Computer logic and organization exam II

Binary (Base 2)

Uses digits 0 and 1. Example: 1011₂ = 11₁₀.

Decimal (Base 10)

Uses digits 0-9. Example: 245₁₀ = 245.

Hexadecimal (Base 16)

Uses digits 0-9 and A-F. Example: 3F₁₆ = 63₁₀.

Octal (Base 8)

Uses digits 0-7. Example: 57₈ = 47₁₀.

Binary to Decimal Conversion

Multiply each bit by 2^position, starting from the right. Example: 1101₂ = (1×2³) + (1×2²) + (0×2¹) + (1×2⁰) = 13₁₀.

Decimal to Binary Conversion

Divide the decimal number by 2, recording the remainders. Example: 13₁₀ → 1101₂.

Binary to Hexadecimal Conversion

Group bits in sets of 4 and convert. Example: 10101100₂ → A C₁₆.

Binary to Octal Conversion

Group bits in sets of 3 and convert. Example: 10101100₂ → 5 3 4₈.

Signed Magnitude Representation

First bit is the sign (0 = positive, 1 = negative). Example: +5 → 00000101, -5 → 10000101.

Two's Complement Representation

Write positive binary equivalent, invert all bits, and add 1 to the result. Example: -5 in two's complement: 5 = 00000101, Invert: 11111010, Add 1: 11111011.

AND Gate

1 if both A and B are 1.

OR Gate

1 if at least one input is 1.

NOT Gate

Flips input (0 ↔ 1).

XOR Gate

1 if inputs differ.

NAND Gate

0 only if both are 1.

NOR Gate

1 only if both are 0.

2-Variable K-map

AB \ CD with values for combinations of inputs.

Steps to Simplify Boolean Expressions

Group adjacent 1s in powers of 2 (1, 2, 4, 8...) and derive a simplified Boolean expression.

SR Latch (Set-Reset Latch)

Inputs: S (Set), R (Reset); Outputs: Q (Stored value), Q' (Complement).

Half-Adder

Adds two single-bit numbers with outputs: Sum (S) & Carry (C).

Full-Adder

Adds two bits plus a carry-in with outputs: Sum (S) & Carry-out (Cout).

Boolean Expression for Half-Adder

Sum = A ⊕ B; Carry = A * B.

Boolean Expression for Full-Adder

Sum = A ⊕ B ⊕ Cin; Cout = (A * B) + (Cin * (A ⊕ B)).