number system understand

Prepared By- Prof. M. P. Raj
AIT 124
COMPUTER ORGANIZATION AND ARCHITECTURE
3(3+0)

UNIT I
Introduction to COA, Basic Computer Model and different units of Computer, Basic Working Principle of a Computer Number System Representation: Representation of Decimal, Binary,
Octal and Hexadecimal Number; Conversation of Numbers, Representation of Unsigned & Sign integer, Representation of real number, Representation of character.
Architecture is those attributes visible to the programmer – Instruction set, number of bits used for data
representation, I/O mechanisms, addressing techniques. – e. g. Is there a multiply instruction?
Computer architecture deals with the functional behavior of a computer system as viewed by a programmer (like the
size of a data type – 32 bits to an integer).
Organization is how features are implemented – Control signals, interfaces, memory technology. – e. g. Is there a
hardware multiply unit or is it done by repeated addition?
Computer organization deals with structural relationships that are not visible to the programmer (like clock frequency
or the size of the physical memory). There is a concept of levels in computer architecture. The basic idea is that there
are many levels at which a computer can be considered, from the highest level, where the user is running programs,
to the lowest level, consisting of transistors and wires.
Levels of Machines
There are a number of levels in a computer, from the user level down to the transistor level. Progressing from the top
level downward, the levels become less abstract as more of the internal structure of the computer becomes visible.
The von Neumann Model
The von Neumann model consists of ﬁve major components: (1) input unit; (2) output unit; (3) arithmetic logic unit;
(4) memory unit; (5) control unit.

The System Bus Model
A reﬁnement of the von Neumann model, the system bus model has a CPU (ALU and control), memory, and an
input/output unit.
Communication among components is handled by a shared pathway called the system bus, which is made up of the
data bus, the address bus, and the control bus. There is also a power bus, and some architectures may also have a
separate I/O bus.

The Motherboard
Moore’s Law
Computing power doubles every 18 months for the same price.
Project planning needs to take this observation seriously: an architectural innovation that is being developed for a
projected beneﬁt that quadruples performance in three years may no longer be relevant: the architectures that exist
by then may already offer quadrupled performance and may look entirely different from what the innovation needs
to be effective.

Abstraction Layers
A typical vision of a computer architecture as a series of abstraction layers: hardware, firmware, assembler, kernel,
operating system and applications.
Firmware
In computing, firmware is software that is embedded in a hardware device. It is often provided on flash ROMs or as a
binary image file that can be uploaded onto existing hardware by a user. Firmware is defined as:
The computer program in a read-only memory (ROM) integrated circuit (a hardware part number or other
configuration identifier is usually used to represent the software); • the erasable programmable read-only memory
(EPROM) chip, whose program may be modified by special external hardware, but not by [a general purpose]
application program. • the electrically erasable programmable read-only memory (EEPROM) chip, whose program
may be modified by special electrical external hardware (not the usual optical light), but not by [a general purpose]
application program.
Assembler
Assembler creates object code by translating assembly instruction mnemonics into opcodes, and by resolving
symbolic names for memory locations and other entities. The use of symbolic references is a key feature of
assemblers, saving tedious calculations and manual address updates after program modifications.
Kernel
In computing, the kernel is the central component of most computer operating systems (OSs). Its responsibilities
include managing the system's resources and the communication between hardware and software components. As a
basic component of an operating system, a kernel provides the lowest-level abstraction layer for the resources
(especially memory, processor and I/O devices) that applications must control to perform their function. It typically
makes these facilities available to application processes through inter-process communication mechanisms and
system calls.
These tasks are done differently by different kernels, depending on their design and implementation. While
monolithic kernels will try to achieve these goals by executing all the code in the same address space to increase the
performance of the system, micro kernels run most of their services in user space, aiming to improve maintainability
and modularity of the code base. A range of possibilities exists between these two extremes.

Operating System
An operating system (OS) is a computer program that manages the hardware and software resources of a computer.
An operating system performs basic tasks such as controlling and allocating memory, prioritizing system requests,
controlling input and output devices, facilitating networking, and managing files. It also may provide a graphical user
interface for higher level functions. It forms a platform for other software.
Application software is a subclass of computer software that employs the capabilities of a computer directly to a task
that the user wishes to perform.
NUMBERING SYSTEMS AND COMPUTERS
We humans use a decimal, or base-10, numbering system, presumably because people have 10 fingers. If we had
three fingers and a thumb on each hand, as does the Extra Terrestrial (E.T.), from the classic movie, and Homer
Simpson of the Simpson’s, then in all probability we would be using the octal numbering system, which has a
base of 8.
Early computers were designed around the decimal numbering system. This approach made the creation of
computer logic capabilities unnecessarily complex and did not make efficient use of resources. (For example, 10
vacuum tubes were needed to represent one decimal digit.) In 1945, as computer pioneers were struggling to
improve this cumbersome approach, John von Neumann suggested that the numbering system used by
computers should take advantage of the physical characteristics of electronic circuitry. To deal with the basic
electronic states of on and off, von Neumann suggested using the binary numbering system. His insight has
vastly simplified the way computers handle data.
Computers operate in binary and communicate to us in decimal. A special program translates decimal into
binary on input, and binary into decimal on output. Under normal circumstances, a programmer would see only
decimal input and output. On occasion, though, he or she must deal with long and confusing strings of 1s and 0s
that represent the content of RAM, the computer’s memory. Occasionally a programmer or computer engineer
takes a snapshot of the contents of RAM (on-bits and off-bits) at a given moment in time. To reduce at least part
of the confusion of seeing only 1s and 0s on the output, the hexadecimal (base-16) numbering system is used as
a shorthand to display the binary contents of both RAM and secondary storage, such as disk.
The decimal equivalents for binary, decimal, and hexadecimal numbers are shown in Figure 1. We know that in
decimal, any number greater than 9 is represented by a sequence of digits. When you count in decimal, you
"carry" to the next position in groups of 10. As you examine Figure 1, notice that you carry in groups of 2 in
binary and in groups of 16 in hexadecimal. Also note that any combination of four binary digits can be
represented by one "hex" digit.

FIGURE 1 Numbering System Equivalence Table
The hexadecimal numbering system is used only for the convenience of the programmer or computer scientist,
or computer engineer when reading and reviewing the binary display of memory. Computers do not operate or
process in hex. During the 1960s and early 1970s, programmers often had to examine the contents of RAM to
debug their programs (that is, to eliminate program errors). Today's programming languages have user friendly
diagnostics (error messages) and computer-assisted tools that help programmers during program development.
These diagnostics and development aids have minimized the need for applications programmers to convert
binary and hexadecimal numbers into their more familiar decimal equivalents. However, if you become familiar
with these numbering systems, you should achieve a better overall understanding of computers. And, someday
you may need to read hex to decode an error message or set the jumpers on an expansion card.
PRINCIPLES OF NUMBERING SYSTEMS
Binary
The binary, or base-2, numbering system is based on the same principles as the decimal, or base-10, numbering system,
with which we are already familiar. The only difference between the two numbering systems is that binary uses only two
digits, 0 and 1, and the decimal numbering system uses 10 digits, 0 through 9. The equivalents for binary, decimal, and
hexadecimal numbers are shown in Figure 1.
The value of a given digit is determined by its relative position in a sequence of digits. Consider the example in
Figure 2. If we want to write the number 124 in decimal, the interpretation is almost automatic because of our
familiarity with the decimal numbering system.

FIGURE 2 Numbering System Fundamentals
FIGURE 3 Representing a Binary Number
To represent 124, we would follow the same thought process as we would in decimal (see Figure 2), but this time we
have only two digits (0 and 1). For ease of understanding, the arithmetic is done in decimal.
Hexadecimal
Perhaps the biggest drawback to using the binary numbering system for computer operations is that programmers may have
to deal with long and confusing strings of 1s and 0s. To reduce the confusion, the hexadecimal, or base-16, numbering
system is used as shorthand to display the binary contents of primary and secondary storage.

Notice that the bases of the binary and hexadecimal numbering systems are multiples of 2: 2 and 24
, respectively. Because
of this, there is a convenient relationship between these numbering systems. The numbering-system equivalence table
shown in Figure 1 illustrates that a single hexadecimal digit represents four binary digits (01112 = 716, 11012 = D16, 10102 =
A16 where subscripts are used to indicate the base of the numbering system). Notice that in hexadecimal, or "hex," letters
are used to represent the six higher order digits.
Two hexadecimal digits can be used to represent the eight-bit byte of an EBCDIC equals sign (=) (011111102 is
the same as 7E16). Figure 4 illustrates how a string of EBCDIC bits can be reduced to a more recognizable form
using hexadecimal.
FIGURE 4 System Expressed in Different Ways
The word System is shown as it would appear in input/output, internal binary notation, and hexadecimal notation.
We will now examine how to convert one number in a numbering system to an equivalent number in another numbering
system. For example, there are occasions when we might wish to convert a hexadecimal number into its binary equivalent.
We shall also learn the fundamentals of numbering-system arithmetic.
CONVERTING NUMBERS FROM ONE BASE INTO ANOTHER
Decimal to Binary or Hexadecimal
A decimal number can be converted easily into an equivalent number of any base by the use of the division/remainder
technique. This two-step technique is illustrated in Figure 5. Follow these steps to convert decimal to binary.
FIGURE 5 Converting a Decimal Number into Its Binary Equivalent
Use the two step division/remainder technique to convert a decimal number into an equivalent number of any base.
Step 1. Divide the number (19, in this example) repeatedly by 2, and record the remainder of each division. In
the first division, 2 goes into 19 nine times with a remainder of 1. The remainder is always one of the
binary digits--0 or 1. In the last division you divide 1 by the base (2) and the remainder is 1.

Step 2. Rotate the remainders as shown in Figure 5; the result (10011) is the binary equivalent of a decimal 19.
Figure 6 illustrates how the same division/remainder technique is used to convert a decimal 453 into its hexadecimal
equivalent (1C5). In a decimal-to-hex conversion, the remainder is always one of the 16 hex digits.
FIGURE 6 Converting a Decimal Number into Its Hexadecimal Equivalent
The two-step division/remainder technique is used to convert a decimal number to its hex equivalent.
Binary to Decimal and Hexadecimal
To convert from binary to decimal, multiply the 1s in a binary number by their position values, then sum the products (see
Figure 7). In Figure 7, for example, binary 11010 is converted into its decimal equivalent (26).
FIGURE 7 Converting a Binary Number into Its Decimal Equivalent
Multiply the 1s in a binary number by their position values.

The easiest conversion is binary to hex. To convert binary to hex, simply begin with the 1s position on the right and
segment the binary number into groups of four digits each (see Figure 8). Refer to the equivalence table in Figure 2, and
assign each group of four binary digits a hex equivalent. Combine your result, and the conversion is complete.
Hexadecimal to Binary
To convert hex numbers into binary, perform the grouping procedure for converting binary to hex in reverse (see Figure 8).
FIGURE 8 Converting a Binary Number into Its Hexadecimal Equivalent
Place the binary digits in groups of four, then convert the binary number directly to hexadecimal.
Hexadecimal to Decimal
Use the same procedure as that used for binary-to-decimal conversions (see Figure 7) to convert hex to decimal. Figure 9
demonstrates the conversion of a hex 3E7 into its decimal equivalent of 999.
FIGURE 9 Converting a Hexadecimal Number into Its Decimal Equivalent
Multiply the digits in a hexadecimal number by their position values.
ARITHMETIC IN BINARY AND HEXADECIMAL
The essentials of decimal arithmetic operations have been drilled into us so that we do addition and subtraction almost by
instinct. We do binary arithmetic, as well as that of other numbering systems, in the same way that we do decimal
arithmetic. The only difference is that we have fewer (binary) or more (hexadecimal) digits to use. Figure 10 illustrates and
compares addition and subtraction in decimal with that in binary and hex. Notice in Figure 10 that you carry to and borrow
from adjacent positions, just as you do in decimal arithmetic.

FIGURE 10 Binary, Decimal, and Hexadecimal Arithmetic Comparison
As you can see, the only difference in doing arithmetic in the various numbering systems is the number of digits used.

Examples of Binary Arithmetic
Example 1,2 Addition
Example 3,4 Subtraction

Examples of Hexadecimal Arithmetic
Example 1 Addition
Example 3 Subtraction

Binary subtraction using 1’s complement
Example 1
1101 1111
- 111  1’s complement - 0111
110 1000
+ 1101
10101 here extra digit 1 indicates that answer is +ve and answer will be obtained by add 1
to the result. So 0101 + 1 = 110 i.e +ve 110
Example 2
1010 11111
- 10101  1’s complement - 10101
- 1011 01010
+ 01010
010100 here extra digit is 0 that indicates that answer is –ve and in the form of 1’s
complement of the result. So 11111 – 10100 = 01011 i.e. –ve 1011
Same way subtraction can be done using 7,9,15 ‘s complement
Binary subtraction using 2’s complement
Example 1
1101 1111
- 111  2’s complement - 0111
110 1000
+ 1
1001
+ 1101
10110 here extra digit 1 indicates that answer is +ve and answer is the result. So 0110 i.e
+ve 110
Example 2
1010 11111
- 10101  1’s complement - 10101
- 1011 01010
+ 1
01011
+ 01010
010101 here extra digit is 0 that indicates that answer is –ve and in the form of 2’s
complement of the result. So 11111
– 10101
01010
+ 1
01011 i.e. –ve 1011
Same way subtraction can be done using 8, 10, 16‘s complement
IMP Rules:
Complement: n = 2, 8, 10, 16
1. For (n-1)’s complement subtract each digit of a number from (n-1)
2. For n’s complement subtract each digit of a number from (n-1) and add 1 to it.
For Conversion:

1. Binary, Octal, Hexadecimal  Decimal:
digit X (base/radix) digit position starting from zero
. digit X (base/radix) digit position starting from -1
2. Decimal  Binary, Octal, Hexadecimal :
For integer portion divide by base and write remainder in reverse order i.e. last reminder digit as first digit of a
number.
3. Octal, Hexadecimal  Binary:
Create pair of three digit for octal and write its equivalent octal digit.
Create pair of three digit for hexadecimal and write its equivalent hexadecimal digit.
For subtraction A- B: n = 2, 8, 10, 16
4. For 1, 7, 9 , 15’s complement
a. Take (n-1)’s complement of B
b. Add it to A
i. If result has extra bit as ‘1’ then answer is +ve and by adding ‘1’ to result we get exact answer.
ii. If result has extra bit as ‘0’ then answer is –ve and by taking (n-1)’s complement of result we get
exact answer.
5. For 2,8,10,16’s complement
a. Take n’s complement of B
b. Add it to A
i. If result has extra bit as ‘1’ then answer is +ve and by adding ‘1’ to result we get exact answer.
ii. If result has extra bit as ‘0’ then answer is –ve and by taking n’s complement of result we get
exact answer.
Self-Check
1. The hex numbering system has a base of ________, and the binary numbering system has a base of ________.
2. The value of a particular digit in a number is determined by its relative position in a sequence of digits. (T/F)
3. A single hexadecimal digit can represent how many binary digits: (a) two, (b) three, or (c) four?
4. The bases of the binary and decimal numbering systems are multiples of 2. (T/F)
5. The binary equivalent of a decimal 255 is ________.
6. The binary equivalent of a hexadecimal 1C is ________.
7. The decimal equivalent of a hexadecimal 1B6 is ________.
8. The hexadecimal equivalent of a decimal 129 is ________.
9. The decimal equivalent of a binary 110101 is ________.
10. The hexadecimal equivalent of a binary 1001 is ________.
11. The binary equivalent of a decimal 28 is ________.
12. The binary equivalent of a hexadecimal 35 is ________.
13. The decimal equivalent of a hexadecimal 7 is ________.
14. The hexadecimal equivalent of a decimal 49 is ________.
15. The decimal equivalent of a binary 110110110 is ________.
16. The hexadecimal equivalent of a binary 1110 is ________.
17. The result of 1012 + 112 is ________ (in binary).
18. The result of A116 + BC16 + 1016 is ________ (in hexadecimal).
19. The result of 6010 + F116 - 10010012 is ________ (in decimal).
20. The result of 112 + 278 + 9310 - B16 is ________ (in decimal).
Self-test answers. 1. 16, 2. 2. T. 3. 4. 4. F. 5. 11111111. 6. 11100. 7. 438. 8. 81. 9. 53. 10. 9. 11. 11100. 12. 110101. 13. 7.
14. 31. 15. 438. 16. E. 17. 10002. 18. 16D16. 19. 22810. 20. 10810.

UNIT II Arithmetic and Logic Unit Logic Gates: AND, OR, NOT, NAND, NOR, XOR, Exclusive, NOR gates Boolean
algebra: Introduction, Explanation of Boolean function, Describe truth table, Simplified Boolean function using
postulates and draw logical diagram of simplified function, Simplified Boolean function using karnaugh map method.
Sequential And Combinational Circuits: Introduction to combinational circuit and sequential circuit, half adder, full
adder, multiplexer, demultiplexer, encoder, decoder
Basic Logic Design
Gate: It is a logic circuit with one or more input signal, but only one output signal.
Logic Circuit: A circuit whose input and output signals are two state either low or high voltage.
OR, AND , NOT are basic logic circuits.
Truth Table: A table that shows all input and output possibilities for a logic circuits is known as truth table.
OR Gate: It has two or more input signals but only one output signal. If any input signal is high then output signal is
high
A B A+B
0 0 0
0 1 1
1 0 1
1 1 1
Use of OR Gate: Octal to Binary encoder
The circuit will convert an octal number into its equivalent binary number. When push button ‘5’ is pressed, the y2
and y0 OR gate have high input. Therefore output word is Y2 Y1Y0 = 101, which is binary of the octal number.
And Gate:
It has two or more input signals but only one output signal. If all input signals are high then output signal will be high
i.e. if any input signal is low then output will be low.
A B A.B
0 0 0
0 1 0
1 0 0
1 1 1
Use of AND gate

UNIT III Memory Concept of Main Memory, Introduction to Flip Flop and explain SR, Clocked SR, D, JK, T flip flops

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