Unit 1 Notes of Basic electronics Jan 2015 - NMIT Bangalore

Dr. H S Prashantha, Professor, Department of ECE, NMIT Bangalore

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UNIT 1

NUMBER SYSTEM AND DIGITAL LOGIC

NUMBER SYSTEM:

Introduction

Decimal, Binary, Octal & hexadecimal number system conversion

Compliments (only 2’s,1’s,10’s & 9’s compliment)

Addition and Subtraction ( Binary and Decimal number system)

Binary coded number (BCD)

1. INTRODUCTION:

There are different numbering systems used in digital electronic circuits and computers. However, the

numbering system used in one type of circuit may be different to that of another type of circuit, for

example, the memory of a computer would use hexadecimal numbers while the keyboard uses decimal

numbers. The different number systems are

Decimal – The decimal numbering system has a base of 10 (MOD-10) and uses the digits from

0 through 9 to represent a decimal number value.

Binary – The binary numbering system has a base of 2 (MOD-2) and uses only two digits a “0” and a

“1” to represent a binary number value.

Octal – The octal numbering system has a base of 8 (MOD-8) and uses 8 digits between 0 and 7 to

represent an octal number value.

Hexadecimal – The Hexadecimal numbering system has a base of 16 (MOD-16) and uses a total of 16

numeric and alphabetic characters to represent a number value. Hexadecimal numbers consist of

digits 0 through 9 and letters A to F.

Long binary numbers are difficult to read or write and are generally converted into a different system

which is user friendly. The two most common derivatives based on binary numbers are the Octal and

the Hexadecimal numbering systems, with both of these limited in length to a byte (8-bits) or a word

(16-bits). Octal numbers can be represented by groups of 3-bits and hexadecimal numbers by groups of

4-bits together, with this grouping of the bits being used in electronic or computer systems in displays or

printouts.

The Comparisons between the various Decimal, Binary, Hexadecimal and Octal numbers are given in

the following table.

Base or radix Byte (8-bits) Word (16-bits)

Decimal 0 to 255(10) 0 to 65,535(10)

Binary 00000000 to 11111111(2) 0000000000000000 to 1111111111111111(2)

Hexadecimal 00 to FF(16) 0000 to FFFF(16)

Octal 000 to 377(8) 000000 to 177777(8)


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We can see from the table above that the Hexadecimal numbering system uses only four digits to express

a single 16-bit word length, and as a result it is the most commonly used Base Numbering System for

digital, micro-electronics and computer systems.

2. NUMBER SYSTEM

2.1 Decimal Number System

In the decimal number system, each integer number column has values of units, tens, hundreds,

thousands, etc as we move along the number from right to left. Mathematically these values are written

as 100, 101, 102, 103 etc. Then each position to the left of the decimal point indicates an increased positive

power of 10. Likewise, for fractional numbers the weight of the number becomes more negative as we

move from left to right, 10-1, 10-2, 10-3 etc.

The value of any decimal number will be equal to the sum of its digits multiplied by their respective

weights. For example: N = 616310 (Six Thousand One Hundred and Sixty Three) in a decimal format is

equal to:

6000 + 100 + 60 + 3 = 6163

Or it can be written reflecting the weight of each digit as:

(6×1000) + (1×100) + (6×10) + (3×1) = 6163

Or it can be written in polynomial form as:

(6×103) + (1×102) + (6×101) + (3×100) = 6163

Where in this decimal numbering system example, the left most digit is the most significant digit, or MSD,

and the right most digit is the least significant digit or LSD. In other words, the digit 6 is the MSD since

its left most position carries the most weight, and the number 3 is the LSD as its right most position

carries the least weight.

2.2 Binary Number System

The Binary Number System is the most fundamental number system in all digital and computer based

systems and binary numbers follow the same set of rules as the decimal numbering system. But unlike

the decimal system which uses powers of ten, the binary numbering system works on powers of two

giving a binary to decimal conversion from base-2 to base-10. Digital logic and computer systems use

just two values or states to represent a condition, a logic level “1” or a logic level “0”, and each “0” and

“1” is considered to be a single digit in a Base-of-2 or “binary number system”.

In electronics, binary numbers is the flow of information in the form of zeros and ones used by digital

computers and systems. Generally, a logic “1” represents a higher voltage, such as 5 volts, which is

commonly referred to as a HIGH value, while a logic “0” represents a low voltage, such as 0 volts or

ground, and is commonly referred to as a LOW value. These two discrete voltage levels representing the

digital values of “1’s” (one’s) and “0’s” (zero’s) are commonly called: BInary digiTS, and in digital and

computational circuits and applications they are normally referred to as binary BITS.

Figure 1: Binary Bits of Zeros and Ones


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Because there are only two valid Boolean values for representing either a logic “1” or a logic “0”, makes

the system of using Binary Numbers ideal for use in digital or electronic circuits and systems.

Digital waveforms or signals consist of discrete or distinctive voltage levels that are changing back and

forth between these two “HIGH” and “LOW” states. But what makes a signal or voltage “Digital” and how

can we represent these “HIGH” and “LOW” voltage levels. Electronic circuits and systems can be divided

into two main categories.

Analog Circuits – Analogue or Linear circuits amplify or respond to continuously varying voltage

levels that can alternate between a positive and negative value over a period of time.

Digital Circuits – Digital circuits produce or respond too two distinct positive or negative voltage

levels representing either a logic level “1” or a logic level “0”.

In the binary number system, a binary number such as 101100101 is expressed with a string of “1’s” and

“0’s” with each digit along the string from right to left having a value twice that of the previous digit.

As the decimal number is a weighted number, converting from decimal to binary (base 10 to base 2) will

also produce a weighted binary number with the right-hand most bit being the Least Significant Bit or LSB,

and the left-hand most bit being the Most Significant Bit or MSB, and we can represent this as:

Representation of a Binary Number

MSB Binary Digit LSB

28 27 26 25 24 23 22 21 20

256 128 64 32 16 8 4 2 1

In the decimal number system, the weight of each digit to the left increases by a factor of 10. In the

binary number system, the weight of each digit increases by a factor of 2 as shown. Then the first digit

has a weight of 1 (20), the second digit has a weight of 2 (21), the third a weight of 4 (22), the fourth a

weight of 8 (23) and so on.

Conversion from Binary to Decimal

Conversion of binary to decimal (base-2 to base-10) numbers and back is an important concept to

understand as the binary numbering system forms the basis for all computer and digital systems. In a

decimal system each digit has a value ten times greater than its previous number and this decimal

numbering system uses a set of symbols, b, together with a base, q, to determine the weight of each

digit within a number.

Any numbering system can be summarized by the following relationship:

N = bi qi

where: N is a real positive number

b is the digit

q is the base value

and integer (i) can be positive, negative or zero

N = bn qn… b3 q3 + b2 q2 + b1 q1 + b0 q0 + b-1 q-1 + b-2 q-2… etc.

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We can convert binary to decimal by finding the decimal equivalent of the binary array of digits

1011001012 and expanding the binary digits into a series with a base of 2 giving an equivalent of 35710 in

decimal or denary.

For example, converting a Binary to Decimal number would be:

Decimal Digit Value 256 128 64 32 16 8 4 2 1

Binary Digit Value 1 0 1 1 0 0 1 0 1

By adding together ALL the decimal number values from right to left at the positions that are represented

by a “1” gives us: (256)+(64)+(32)+(4)+(1) = 35710 or three hundred and fifty seven as a decimal

number.

Conversion from Decimal to binary

Repeated Division-by-2 Method

An easy method of converting decimal to binary number equivalents is to write down the decimal number

and to continually divide-by-2 (two) to give a result and a remainder of either a “1” or a “0” until the final

result equals zero.

So for example. Convert the decimal number 29410 into its binary number equivalent.

Number 294

Dividing each decimal number by “2”

as shown will give a result plus a

remainder.

If the decimal number being divided

is even then the result will be whole

and the remainder will be equal to

“0”. If the decimal number is odd

then the result will not divide

completely and the remainder will be

a “1”.

The binary result is obtained by

placing all the remainders in order

with the least significant bit (LSB)

being at the top and the most

significant bit (MSB) being at the

bottom.

divide by 2

result 147 remainder 0 (LSB)

divide by 2

result 73 remainder 1

divide by 2


divide by 2


divide by 2


divide by 2


divide by 2


divide by 2


divide by 2

result 0 remainder 1 (MSB)

This divide-by-2 decimal to binary conversion technique gives the decimal number 29410 an equivalent

of 1001001102 in binary, reading from right to left. This divide-by-2 method will also work for conversion

to other number bases.


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The main characteristics of a Binary Number System is that each “binary digit” or “bit” has a value of

either “1” or “0” with each bit having a weight or value double that of its previous bit starting from the

lowest or least significant bit (LSB) and this is called the “sum-of-weights” method.

Hence we can convert a decimal number into a binary number either by using the sum-of-weights method

or by using the repeated division-by-2 method, and convert binary to decimal by finding its sum-of-

weights.

Note: Binary Number Names & Prefixes

1. Binary numbers can be added together and subtracted just like decimal numbers with the result

being combined into one of several size ranges depending upon the number of bits being used.

Binary numbers come in three basic forms – a bit, a byte and a word, where a bit is a single binary

digit, a byte is eight binary digits, and a word is 16 binary digits.

The classification of individual bits into larger groups is generally referred to by the following more

common names

Number of Binary Digits (bits) Common Name

1 Bit

4 Nibble

8 Byte

16 Word

32 Double Word

64 Quad Word

2. Also, when converting from Binary to Decimal or even from Decimal to Binary, we need to be

careful that we do not mix up the two sets of numbers. For example, if we write the digits 10 on

the page it could mean the number “ten” if we assume it to be a decimal number, or it could

equally be a “1” and a “0” together in binary, which is equal to the number two in the weighted

decimal format from above. One way to overcome this problem when converting binary to decimal

numbers and to identify whether the digits or numbers being used are decimal or binary is to write

a small number called a “subscript” after the last digit to show the base of the number system

being used.

For example, if we were using a binary number string we would add the subscript “2” to denote a

base-2 number so the number would be written as 102. Likewise, if it was a standard decimal

number we would add the subscript “10” to denote a base-10 number (written as 1010)

3. Today, as micro-controller or microprocessor systems become increasingly larger, the individual

binary digits (bits) are now grouped together into 8’s to form a single BYTE with most computer

hardware such as hard drives and memory modules commonly indicate their size in Mb or Gb.

Number of Bytes Common Name

1,024 (210) kilobyte (kb)

1,048,576 (220) Megabyte (Mb)

1,073,741,824 (230) Gigabyte (Gb)

A very long number! (240) Terabyte (Tb)


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Binary to Decimal Summary

A “BIT” is the abbreviated term derived from BInary digiT

A Binary system has only two states, Logic “0” and Logic “1” giving a base of 2

A Decimal system uses 10 different digits, 0 to 9 giving it a base of 10

A Binary number is a weighted number who’s weighted value increases from right to left

The weight of a binary digit doubles from right to left

A decimal number can be converted to a binary number by using the sum-of-weights method or the

repeated division-by-2 method

When we convert numbers from binary to decimal, or decimal to binary, subscripts are used to avoid

errors. Converting binary to decimal (base-2 to base-10) or decimal to Binary Numbers (base10 to

base-2) can be done in a number of different ways as shown above. When converting decimal numbers

to binary numbers it is important to remember which the least significant bit (LSB) is, and which the

most significant bit (MSB) is.

2.3 Hexadecimal Number System

The one main disadvantage of binary numbers is that the binary string equivalent of a large decimal base-

10 number can be quite long. When working with large digital systems, such as computers, it is common

to find binary numbers consisting of 8, 16 and even 32 digits which makes it difficult to both read and

write without producing errors especially when working with lots of 16 or 32-bit binary numbers. One

common way of overcoming this problem is to arrange the binary numbers into groups or sets of four

bits (4-bits). These groups of 4-bits uses another type of numbering system also commonly used in

computer and digital systems called Hexadecimal Numbers.

The “Hexadecimal” or simply “Hex” numbering system uses the Base of 16 system and are a popular

choice for representing long binary values because their format is quite compact and much easier to

understand compared to the long binary strings of 1’s and 0’s. Being a Base-16 system, the hexadecimal

numbering system therefore uses 16 (sixteen) different digits with a combination of numbers

from 0 through to 15. In other words, there are 16 possible digit symbols.

However, there is a potential problem with using this method of digit notation caused by the fact that the

decimal numerals of 10, 11, 12, 13, 14 and 15 are normally written using two adjacent symbols. For

example, if we write 10 in hexadecimal, do we mean the decimal number ten, or the binary number of

two (1 + 0). To get around this tricky problem hexadecimal numbers that identify the values of ten,

eleven, . . . , fifteen are replaced with capital letters of A, B, C, D, E and F respectively.

The binary strings can be quite long and difficult to read, but we can make life easier by splitting these

large binary numbers up into even groups to make them much easier to write down and understand. For

example, the following group of binary digits 1101 0101 1100 11112 are much easier to read and

understand than 11010101110011112 when they are all bunched up together.

Hexadecimal Numbers is a more complex system than using just binary or decimal and is mainly used

when dealing with computers and memory address locations. By dividing a binary number up into groups

of 4 bits, each group or set of 4 digits can now have a possible value of between “0000” (0) and “1111”

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( 8+4+2+1 = 15 ) giving a total of 16 different number combinations from 0 to 15. Don’t forget that “0”

is also a valid digit.

We remember from our first tutorial about Binary Numbers that a 4-bit group of digits is called a “nibble”

and as 4-bits are also required to produce a hexadecimal number, a hex digit can also be thought of as

a nibble, or half-a-byte. Then two hexadecimal numbers are required to produce one full byte ranging

from 00 to FF. Also, since 16 in the decimal system is the fourth power of 2 ( or 24 ), there is a direct

relationship between the numbers 2 and 16 so one hex digit has a value equal to four binary digits so

now q is equal to “16”. Because of this relationship, four digits in a binary number can be represented

with a single hexadecimal digit. This makes conversion between binary and hexadecimal numbers very

easy, and hexadecimal can be used to write large binary numbers with much fewer digits.

Hexadecimal Numbers

Decimal Number 4-bit Binary Number Hexadecimal Number

0 0000 0

1 0001 1

2 0010 2

3 0011 3

4 0100 4

5 0101 5

6 0110 6

7 0111 7

8 1000 8

9 1001 9

10 1010 A

11 1011 B

12 1100 C

13 1101 D

14 1110 E

15 1111 F

16 0001 0000 10 (1+0)

17 0001 0001 11 (1+1)

Continuing upwards in groups of four

Using the original binary number from above 1101 0101 1100 11112 this can now be converted into an

equivalent hexadecimal number of D5CF which is much easier to read and understand than a long row


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of 1’s and 0’s that we had before. So by using hexadecimal notation, the numbers can be written with

fewer digits and much less likelihood of error. Similarly, converting hexadecimal based numbers back into

binary is simply the reverse operation.

Then the main characteristics of a Hexadecimal Numbering System is that there are 16 distinct

counting digits from 0 to F with each digit having a weight or value of 16 starting from the least significant

bit (LSB). In order to distinguish Hexadecimal numbers from Denary numbers, a prefix of either a “#”,

(Hash) or a “$” (Dollar sign) is used before the actual Hexadecimal Number value,#D5CF or $D5CF.

As the base of a hexadecimal system is 16, which also represents the number of individual symbols used

in the system, the subscript 16 is used to identify a number expressed in hexadecimal. For example, the

previous hexadecimal number is expressed as: D5CF16

Counting using Hexadecimal Numbers

So we now know how to convert 4 binary digits into a hexadecimal number. But what if we had more

than 4 binary digits how would we count in hexadecimal beyond the final letter F. The simple answer is

to start over again with another set of 4 bits as follows.

0…to…9, A, B, C, D, E, F, 10…to…19, 1A, 1B, 1C, 1D, 1E, 1F, 20, 21….etc

Representation of a Hexadecimal Number is

MSB Hexadecimal Number LSB

168 167 166 165 164 163 162 161 160

4.3G 2.6G 16M 1M 65k 4k 256 16 1

This adding of additional hexadecimal digits to convert both decimal and binary numbers into an

Hexadecimal Number is very easy if there are 4, 8, 12 or 16 binary digits to convert. But we can also

add zero’s to the left of the most significant bit, the MSB if the number of binary bits is not a multiple of

four.

For example, 110010110110012 is a fourteen bit binary number that is to large for just three hexadecimal

digits only, yet too small for a four hexadecimal number. The answer is to ADD additional zeros to the

left most bit until we have a complete four bit binary number or multiples thereof.

Adding of Additional 0’s to a Binary Number

Binary Number 0011 0010 1101 1001

Hexadecimal Number 3 2 D 9

The main advantage of a Hexadecimal Number is that it is very compact and by using a base of 16

means that the number of digits used to represent a given number is usually less than in binary or

decimal. Also, it is quick and easy to convert between hexadecimal numbers and binary.

Hexadecimal Numbers Example No1

Convert the following Binary number 1110 10102 into its Hexadecimal number equivalent.


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Binary Number = 111010102

Group the bits into four’s starting from the right hand side

=

1110

1010

=

14

10 (in decimal)

=

E

A (in Hex)

Then, the hexadecimal equivalent of the binary number 1110 10102 is #EA16

Hexadecimal Numbers Example No2

Convert the following Hexadecimal number #3FA716 into its Binary equivalent, and also into its Decimal

or Denary equivalent using subscripts to identify each numbering system.

#3FA716

= 0011 1111 1010 01112

= (8192 + 4096 + 2048 + 1024 + 512 + 256 + 128 + 32 + 4 + 2 + 1)

= 16,29510

Then, the Decimal number of 16,295 can be represented as:- #3FA716 in Hexadecimal

or

0011 1111 1010 01112 in Binary.

2.4 Octal Number System

The Octal Number System is another type of computer and digital base number system. The Octal Number

System is very similar in principle to the previous hexadecimal numbering system except that in Octal, a

binary number is divided up into groups of only 3 bits, with each group or set of bits having a distinct

value of between 000 (0) and 111 ( 4+2+1 = 7 ). Octal numbers therefore have a range of just “8” digits,

(0, 1, 2, 3, 4, 5, 6, 7) making them a Base-8 numbering system and therefore, q is equal to “8”.

Then the main characteristics of an Octal Numbering System is that there are only 8 distinct counting

digits from 0 to 7 with each digit having a weight or value of just 8 starting from the least significant bit

(LSB). In the earlier days of computing, octal numbers and the octal numbering system was very popular

for counting inputs and outputs because as it works in counts of eight, inputs and outputs were in counts

of eight, a byte at a time. As the base of an Octal Numbers system is 8 (base-8), which also represents

the number of individual numbers used in the system, the subscript 8 is used to identify a number

expressed in octal. For example, an octal number is expressed as: 2378

Just like the hexadecimal system, the “octal number system” provides a convenient way of converting

large binary numbers into more compact and smaller groups. However, these days the octal numbering

system is used less frequently than the hexadecimal number system

Representation of an Octal Number is

MSB Octal Number LSB

88 87 86 85 84 83 82 81 80

16M 2M 262k 32k 4k 512 64 8 1


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As the octal number system uses only eight digits (0 through 7) there are no numbers or letters used

above 8, but the conversion from decimal to octal and binary to octal follows the same pattern as we

have seen previously for hexadecimal.

To count above 7 in octal we need to add another column and start over again in a similar way to

hexadecimal.

0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21….etc

Again do not get confused, 10 or 20 is NOT ten or twenty it is 1 + 0 and 2 + 0 in octal exactly the same

as for hexadecimal. The relationship between binary and octal numbers is given below.

Octal Numbers

Decimal Number 3-bit Binary Number Octal Number

0 000 0

1 001 1

2 010 2

3 011 3

4 100 4

5 101 5

6 110 6

7 111 7

8 001 000 10 (1+0)

9 001 001 11 (1+1)

Continuing upwards in groups of three

Then we can see that 1 octal number or digit is equivalent to 3 bits, and with two octal number, 778we

can count up to 63 in decimal, with three octal numbers, 7778 up to 511 in decimal and with four octal

numbers, 77778 up to 4095 in decimal and so on.

Octal Numbers Example No1

Using our previous binary number of 11010101110011112 convert this binary number to its octal

equivalent, (base-2 to base-8).

Binary Digit Value 001101010111001111

Group the bits into three´s starting

from the right hand side

001 101 010 111 001 111

Octal Number form 1 5 2 7 1 78

Thus, 0011010101110011112 in its Binary form is equivalent to 1527178 in Octal form or 54,735 in

denary.


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Octal Numbers Example No2

Convert the octal number 23228 to its decimal number equivalent, (base-8 to base-10).

Octal Digit Value 23228

In polynomial form = ( 2×83 ) + ( 3×82 ) + ( 2×81 ) + ( 2×80 )

Add the results = ( 1024 ) + ( 192 ) + ( 16 ) + ( 2 )

Decimal number form equals: 123410

Then, converting octal to decimal shows that 23228 in its Octal form is equivalent to 123410 in its Decimal

form.

While Octal is another type of digital numbering system, it is little used these days instead the more

commonly used Hexadecimal Numbering System is used as it is more flexible.

1.3 COMPLIMENTS

In mathematics, positive numbers (including zero) are represented as unsigned numbers. That is we do

not put the +ve sign in front of them to show that they are positive numbers. However, when dealing

with negative numbers we do use a -ve sign in front of the number to show that the number is negative

in value and different from a positive unsigned value, and the same is true with signed binary numbers.

However, in digital circuits there is no provision made to put a plus or even a minus sign, since digital

systems operate with binary numbers that are represented in terms of “0’s” and “1’s”. We have seen

previously that an 8-bit byte can have a value from 0 to 255, which are 28 = 256 different combinations

of bits forming a single 8-bit byte. So for example an unsigned binary number such

as:010011012 = 64 + 8 + 4 + 1 = 7710 in decimal. But Digital Systems and computers must also be able

to use and to manipulate negative numbers as well as positive numbers.

Mathematical Numbers are generally made up of a sign and a value (magnitude) in which the sign

indicates whether the number is positive, ( + ) or negative, ( – ) with the value indicating the size of the

number, for example 23, +156 or -274. Presenting numbers is this fashion is called “sign-magnitude”

representation since the left most digit can be used to indicate the sign and the remaining digits the

magnitude or value of the number.

Sign-magnitude notation is the simplest and one of the most common methods of representing positive

and negative numbers either side of zero, (0). Thus negative numbers are obtained simply by changing

the sign of the corresponding positive number as each positive or unsigned number will have a signed

opposite, for example, +2 and -2, +10 and -10, etc.

But how do we represent signed binary numbers if all we have is a bunch of one’s and zero’s. We know

that binary digits, or bits only have two values, either a “1” or a “0”, and conveniently a sign also has

only two values, a “+” or a “–“. Then we can use a single bit to identify the sign of a signed binary

number.

So to represent a positive (N) and a negative (-N) binary number we can use the binary numbers with

sign. For signed binary numbers the most significant bit (MSB) is used as the sign. If the sign bit is “0”,

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this means the number is positive. If the sign bit is “1”, then the number is negative. The remaining bits

are used to represent the magnitude of the binary number in the usual unsigned binary number format.

Then we can see that the Sign-and-Magnitude (SM) notation stores positive and negative values by

dividing the “n” total bits into two parts: 1 bit for the sign and n–1 bits for the value which is a pure

binary number. For example, the decimal number 53 can be expressed as an 8-bit signed binary number

as follows.

Positive Signed Binary Numbers

Negative Signed Binary Numbers

The disadvantage here is that whereas before we had a n-bit unsigned binary number we now have a n-

1 bit signed binary number giving a range of digits from:

-(2(n-1) - 1) to +(2(n-1) - 1)

So for example: if we have 4 bits to represent a signed binary number, (1-bit for the Sign bit and 3-bits

for the Magnitude bits), then the actual range of numbers we can represent in sign-magnitude notation

would be:

-(2(4-1) - 1) to +(2(4-1) - 1)

-2(3) - 1 to +2(3) - 1

-7 to +7

Signed Binary Numbers Example No1

Convert the following decimal values into signed binary numbers using the sign-magnitude format:

-1510 as a 6-bit number

⇒ 1 0 1 1 1 1 2

+2310 as a 6-bit number

⇒ 0 1 0 1 1 1 2


⇒ 1 0 1 1 1 0 0 0 2

+8510 as a 8-bit number

⇒ 0 1 0 1 0 1 0 1 2


⇒ 1 1 1 1 1 1 1 1 2


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Note that for a 4-bit, 6-bit, 8-bit, 16-bit or 32-bit signed binary number all the bits MUST have a value,

therefore “0’s” are used to fill the spaces between the leftmost sign bit and the first or highest value “1”.

The sign-magnitude representation of a binary number is a simple method to use and understand for

representing signed binary numbers, as we use this system all the time with normal decimal (base 10)

numbers in mathematics. Adding a “1” to the front of it if the binary number is negative and a “0” if it is

positive.

However, using this sign-magnitude method can result in the possibility of two different bit patterns

having the same binary value. For example, +0 and -0 would be 0000 and 1000 respectively as a signed

4-bit binary number. So we can see that using this method there can be two representations for zero, a

positive zero ( 00002 ) and also a negative zero ( 10002 ) which can cause big complications for computers

and digital systems.

One’s Complement of a Signed Binary Number

One’s Complement or 1’s Complement as it is also termed, is another method which we can use to

represent negative binary numbers in a signed binary number system. In one’s complement, positive

numbers (also known as non-complements) remain unchanged as before with the sign-magnitude

numbers.

Negative numbers however, are represented by taking the one’s complement (inversion, negation) of the

unsigned positive number. Since positive numbers always start with a “0”, the complement will always

start with a “1” to indicate a negative number.

The one’s complement of a negative binary number is the complement of its positive counterpart, so to

take the one’s complement of a binary number, all we need to do is change each bit in turn. Thus the

one’s complement of “1” is “0” and vice versa, then the one’s complement of 100101002 is

simply011010112 as all the 1’s are changed to 0’s and the 0’s to 1’s.

The easiest way to find the one’s complement of a signed binary number when building digital arithmetic

or logic decoder circuits is to use Inverters. The inverter is naturally a complement generator and can be

used in parallel to find the 1’s complement of any binary number as shown.

1’s Complement Using Inverters

Then we can see that it is very easy to find the one’s complement of a binary number N as all we need

do is simply change the 1’s to 0’s and the 0’s to 1’s to give us a -N equivalent. Also just like the previous

sign-magnitude representation, one’s complement can also have n-bit notation to represent numbers in

the range from: -2(n-1) - 1 and +2(n-1) - 1. For example, a 4-bit representation in the one’s complement

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format can be used to represent decimal numbers in the range from -7 to +7 with two representations

of zero: 0000 (+0) and 1111 (-0) the same as before.

Subtraction Using One’s Complement

One of the main advantages of One’s Complement is in the addition and subtraction of two binary

numbers. In mathematics, subtraction can be implemented in a variety of different ways as A – B, is the

same as saying A + (-B) or -B + A etc. Therefore, the complication of subtracting two binary numbers

can be performed by simply using addition.

The binary addition follows the same rules as for the normal addition except that in binary there are only

two bits (digits) and the largest digit is a “1”, (just as “9” is the largest decimal digit) thus the possible

combinations for binary addition are as follows:

0 0 1 1

+ 0 + 1 + 0 + 1

0 1 1 1← 0 ( 0 plus a carry 1 )

When the two numbers to be added are both positive, the sum A + B, they can be added together by

means of the direct sum (including the number and bit sign), because when single bits are added together,

“0 + 0″, “0 + 1″, or “1 + 0″ results in a sum of “0” or “1”. This is because when the two bits to be added

together are odd (“0″ + “1” or “1 + 0″), the result is “1”. Likewise when the two bits to be added together

are even (“0 + 0″ or “1 + 1″) the result is “0” until you get to “1 + 1″ then the sum is equal to “0” plus

a carry “1”. Let’s look at a simple example.

Subtraction of Two Binary Numbers

An 8-bit digital system is required to subtract the following two numbers 115 and 27 from each other

using one’s complement. So in decimal this would be: 115 - 27 = 88.

First we need to convert the two decimal numbers into binary and make sure that each number has the

same number of bits by adding leading zero’s to produce an 8-bit number (byte). Therefore:

11510 in binary is: 0 1 1 1 0 0 1 1 2

2710 in binary is: 0 0 0 1 1 0 1 1 2

Now we need to find the complement of the second binary number, (00011011) while leaving the first

number (01110011) unchanged.

By changing all the 1’s to 0’s and 0’s to 1’s, the one’s complement of 00011011 is equal to 11100100.

Adding the first number and the complement of the second number gives:

0 1 1 1 0 0 1 1

+ 1 1 1 0 0 1 0 0

O v e r f l o w → 1 0 1 0 1 0 1 1 1

Since the digital system is to work with 8-bits, only the first eight digits are used to provide the answer

to the sum, and we simply ignore the last bit (bit 9). This bit is call an “overflow” bit. Overflow occurs

when the sum of the most significant (left-most) column produces a carry forward. This overflow or carry

bit can be ignored completely or passed to the next digital section for use in its calculations. Overflow

indicates that the answer is positive. If there is no overflow then the answer is negative.


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The 8-bit result from above is: 01010111 (the overflow “1” cancels out) and to convert it back from a

one’s complement answer to the real answer we now have to add “1” to the one’s complement result,

therefore:

0 1 0 1 0 1 1 1

+ 1

0 1 0 1 1 0 0 0

So the result of subtracting 27 (000110112 ) from 115 (011100112 ) using 1’s complement in binary gives

the answer of: 010110002 or (64 + 16 + 8) = 8810 in decimal.

Then we can see that signed or unsigned binary numbers can be subtracted from each other using One’s

Complement through the process of addition.

Two’s Complement of a Signed Binary Number

Two’s Complement or 2’s Complement as it is also termed is another method like the previous sign-

magnitude and one’s complement form, which we can use to represent negative binary numbers in a

signed binary number system. In two’s complement, the positive numbers are exactly the same as before

for unsigned binary numbers. A negative number, however, is represented by a binary number, which

when added to its corresponding positive equivalent results in zero.

In two’s complement form, a negative number is the 2’s complement of its positive number with the

subtraction of two numbers being A – B = A + ( 2’s complement of B ) using much the same process as

before as basically, two’s complement is one’s complement + 1.

The main advantage of two’s complement over the previous one’s complement is that there is no double-

zero problem plus it is a lot easier to generate the two’s complement of a signed binary number.

Therefore, arithmetic operations are relatively easier to perform when the numbers are represented in

the two’s complement format.

Let’s look at the subtraction of our two 8-bit numbers 115 and 27 from above using two’s complement,

and we remember from above that the binary equivalents are:

11510 in binary is: 0 1 1 1 0 0 1 1 2

2710 in binary is: 0 0 0 1 1 0 1 1 2

Our numbers are 8-bits long, then there are 28 digits available to represent our values and in binary this

equals: 1000000002 or 25610. Then the two’s complement of 2710 will be:

(28)2 – 00011011 = 100000000 – 00011011 = 111001012

The complementation of the second negative number means that the subtraction becomes a much easier

addition of the two numbers so therefore the sum is: 115 + ( 2’s complement of 27 ) which is:

01110011 + 11100101 = 1 010110002

As previously, the 9th overflow bit is disregarded as we are only interested in the first 8-bits, so the result

is: 010110002 or (64 + 16 + 8) = 8810 in decimal the same as before.


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4-bit Signed Binary Number Comparison

Decimal Signed

Magnitude

Signed One’s

Complement

Signed Two’s

Complement

+7 0111 0111 0111

+6 0110 0110 0110

+5 0101 0101 0101

+4 0100 0100 0100

+3 0011 0011 0011

+2 0010 0010 0010

+1 0001 0001 0001

+0 0000 0000 0000

-0 1000 1111 –

-1 1001 1110 1111

-2 1010 1101 1110

-3 1011 1100 1101

-4 1100 1011 1100

-5 1101 1010 1011

-6 1110 1001 1010

-7 1111 1000 1001

Signed-complement forms of binary numbers can use either 1’s complement or 2’s complement. The 1’s

complement and the 2’s complement of a binary number are important because they permit the

representation of negative numbers. The method of 2’s complement arithmetic is commonly used in

computers to handle negative numbers.


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DIGITAL LOGIC

Boolean Algebra

Logic Gates

Introduction to Combinational Logic : Half Adder, Full adder

Introduction to Sequential circuits : Flip Flops ( RS FF , JK FF, D FF , T FF ) (only truth table)

Introduction to Boolean

In 1854, George Boole performed an investigation into the “laws of thought” which were based on a

simplified version of the “group” or “set” theory, and from this Boolean or “Switching” algebra was

developed. Boolean algebra deals mainly with the theory that both logic and set operations are either

“TRUE” or “FALSE” but not both at the same time.

For example, A + A = A and not 2A as it would be in normal algebra. Boolean Algebra is a simple and

effective way of representing the switching action of standard Logic Gates and the basic logic statements

which concern us here are given by the logic gate operations of the AND, the OR and the NOT gate

functions.

As well as a standard Boolean Expression, the input and output information of any Logic Gate or circuit

can be plotted into a standard table to give a visual representation of the switching function of the system.

The table used to represent the Boolean expression of a logic gate function is commonly called a Truth

Table. A logic gate truth table shows each possible input combination to the gate or circuit with the

resultant output depending upon the combination of these input(s).

Logic Gate Truth Tables

For example, consider a single 2-input logic circuit with input variables labeled as A and B. There are

“four” possible input combinations or 22 of “OFF” and “ON” for the two inputs. However, when dealing

with Boolean expressions and especially logic gate truth tables, we do not general use “ON” or “OFF” but

instead give them bit values which represent a logic level “1” or a logic level “0” respectively.

Then the four possible combinations of A and B for a 2-input logic gate is given as:

Input Combination 1. – “OFF” – “OFF” or ( 0, 0 )

Input Combination 2. – “OFF” – “ON” or ( 0, 1 )

Input Combination 3. – “ON” – “OFF” or ( 1, 0 )

Input Combination 4. – “ON” – “ON” or ( 1, 1 )

Therefore, a 3-input logic circuit would have 8 possible input combinations or 23 and a 4-input logic circuit

would have 16 or 24, and so on as the number of inputs increases. Then a logic circuit with “n” number

of inputs would have 2n possible input combinations of both “OFF” and “ON”. In order to keep things

simple to understand, we will only deal with simple 2-input logic gates, but the principals are still the

same for gates with more inputs.

The logical AND Function

The Logic AND Function function states that two or more events must occur together and at the same

time for an output action to occur. The order in which these actions occur is unimportant as it does not

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affect the final result. For example, A & B = B & A. In Boolean algebra the Logic AND Function follows

the Commutative Law which allows a change in position of either variable.

The AND function is represented in electronics by the dot or full stop symbol ( . ) Thus a 2-input (A B)

AND Gate has an output term represented by the Boolean expression A.B or just AB.

Switch Representation of the AND Function

Here the two switches, A and B are connected together to

form a series circuit. Therefore, in the circuit above, both

switch A AND switch B must be closed (Logic “1”) in order

to put the lamp on. In other words, both switches must

be closed, or at logic “1” for the lamp to be “ON”.

AND Gate produces an output when “ALL” of its inputs are

present. In Boolean algebra terms the output will

be TRUE only when all of its inputs are TRUE. In electrical

terms, the logic AND function is equal to a series circuit as shown above.

As there are only two Switches, each with two possible

states “open” or “closed”. Defining a Logic “0” as being

when the switch is open and a Logic “1” when the switch

is closed, there are then four different ways or

combinations of arranging the two switches together as

shown.

AND Function Truth Table

Switch A Switch B Output Description Symbol

0 0 0 A and B are both open, lamp OFF

0 1 0 A is open and B is closed, lamp OFF

1 0 0 A is closed and B is open, lamp OFF

1 1 1 A is closed and B is closed, lamp ON

Boolean Expression (A AND B) A . B

The Logical OR Function

The Logic OR Function function states that an output action will occur or become TRUE if either one “OR”

more events are TRUE, but the order at which they occur is unimportant as it does not affect the final

result. For example, A + B = B + A. In Boolean algebra the Logic OR Function follows the Commutative

Law the same as for the logic AND function, allowing a change in position of either variable.

The OR function is sometimes called by its full name of “Inclusive OR” in contrast to the Exclusive-OR

function.

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Switch Representation of the OR Function is

Here the two switches A and B are connected in parallel

and either Switch A OR Switch B can be closed in order

to put the lamp on. In other words, either switch can be

closed, or at logic “1” for the lamp to be “ON”. OR Gate

produces HIGH output when “ANY” of its inputs are

HIGH. In electrical terms, the logic OR function is equal

to a parallel circuit. Again as with the AND function there

are two switches, each with two possible positions open

or closed so therefore there will be 4 different ways of

arranging the switches.

OR Function Truth Table

Switch

A

Switch

B Output Description

Symbol

0 0 0 A and B are both open, lamp OFF

0 1 1 A is open and B is closed, lamp ON

1 0 1 A is closed and B is open, lamp ON

1 1 1 A is closed and B is closed, lamp

ON

Boolean Expression (A OR B)

A + B

The Logic NOT Function

The Logic NOT Function is simply a single input inverter that changes the input of a logic level “1” to an

output of logic level “0” and vice versa. The logic NOT function is so called because its output state

is NOT the same as its input state. The “logical NOT function” is generally denoted by a bar or overline

( ¯ ) over its input symbol which denotes the inversion operation, (hence its name as an inverter).

As NOT gates perform the logic INVERT or COMPLEMENTATION function they are more commonly known

as Inverters because they invert the signal. In logic circuits this negation can be represented by a normally

closed switch.

Switch Representation of the NOT Function is

If A means that the switch is closed, then NOT A or

simply A says that the switch is NOT closed or in other words,

it is open. The logic NOT function has a single input and a

single output as shown.


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NOT Function Truth Table is

Switch Output Symbol

1 0

0 1

Boolean Expression not-A or A

The inversion indicator for a logic NOT function is a “bubble”, (O) symbol on the output (or input) of the

logic elements symbol. In Boolean algebra the inverting Logic NOT Function follows the Complementation

Law producing inversion.

Logic NOT Gates or “Inverters” as they are more commonly called, can be connected with standard

AND and OR gates to produce NAND and NOR gates respectively. Logic NOT gates can also be used to

produce “Complementary” signals in more complex decoder/logic circuits for example, the complement

of logic A is A and two Inverters connected together in series will give a double inversion which produces

at its output the original value of A.

When designing logic circuits and you may only need one or two logic NOT gates within your design, but

do not have the space or the money for a dedicated Inverter chip such as the 74LS04. The

logic NOT function can be easily implemented by using any NAND or NOR gates by simply connecting

their inputs together as shown below.

NOT Function Equivalents is

2-Input NAND (Not AND) Gate

For a 2-input NAND gate, the output Q is true if BOTH input A and input B are NOT true, giving the

Boolean Expression of: (Q = not (A and B)).

Symbol Truth Table

A B Q

0 0 1

0 1 1

1 0 1

1 1 0

Boolean Expression Q = A .B Read as A AND B gives NOT-Q

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2-Input NOR (Not OR) Gate

For a 2-input NOR gate, the output Q is true if BOTH input A and input B are NOT true, giving the Boolean

Expression of: (Q = not (A or B)).

Symbol Truth Table

A B Q

0 0 1

0 1 0

1 0 0

1 1 0

Boolean Expression Q = A+B Read as A OR B gives NOT-Q

As well as the standard logic gates there are also two special types of logic gate function called an

Exclusive-OR Gate and an Exclusive-NOR Gate.

2-Input EX-OR (Exclusive OR) Gate

For a 2-input Ex-OR gate, the output Q is true if EITHER input A or if input B is true, but NOT both giving

the Boolean Expression of: (Q = (A and NOT B) or (NOT A and B)).

Symbol Truth Table

A B Q

0 0 0

0 1 1

1 0 1

1 1 0

Boolean Expression Q = A B

2-Input EX-NOR (Exclusive NOR) Gate For a 2-input Ex-NOR gate, the output Q is true if BOTH input A and input B are the same, either true or

false, giving the Boolean Expression of: ( Q = (A and B) or (NOT A and NOT B) ).

Symbol Truth Table

A B Q

0 0 1

0 1 0

1 0 0

1 1 1

Boolean Expression Q = A B


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Summary of 2-input Logic Gates is

Inputs Truth Table Outputs For Each Gate

A B AND NAND

OR NOR EX-OR EX-NOR

0 0 0 1 0 1 0 1

0 1 0 1 1 0 1 0

1 0 0 1 1 0 1 0

1 1 1 0 1 0 0 1

Logic Function Boolean Notation

AND A.B

OR A+B

NOT A

NAND A .B

NOR A+B

EX-OR (A.B) + (A.B) or A B

EX-NOR (A.B) + or A B

NOTE : 2-input logic gate truth tables are given here as examples of the operation of each logic function,

but there are many more logic gates with 3, 4 even 8 individual inputs. The multiple input gates logic

functions are same as 2-input gates discussed above.

Boolean algebra

The digital logic functions can be defined and displayed as either a Boolean algebra expression or as a

logic gate truth table. Few examples of how we can use Boolean algebra to simplify larger digital logic

circuits.

Boolean algebra Example No1:

Construct a Truth Table for the logical functions at points C, D and Q in the following circuit and identify

a single logic gate that can be used to replace the whole circuit.

First observations indicates that the circuit consists of a 2-

input NAND gate, a 2-input EX-OR gate and finally a 2-

input EX-NOR gate at the output. As there are only 2 inputs to

the circuit labelled A and B, there can only be 4 possible

combinations of the input ( 22 ) and these are: 0-0, 0-1, 1-0 and finally 1-1.

Inputs Output at

A B C D Q

0 0 1 0 0

0 1 1 1 1

1 0 1 1 1

1 1 0 0 1


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From the truth table above, column C represents the output function generated by the NAND gate, while

column D represents the output function from the Ex-OR gate. Both of these two output expressions then

become the input condition for the Ex-NOR gate at the output.

It can be seen from the truth table that an output at Q is present when any of the two inputs A or Bare

at logic 1. The only truth table that satisfies this condition is that of an OR Gate. Therefore, the whole of

the above circuit can be replaced by just one single 2-input OR Gate.

Boolean algebra Example No 2

Find the Boolean algebra expression for the following system.

The system consists of an AND Gate, a NOR Gate and finally an OR Gate. The expression for the AND

gate is A.B, and the expression for the NOR gate is A+B. Both these expressions are also separate inputs

to the OR gate which is defined as A+B. Thus the final output expression is given as:

The output of the system is given as Q = (A.B) + (A+B), but the notation A+B is the same as the De

Morgan´s notation A.B, Then substituting A.B into the output expression gives us a final output notation

of Q = (A.B)+(A.B), which is the Boolean notation for an Exclusive-NOR Gate as seen in the previous

section.

Inputs Intermediates Output

B A A.B 𝑨+𝑩→ Q

0 0 0 1 1

0 1 0 0 0

1 0 0 0 0

1 1 1 0 1

Then, the whole circuit above can be replaced by just one single Exclusive-NOR Gate and indeed an

Exclusive-NOR Gate is made up of these individual gate functions.

Boolean algebra Example No3 is

Find the Boolean algebra expression for the following system.


26

This system may look more complicated than the other two to analyse but again, the logic circuit just

consists of simple AND, OR and NOT gates connected together.

As with the previous Boolean examples, we can simplify the circuit by writing down the Boolean notation

for each logic gate function in turn in order to give us a final expression for the output at Q.

The output from the 3-input AND gate is only at logic “1” when ALL the gates inputs are HIGH at logic

level “1” (A.B.C). The output from the lower OR gate is only a “1” when one or both inputs B or C are at

logic level “0”. The output from the 2-input AND gate is a “1” when input A is a “1” and inputs B or Care

at “0”. Then the output at Q is only a “1” when inputs A.B.C equal “1” or A is equal to “1” and both

inputs B or C equal “0”, A.(B+C).

By using “de Morgan’s theorem” inputs B and input C cancel out as to produce an output at Q they can

be either at logic “1” or at logic “0”. Then this just leaves input A as the only input needed to give an

output at Q as shown in the table below.

Inputs Intermediates Output

C B A A.B.C B C B+C A.(B+C) Q

0 0 0 0 1 1 1 0 0

0 0 1 0 1 1 1 1 1

0 1 0 0 0 1 1 0 0

0 1 1 0 0 1 1 1 1

1 0 0 0 1 0 1 0 0

1 0 1 0 1 0 1 1 1

1 1 0 0 0 0 0 0 0

1 1 1 1 0 0 0 0 1

Then we can see that the entire logic circuit above can be replaced by just one single input labelled A

thereby reducing a circuit of six individual logic gates to just one single piece of wire, (or Buffer). This

type of circuit analysis using Boolean Algebra can be very powerful and quickly identify any unnecessary

logic gates within a digital logic design thereby reducing the number of gates required, the power

consumption of the circuit and of course the cost.

The Laws of Boolean

As well as the logic symbols “0” and “1” being used to represent a digital input or output, we can also

use them as constants for a permanently “Open” or “Closed” circuit or contact respectively. A set of rules

or Laws of Boolean Algebra expressions have been invented to help reduce the number of logic gates




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needed to perform a particular logic operation resulting in a list of functions or theorems known commonly

as the Laws of Boolean Algebra.

Boolean algebra is the mathematics we use to analyze digital gates and circuits. We can use these “Laws

of Boolean” to both reduce and simplify a complex Boolean expression in an attempt to reduce the number

of logic gates required. Boolean algebra is therefore a system of mathematics based on logic that has its

own set of rules or laws which are used to define and reduce Boolean expressions.

The variables used in Boolean Algebra only have one of two possible values, a logic “0” and a logic “1”but

an expression can have an infinite number of variables all labelled individually to represent inputs to the

expression, For example, variables A, B, C etc, giving us a logical expression of A + B = C, but each

variable can ONLY be a 0 or a 1.

Examples of these individual laws of Boolean, rules and theorems for Boolean algebra are given in the

following table.

Truth Tables for the Laws of Boolean is

Boolean

Expression Description

Equivalent

Switching Circuit

Boolean Algebra

Law or Rule

A + 1 = 1 A in parallel with closed

= "CLOSED"

Annulment

A + 0 = A A in parallel with open

= "A"

Identity

A . 1 = A A in series with closed

= "A"

Identity

A . 0 = 0 A in series with open

= "OPEN"

Annulment

A + A = A A in parallel with A = "A"

Indempotent

A . A = A A in series with A = "A"

Indempotent

NOT A = A NOT NOT A

(double negative) = "A" Double Negation

A + A = 1 A in parallel with not A

= "CLOSED"

Complement



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A . A = 0 A in series with not A

= "OPEN"

Complement

A+B = B+A A in parallel with B =

B in parallel with A

Commutative

A.B = B.A A in series with B =

B in series with A

Commutative

A+B = A.B invert and replace OR with AND de Morgan’s Theorem

A.B = A+B invert and replace AND with OR de Morgan’s Theorem

The basic Laws of Boolean Algebra that relate to the Commutative Law allowing a change in position

for addition and multiplication, the Associative Law allowing the removal of brackets for addition and

multiplication, as well as the distributive Law allowing the factoring of an expression, are the same as

in ordinary algebra. Each of the Boolean Laws above are given with just a single or two variables, but the

number of variables defined by a single law is not limited to this as there can be an infinite number of

variables as inputs too the expression. These Boolean laws detailed above can be used to prove any given

Boolean expression as well as for simplifying complicated digital circuits.

NOTE:

The Logic NAND Function

Switch Representation of the NAND Function

The Logic NOR Function

Switch Representation of the NOR Function



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Combinational Logic

Combinational Logic Circuits are made up from basic

logic NAND, NOR or NOT gates that are “combined” or

connected together to produce more complicated

switching circuits. These logic gates are the building

blocks of Combinational Logic Circuits. An example of

a combinational circuit is a decoder, which converts

the binary code data present at its input into a number

of different output lines, one at a time producing an equivalent decimal code at its output.

Combinational logic circuits can be very simple or very

complicated and any combinational circuit can be

implemented with only NAND and NOR gates as these

are classed as “universal” gates. Common

combinational circuits made up from individual logic

gates that carry out a desired application

include Multiplexers, De-multiplexers, Encoders, Decoders, Full and Half adders, etc..

The Binary Adder

Another common and very useful combinational logic circuit which can be constructed using just a few

basic logic gates and adds together binary numbers is the Binary Adder circuit. The Binary Adder is made

up from standard AND and Ex-OR gates and allow us to “add” together single bit binary

numbers, a and b to produce two outputs called the SUM of the addition and a CARRY called the Carry-

out, ( C out ) bit. One of the main uses for the Binary Adder is in arithmetic and counting circuits.

Binary Addition

Binary Addition follows the same basic rules as for the denary addition above except in binary there are

only two digits and the largest digit is “1”, so any “SUM” greater than 1 will result in a “CARRY”. This

carry 1 is passed over to the next column for addition and so on. Consider the single bit addition below.

0 0 1 1

+ 0 + 1 + 0 + 1

0 1 1 10

The single bits are added together and “0 + 0″, “0 + 1″, or “1 + 0″ results in a sum of “0” or “1” until

you get to “1 + 1″ then the sum is equal to “2”, (a zero plus a carry). For a simple 1-bit addition problem

like this, the resulting carry bit could be ignored which would result in an output truth table resembling

that of an Ex-OR Gate as seen in the Logic Gates section and whose result is the sum of the two bits but

without the carry.

An Ex-OR gate only produces an output “1” when either input is at logic “1”, but not both. However, all

microprocessors and electronic calculators require the carry bit to correctly calculate the equations so we

need to rewrite them to include 2 bits of output data as shown below.

00 00 01 01

+ 00 + 01 + 00 + 01

00 01 01 10




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From the above equations we know that an Ex-OR gate will only produce an output “1” when “EITHER”

input is at logic “1”, so we need an additional output to produce a carry output, “1” when “BOTH” inputs

“A” and “B” are at logic “1” and a standard AND Gate fits the bill nicely. By combining the Ex-OR gate

with the AND gate results in a simple digital binary adder circuit known commonly as the “Half Adder”

circuit.

The Half Adder Circuit - 1-bit Adder with Carry-Out

Symbol Truth Table

A B SUM CARRY

0 0 0 0

0 1 1 0

1 0 1 0

1 1 0 1

Boolean Expression: Sum = A ⊕ B Carry = A . B

From the truth table we can see that the SUM (S) output is the result of the Ex-OR gate and the Carry-

out (Cout) is the result of the AND gate. One major disadvantage of the Half Adder circuit when used as

a binary adder, is that there is no provision for a “Carry-in” from the previous circuit when adding together

multiple data bits.

For example, suppose we want to add together two 8-bit bytes of data, any resulting carry bit would need

to be able to “ripple” or move across the bit patterns starting from the least significant bit (LSB). The

most complicated operation the half adder can do is “1 + 1″ but as the half adder has no carry input the

resultant added value would be incorrect. One simple way to overcome this problem is to use a Full

Adder type binary adder circuit.

Complete the following

1. Realize half adder using basic gates

2. Realize half adder using Exor Gates

3. Realize Half adder using universal gates (Both NAND and NOR)

The Full Adder Circuit

The main difference between the Full Adder and the previous seen Half Adder is that a full adder has

three inputs, the same two single bit binary inputs A and B as before plus an additional Carry-In(C-in)

input as shown below.




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Full Adder with Carry-In

Symbol Truth Table<

A B C-in Sum C-out

0 0 0 0 0

0 1 0 1 0

1 0 0 1 0

1 1 0 0 1

0 0 1 1 0

0 1 1 0 1

1 0 1 0 1

1 1 1 1 1

Boolean Expression: Sum = A ⊕ B ⊕ C-in

The 1-bit Full Adder circuit above is basically two half adders connected together and consists of

three Ex-OR gates, two AND gates and an OR gate, six logic gates in total. The truth table for the full

adder includes an additional column to take into account the Carry-in input as well as the summed output

and carry-output.

Complete the following

1. Realize half adder using basic gates

2. Realize half adder using Exor Gates

3. Realize Half adder using universal gates (Both NAND and NOR)


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Unit 1 Notes of Basic electronics Jan 2015 - NMIT Bangalore

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