representation using coding schemes

• In computing, a single character such as a letter, a number or a symbol is represented by a group of bits. The number of bits per character depends on the coding scheme used.
• The most common coding schemes are:
• Binary Coded Decimal (BCD),
• Extended Binary Coded Decimal Interchange Code (EBCDIC) and
• American Standard Code for Information Interchange (ASCII).

Binary Coded Decimal

• Binary Coded Decimal is a 4-bit code used to represent numeric data only. For example, a number like 9 can be represented using Binary Coded Decimal as 10012 .
• Binary Coded Decimal is mostly used in simple electronic devices like calculators and microwaves. This is because it makes it easier to process and display individual numbers on their Liquid Crystal Display (LCD) screens.
• A standard Binary Coded Decimal, an enhanced format of Binary Coded Decimal, is a 6-bit representation scheme which can represent non-numeric characters. This allows 64 characters to be represented. For letter A can be represented as 1100012 using standard Binary Coded Decimal

Extended Binary Coded Decimal Interchange code (EBCDIC)

• Extended Binary Coded Decimal Interchange code (EBCDIC) is an 8-bit character-coding scheme used primarily on IBM computers. A total of 256 (28) characters can be coded using this scheme. For example, the symbolic representation of letter A using Extended Binary Coded Decimal Interchange code is 110000012.

American standard code for information interchange (ASCII)

• American standard code for information interchange (ASCII) is a 7-bit code, which means that only 128 characters i.e. 27 can be represented. However, manufactures have added an eight bit to this coding scheme, which can now provide for 256 characters.
• This 8-bit coding scheme is referred to as an 8-bit American standard code for information interchange. The symbolic representation of letter A using this scheme is 10000012..

# Binary arithmetic operations

• In mathematics, the four basic arithmetic operations applied on numbers are addition, subtraction, multiplications and division.
• In computers, the same operations are performed inside the central processing unit by the arithmetic and logic unit (ALU). However, the arithmetic and logic unit cannot perform binary subtractions directly. It performs binary subtractions using a process known as For multiplication and division, the arithmetic and logic unit uses a method called shifting before adding the bits.

Representation of signed binary numbers

• In computer technology, there are three common ways of representing a signed binary number.
1. Prefixing an extra sign bit to a binary number.
2. Using ones compliment.
3. Using twos compliment.

Prefixing an extra sign bit to a binary number

• In decimal numbers, a signed number has a prefix “+” for a positive number e.g. +2710 and “-“ for a negative number e.g.-27
• However, in binary, a negative number may be represented by prefixing a digit 1 to the number while a positive number may be represented by prefixing a digit 0. For example, the 7-bit binary equivalent of 127 is 11111112. To indicate that it is positive, we add an extra bit (0) to the left of the number i.e. (0)11111112.
• To indicate that it is negative number we add an extra bit (1) i.e. (1)11111112.
• The problem of using this method is that the zero can be represented in two ways i.e.(0)00000002 and (1)00000002.

# Ones compliment

• The term compliment refers to a part which together with another makes up a whole. For example in geometry two complimentary angle (900).
• The idea of compliment is used to address the problem of signed numbers i.e. positive and negative.
• In decimal numbers (0 to 9), we talk of nine’s compliment. For example the nines compliment
• Of 9 is 0, that of 5 is 4 while 3 is 6.
• However, in binary numbers, the ones compliment is the bitwise NOT applied to the number. Bitwise NOT is a unary operator (operation on only one operand) that performs logical negation on each bit. For example the bitwise NOT of 11002 is 00112e.
• 0s are negated to 1s while 1s are negated to 0s.

Twos compliment

• Twos compliment, equivalent to tens compliment in decimal numbers, is the most popular way of representing negative numbers in computer systems. The advantages of using this method are:
1. There are no two ways of representing a zero as in the case with other two methods.
2. Effective addition and subtraction can be done even with numbers that are represented with a sign bit without a need for circuitries to examine the sign of an operand.
• The twos compliment of a number is obtained by getting the ones compliment then adding a 1. For example, to get the twos compliment of a decimal number 4510,
• First convert it to its binary equivalent then find its ones compliment. Add a 1 to ones compliment i.e.

4510=001011012

Bitwise NOT (00101101) =11010010

Two’s compliment = 110100102+12

= 110100112

The five possible additions in binary are

1. 0 + 0 = 0
2. 0+ 12 = 12
3. 12 + 0 = 12
4. 12 + 12 = 102 (read as 0, carry 1)
5. 12 + 12 + 12 = 112 (read as 1, carry 1)

Example 1

Find the sum of 1112 + 0112

Solution

Arrange the bits vertically. 111

Working from the right to the left, we proceed as follows: + 011

Step 1 12 + 12 = 102, (write down 0 and carry 1) 10102

Step 2 12 + 12 + 12 = 112, (add and carry over digit to 1 + 1 in order to get 1 + 1

+1. From the sum, write down digit one the carry

Forward)

Step 3 12 + 12 + 02 = 102, (add the carry over digit to 1 + 0 in order to get

1 + 1 + 0.since this is the last step, write down 10)

Therefore 1112 + 0112 = 10102

This can be summarized in the table

 1st number 1 1 1 2nd number 0 1 1 Carry digit - 1 1 sum 10 1 0

Example 2

101102

10112

+ 1112

Solution

Add the first two numbers and then add the sum to the third number a follows:

Step 1 Step 2

101102 1000012

+ 10112 + 1112

1000012 1010002

# Binary subtraction

Direct subtraction

The four possible subtractions in binary are:

1. 0 – 0 = 0
2. 12 – 0 = 12
3. 12 – 12 = 0
4. 102 – 12 = 12 ( borrow 1 from the next most significant digit to make 0 become 102,

hence 102 – 12 = 12)

Subraction using ones compliment

The main purpose of using ones compliment in computers is to perform binary subtraction. For example to get the difference in 5 – 3, using the ones compliment, we proceed as follows:

1. Rewrite the problem as 5 + (-3) to show that he computer binary subtraction by adding the binary equivalent of 5 to ones compliment of 3.
2. Convert the absolute value of 3 into 8-bits equivalent i.e. 000000112.
3. Take the ones compliment of 000000112e. 111111002 which is the binary representation of -310.
4. Add the binary equivalent of 5 to ones compliment of 3 i.e.

00000101

+ 11111000

(1)00000001

# Subtraction using twos compliments.

Like in ones compliment, the twos compliment of a number is obtained by negating a positive number to is negative counterpart. For example to get the difference in 5-3, using twos compliment, we proceed as follow:

1. rewrite the problem as 5 + (-3)
2. Convert the absolute value of 3 into 8-bit binary equivalent i.e. 00000011.
3. Take the ones compliment of 00000011 i.e. 11111100.
4. add a 1 to the ones compliment i.e. 11111100 to get 11111101
5. add he binary equivalent of 5 to the twos compliment of 3 i.e.

00000101

+ 11111001

(1)00000010 Ignoring the overflow bit, the resulting number is 00000010, which is directly read as a binary equivalent of +2.

Example

Using twos compliment

3110 - 1710 in binary form.

Solution

1710 in binary 00010001

1’s compliment 11101110

2’s compliment 11101111

3110 = 000111112

00011111 + 11101111 = (1)000011102

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