Bases, conversions and negative numbers

Conversions

• ›binary to decimal — add the place values: (101101)₂ = 32 + 8 + 4 + 1 = 45
• ›decimal to binary — divide by 2 and read the remainders upwards; for a fraction, multiply by 2 and read the integer parts downwards: (0.625)₁₀ = (0.101)₂
• ›binary to octal or hex — group bits in threes or fours from the binary point: 101101 = 101 101 = (55)₈ = 0010 1101 = (2D)₁₆

Negative integers — in an n-bit word:

• ›one's complement flips every bit
• ›two's complement flips every bit and adds 1; −5 in 8 bits is 11111011
• ›the n-bit two's complement range is −2ⁿ⁻¹ to 2ⁿ⁻¹ − 1 — for 8 bits, −128 to +127 — and there is only one zero

The trap: an 8-bit two's complement word reaches −128 but only +127 — the range is not symmetric.

Encoding characters — ASCII, ISCII and Unicode

Values worth knowing in ASCII — 'A' = 65, 'a' = 97, '0' = 48, space = 32; a capital and its small letter differ by 32, a single bit.

Unicode's encodings

• ›UTF-8 — variable, 1 to 4 bytes; ASCII text is unchanged, which is why it dominates the web
• ›UTF-16 — 2 or 4 bytes
• ›UTF-32 — a fixed 4 bytes for every character

• ›a Devanagari letter such as क takes 3 bytes in UTF-8
• ›storing text in Unicode is what lets one document mix Hindi, Urdu and English without garbling

Units — 4 bits = a nibble · 8 bits = a byte · 1 KB = 1024 bytes · 1 MB = 1024 KB.

The trap: ASCII is a 7-bit code with 128 characters — the 8-bit 'extended ASCII' tables are vendor additions, not the standard.

Boolean algebra — the laws and simplification

• ›involution — (A′)′ = A
• ›duality — swapping + with ·, and 0 with 1, turns any law into another valid law

Simplifying

• ›A + A′B = (A + A′)(A + B) = A + B
• ›AB + AB′ = A(B + B′) = A
• ›(A + B)(A + B′) = A + BB′ = A

Universal gates — NAND and NOR can each build every other gate: NOT A is A NAND A; AND is a NAND followed by a NOT.

The trap: the distributive law A + BC = (A + B)(A + C) holds in Boolean algebra but not in ordinary arithmetic — students who refuse it cannot simplify A + A′B.

Gates, adders and Karnaugh maps

Adders

• ›half adder — sum = A ⊕ B, carry = AB
• ›full adder — adds three bits: sum = A ⊕ B ⊕ Cᵢₙ, carry = AB + Cᵢₙ(A ⊕ B); two half adders and an OR gate make one

Canonical forms — a function can be written as a sum of minterms (SOP) or a product of maxterms (POS).

Karnaugh maps — cells are arranged in Gray code so neighbours differ in one variable; group 1s in blocks of 1, 2, 4 or 8, as large as possible, wrapping round the edges.

• ›F(A, B, C) = Σm(0, 1, 2, 3) covers every cell with A = 0, so F = A′
• ›F(A, B, C) = Σm(1, 3, 5, 7) covers every cell with C = 1, so F = C

The trap: a K-map group must contain a power of two cells — a group of three or six is never allowed.

Points to Remember

• ›(101101)₂ = 45 = (55)₈ = (2D)₁₆ · (0.625)₁₀ = (0.101)₂
• ›group bits in threes for octal, fours for hex
• ›two's complement = invert + 1 · −5 = 11111011 · 8-bit range −128 to +127 · one zero
• ›ASCII 7-bit, 128 · 'A' 65, 'a' 97, '0' 48 · ISCII 1991 · Unicode — Devanagari U+0900–U+097F · UTF-8 1–4 bytes
• ›De Morgan: (A + B)′ = A′B′, (AB)′ = A′ + B′ · absorption A + AB = A · A + A′B = A + B
• ›NAND, NOR universal · XOR = A′B + AB′
• ›half adder: sum A ⊕ B, carry AB · full adder = two half adders + OR
• ›K-map — Gray code order · groups of 1, 2, 4, 8 · Σm(0–3) = A′ · Σm(1, 3, 5, 7) = C
• ›Exam tip: before simplifying, try the absorption and A + A′B patterns — most textbook expressions collapse in one of those two steps