Number System. Binary Number System (Base 2) octal numeral system(base-8 )

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• A computer can understand only the “on” and “off” state of a switch. • These two states are represented by 1 and 0.

Binary Number System (Base 2)

Decimal Number System (Base 10)

• In this number system, the digits 0 to 9 represents numbers.

octal numeral system(base-8 )

• The octal numeral system, uses the digits 0 to 7.

Number System

The number system is a way to represent or express numbers. in the context of computers, we define other types of number systems. They are:

•The decimal number system •The binary number system •The octal number system and •The hexadecimal number system

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DECIMAL to BINARY conversion

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• The following steps are needed to convert from Octal to Binary:

• Convert each octal digit to its 3-digit binary representation. Each of the digits must be

treated as a decimal value.

• Combine these binary representations to form a single binary number.

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• If we have an Octal value of 305305₈, its binary representation would be:

• = 305₈

• = (011 000 101)₂ • = (011000101)₂

• Hence, we can form the following table for octal to binary conversion, where each digit is

represented with its corresponding binary representation.

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• What is Binary Addition

• The binary addition operation works similarly to the base 10 decimal system, except that it is a base 2 system.

• The binary system consists of only two digits, 1 and 0. Most of the functionalities of the computer system use the binary number system.

• The binary code uses the digits 1’s and 0’s to make certain processes turn off or on.

• Rules for Binary addition

• Binary addition is much easier than the decimal addition when you remember the following tricks or rules. The four rules of binary addition are:

• 0 + 0 = 0 • 0 + 1 = 1 • 1 + 0 = 1 • 1 + 1 =10

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• Example: 101 + 101

How To Do Binary Addition?

Tips:

When you cross-check the binary value with the decimal value, the resultant value should be the same.

The binary value 101 is equal to the decimal value 5

So, 5 + 5 = 10

The decimal number 10 is equal to the binary number 1010.

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BOOLEAN ALGEBRA

Objective: To represent logical operation in Boolean algebra using truth values. BINARY DECISION:

The decision which results into either YES (TRUE) or NO(FALSE) is called a binary decision. TRUTH VALUES:

Values true and false are called Truth values. LOGICAL STATEMENTS:

The sentences which can be determined to be true or false are called logical statements or truth functions.

LOGICAL VARIABLES:

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LOGICAL FUNCTIONS

• Logical Functions:

Algebraic variables like a, b,c or x, y, z ..,are combined with the help of mathematical operations like +, -, x, / to form algebraic expressions.

Ex:

• Similarly logic statements or truth functions are combined with the help of Logical Operators like, AND, OR, NOT to form

compound statements or logical functions.

3 x A + 4 x B – 7 x D =( 11 x Z) / 2 x Y

X NOT Y OR Z Y NOT X OR Z

Digital systems are said to be constructed by using logic gates. These gates are the AND, OR, NOT. LOGICAL OPERATORS

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OR OPERATOR

• It denotes logical addition.

• The symbol is +.

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AND OPERATOR

• It denotes logical Multiplication.

• The symbol is (

.)

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NOT operator

The Logic NOT Function output is true when

its single input is false, and false when its

single input is true

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Objective: To evaluate Boolean Expressions using Truth Table

Give truth table for the Boolean expressions 1. (X + Y )+ (Y+Z)

A truth table of n input variables will have 2n

input combinations, 2n

rows.

Eg: _{if 3 variables, then}

how many rows?

X Y Z X+Y Y+Z (X + Y )+ (Y+Z)

0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 1 1 1 0 1 1 1 1 1 1 0 0 1 0 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1

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Give truth table for the Boolean expressions

1. X + YZ

X Y Z 0 0 0 0 1 1 1 1

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LOGIC GATES

Objective: To analyze the application of Boolean

algebra to computer electronics.

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De Morgan’s Theorem

De Morgan’s theorems allow the input variables to be negated and converted from one form of a Boolean function into an opposite form

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Logic circuits

1. Write the Boolean expression for the given circuit:

A(B+C)

((U+V’)(U+W)).(V+W’)

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Logic circuits

2. Draw a circuit for the given Boolean expression:

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NOT + AND= NAND gate

• The NAND gate operates as an AND gate followed by a NOT gate.

• It acts in the manner of the logical operation "and" followed by negation. • The output is "false" if both inputs are "true." Otherwise, the output is "true."

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NOT + OR=NOR gate

• The NOR gate is a combination OR gate followed by an inverter.

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Memory Unit

Memory unit is the amount of data that can be stored in the storage unit. This storage capacity is expressed in terms of Bytes.

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S.No. Unit & Description 1 Kilobyte (KB)_{1 KB = 1024 Bytes} 2 Megabyte (MB)_{1 MB = 1024 KB} 3 GigaByte (GB)_{1 GB = 1024 MB} 4 TeraByte (TB)_{1 TB = 1024 GB} 5 PetaByte (PB)_{1 PB = 1024 TB}