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Text Book:

Peter Norton (2011), Introduction to Computers, 7 /e, McGraw-Hill

Reference Book:

Gary B (2012), Discovering Computers, 1/e, South Western

Deborah (2013), Understanding Computers, 14/e, Cengage Learning

June P & Dan O (2014), New Perspective on Computer, 16/e




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Presented by: Asma Khan



Learning Outcome

◻ What is a Number System

◻ Decimal & Binary Number System

◻ How Data Represented in Computer

◻ Binary & Switches, Bits & Bytes ◻ Binary Number System Explained

◻ Writing Binary Numbers, Expanded Notation

◻ Conversions

Binary to Decimal Decimal to Binary ◻ Exercise


What is a Number System ?

◻ A number system is a way to represent numbers

◻ Any system using a range of digits organized in a series of columns or "places" that represents a specific quantity.

◻ The most common numbering systems are decimal, binary, octal, and hexadecimal.


Decimal Number System : Base 10

◻ The decimal numeral system (also called base 10 or occasionally denary) has Ten as its base. It is the numerical base most widely used by modern civilizations.

◻ The decimal number system consists of ten

single- digit numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

(28)10 = (011100)2 48-D = 110000-B


Binary Number System : Base 2

◻ Also called as base-2 number system. This system uses

only 0s and 1s to represent letters, numbers and other characters and it is used by almost all modern computers.

◻ Numbering system that represents numeric values using

two unique digits (0 and 1).

(1100)2 = (12)10 (0101)2 = (5)10 (1010)2 = (10)10 (1111)2 = (15)10


How Data Represented in Computer

Computers Are Electronic Machines

◻ The computer uses electricity (CPU, RAM, MB etc) not mechanical parts

◻ Electricity moves very fast through wires

◻ Electrical parts fail less frequently than mechanical parts

◻ Internal data processing and storage is electronic


How Data Represented in Computer

◻ Electricity can flow through switches

◻ If the switch is closed, the electricity flows

◻ If the switch is open, the electricity does not flow

◻ We need a way to represent the data in switches

◻ Computers do this representation using a binary coding system


How Data Represented in Computer

Binary and Switches. Binary is a mathematical number system, a way of counting

◻ The computer has switches to represent data

◻ Switches have only two states: ON and OFF

◻ Binary has two digits to do the counting: two states of a switch (0 = OFF, 1 = ON).


How Data Represented in Computer

Bits and Bytes One binary digit (0 or 1) is referred to as a bit, which is short for binary digit. Thus, one bit can be implemented by one switch

Implementing a Byte


How Data Represented in Computer

Representing Data In Bytes

A single byte can represent many different kinds of data. What data it actually represents depends on how the computer uses the byte.

◻ For instance, the byte: 01000011

Can represent the integer 67 (to ALU)

The character ‘C’ (to Monitor)

The 67th decibel level for a part of a sound (to Speaker)

The 67th level of darkness for a dot in a picture (to Graphics)


How Data Represented in Computer

Characters. The computer also uses a single byte to represent a single character

◻ American Standard Code for Information Interchange

ASCII-8, also called extended ASCII

◻ Uses 8 bits per character and can represent 256

different characters

ASCII representation has been adopted as a

standard particularly minicomputers and microcomputers


Binary Number System

Binary Numbers

◻ A binary number is a sequence of the digits 0 and 1, such as


◻ The number shown has no fractional part and so is called a

binary integer.

◻ A binary number having a fractional part contains a binary point (also called a radix point), as in the number


Binary Number System

Base or Radix

The base of a number system (also called the radix) is equal to the number of digits used in the system.

Bits, Bytes, and Words

Each of the digits is called a bit, from binary digit. A byte is a group of 8 bits A word is the largest string of bits that a computer can handle in one operation The number of bits in a word is called the word length.

Different computers have different word lengths, with 8, 16, or 32 bits Half a byte (4 bits) is called a Nibble.

A kilobyte (Kbyte or KB) is 1024 (2100) bytes

A megabyte (MByte or MB) is 1,048,575 (2200) bytes.


Binary Number System

Writing Binary Numbers

A binary number is sometimes written with a subscript 2 when there is a chance that the binary number would otherwise be mistaken for a decimal number.

Example: The binary number 110 could easily be mistaken for the decimal number 110, unless we write it

■ 110


Similarly, a decimal number that may be mistaken for binary is often written with a subscript 10, as in

■ 110



Binary Number System

◻ The number 100100010100.001001 is easier to read when written as

1001 0001 0100.0010 01

◻ The leftmost bit in a binary number is called the high-order or Most significant bit (MSB).

◻ The bit at the extreme right of the number is the low-order or Least significant bit (LSB).


Binary Number System

Place Value

◻ A positional number system is one in which the position of a digit determines its value,

◻ And each position in a number has a place value equal to the base of the number system raised to the position number.


Converting Binary Numbers to Decimal

◻ The place values in the binary number system are


Binary Number System

◻ Expanded Notation

◻ Example :

◻ The binary number 1011 can be expressed as


Converting Binary Numbers to Decimal

◻ To convert a binary number to decimal, simply write the binary number in expanded notation

(omitting those where the bit is 0), and add the

resulting values.

◻ Example : Convert the binary number

1001.011to decimal.


Converting Binary Numbers to Decimal


Converting Decimal Integers to Binary

◻ To convert a decimal integer to binary

◻ First divide it by 2, obtaining a Quotient and a


◻ Write down the remainder and divide the quotient by 2, getting a new quotient and remainder

◻ Repeat this process until the quotient is zero.


Converting Decimal Integers to Binary

◻ Example : Convert the decimal integer 59 to binary.

Solution: We divide 59 by 2, getting a quotient of 29 and a remainder of 1. Then

Dividing 29 by 2 gives a quotient of 14 and a remainder of 1.

These calculations, can be arranged in a table, as follows:


Converting Decimal Integers to Binary

◻ Our binary number then consists of the digits in the remainders

◻ Thus



Converting Decimal Fraction to Binary

◻ To convert a decimal fraction to binary, we first multiply it by 2, remove the integer part of the product, and multiply by 2 again.

◻ We then repeat the procedure

◻ Example : Convert the decimal fraction 0.546875 to binary.


Converting Decimal Fraction to Binary

Solution: We multiply the given

number by 2, getting 1.09375.

◻ We remove the integer part, 1, leaving 0.09375, which we again multiply by 2, getting 0.1875.

◻ We repeat the computation until we get a product that has a fractional part of zero (Not possible always), as shown on R.H.S, so

0.54687510 = 0.1000 112


Converting Decimal to Binary

◻ To convert a decimal number having both an integer part

and a fractional part to binary, convert each part

separately, and then combine

◻ Example : Convert the number 59.546875

10 to binary.

Solution: From the preceding examples,

◻ 5910 = 1110112

◻ 0.546875

10 = 0.1000 112

◻ So, 59.546875 = 11 1011.1000 11




◻ Convert each binary number to decimal.

◻ Convert each decimal number to binary.

1001 0110 1111

0111.111 1100.001 1101 0010.1011

21 64 93


Any Questions !!!






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