Digital Electronics & Computer Organization Unit 1

Study Notes

A binary number is expressed in base 2, using two symbols: 0 and 1.
Each digit in a binary number is called a "bit."
Example of binary representation: Decimal 4 is (100)2 in binary.
To convert decimal to binary, repeatedly divide by 2 and record remainders.
For the number 4:
- 4/2 = 2 (remainder 0)
- 2/2 = 1 (remainder 0)
- 1/2 = 0 (remainder 1)
Write the remainders in reverse order: LSB (0) to MSB (1).
The number 4 in binary has 3 bits (100), comprised of 2 zeroes and 1 one.

A "bit" represents a single binary digit.
Examples of bit counts in binary:
- 10101 is a five-bit binary number.
- 101 is a three-bit binary number.
- 100001 is a six-bit binary number.

Binary numbers consist only of 0s and 1s.
Represented using base-2.
A list of binary equivalents for decimal numbers 1 to 30 is provided, highlighting their patterns.

Forms the foundation of electronic systems like computers and mobile devices.
Operates on binary values (0s and 1s).
Involves operations such as addition, subtraction, multiplication, and division of binary digits.
Key logic gate functions include AND, OR, and NOT.

A specific algebraic structure focusing on binary conditions of TRUE and FALSE.
Utilizes logical operators:
- AND
- OR
- NOT
Boolean logic is crucial for the functioning of electronic systems, where the equivalent of binary logic is employed.

Both logic systems handle binary values.
TRUE equates to '1' and FALSE equates to '0' in binary contexts, reinforcing their relationship in digital electronics.

Basic components in digital circuits that execute logical operations.
AND Gate: Outputs true (1) only if all inputs are true.
OR Gate: Outputs true if at least one input is true.
NOT Gate: Inverts input; outputs true if input is false and vice versa.
NAND Gate: Outputs false (0) only if all inputs are true; combines AND and NOT functions.
NOR Gate: Outputs true if all inputs are false; combines OR and NOT functions.
XOR Gate: Outputs true if the input values differ; known as exclusive OR.
XNOR Gate: Outputs true if the input values are the same; known as exclusive NOR.
Each logic gate has a specific symbol for representation in circuit diagrams.
Truth tables illustrate the output results of logic gates for all possible input combinations.

A mathematical framework essential for analyzing and simplifying digital logic circuits.
Basic Operators:
- AND (·): Represents multiplication; true if both A and B are true.
- OR (+): Represents addition; true if at least one of A or B is true.
- NOT (¬): Represents inversion; true if A is false.
Laws of Boolean Algebra:
- Identity Law: A + 0 = A and A · 1 = A confirm the role of identity elements.
- Null Law: A + 1 = 1 indicates dominance of true; A · 0 = 0 indicates nullifying effect.
- Idempotent Law: Simplifies expressions; A + A = A and A · A = A confirm redundancy.
- Complement Law: A + ¬A = 1 asserts totality of possibilities; A · ¬A = 0 guarantees contradiction.
- Distributive Law: A · (B + C) = (A · B) + (A · C) allows expansion of expressions.
De Morgan's Theorems:
- ¬(A · B) = ¬A + ¬B indicates inversion of AND results.
- ¬(A + B) = ¬A · ¬B reveals inversion of OR results.
Applications of Boolean algebra include circuit design, logical expression simplification, and digital circuit analysis.