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Unit 2: Number System and Conversion & Boolean Logic - Coggle Diagram
Unit 2: Number System and Conversion & Boolean Logic
Computer Number System
Types
Non-Positional Number System
Each symbol has the same value regardless of its position (e.g., Roman Numerals).
Positional Number System
The value of a digit is based on its position, its digit value, and the base (radix). This is the system used by computers.
Definition
An organized and systematic way of representing numbers in computer calculations.
Number Systems Details
Radix / Base
Definition
The number of different symbols used in a number system.
Decimal (Base 10): Uses digits 0-9.
Binary (Base 2): Uses only bits 0 and 1. This is the fundamental system for computers due to their ON/OFF circuits.
Hexadecimal (Base 16): Uses digits 0-9 and letters A-F. Used in memory management.
Octal (Base 8): Uses digits 0-7. Often used as a shorthand for binary.
Number Conversion Rules
Binary to Octal and Vice-versa
Use 3-bit groupings (421 rule).
Binary to Hexa-Decimal and Vice-versa
Use 4-bit groupings (8421 rule).
Others to Decimal
Use the multiplication method, multiplying each digit by the base raised to the power of its position.
Octal to Hexa-decimal and Vice-versa:
First convert to binary (3-bit) and then convert to the new base (4-bit).
Decimal to others
Use the division method by the new base.
Boolean Algebra
Truth Table
A table that shows all possible outcomes for a given logical expression.
Logic Gates
The fundamental building blocks of digital circuits.
Types
NAND Gate
The inverse of an AND gate.
NOR Gate
The inverse of an OR gate.
NOT Gate (Inverter)
Inverts the input (1 becomes 0, and 0 becomes 1).
Exclusive (X-OR) Gate
Output is 1 if inputs are different.
OR Gate
Output is 1 if at least one input is 1.
Exclusive-NOR (X-NOR) Gate
Output is 1 if inputs are the same.
AND Gate
Output is 1 only if all inputs are 1.
Definition
A branch of mathematics dealing with binary variables (true/false or 1/0).
Boolean Laws and Principles
Principle of Duality
Replace AND with OR, OR with AND, and 1 with 0, and 0 with 1.
Other Boolean Laws
Includes the Associative and Distributive laws, used to simplify complex logical expressions.
De Morgan's Laws:
(A+B)' = A',B'
(A.B)' = A' + B'
Applications of Boolean Algebra and Gates
Boolean Algebra
Used by hardware and software designers for tasks like logical decision-making and error checking.
Logic Gates
The physical components used to construct digital electronic circuits in computers.