Digital System UNIT-1 One Shot 🔥 | Number System @Engineerdude-g6m Complete ONESHOT 28 marks 👌

Main ideas / lessons

1) Digital System Unit 1: Number systems (core exam focus)

The video frames Digital System Unit-1 as a one-shot topic and claims it covers ~28 marks.

The key concept is number systems, mainly asked in four standard bases:

• Decimal (base 10)
• Binary (base 2)
  • digits: only 0 and 1
• Octal (base 8)
  • digits: 0–7
• Hexadecimal (base 16)
  • digits: 0–9 plus A–F representing 10–15

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2) Converting between number systems (method repeated throughout)

A common exam pattern: you’re given a number in one base (e.g., binary) and asked to convert it to another (e.g., decimal, octal, hex).

General rules emphasized:

• For conversion to a base: repeatedly divide by the target base (or follow the appropriate source→target procedure).
• Remainders/digits
  • Integer conversion: read remainders from bottom to top
  • Fractional/point conversion:
    • repeatedly multiply the fractional part by the target base
    • take the integer part each time as the next digit

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Detailed conversion methodologies (as taught)

A) Decimal → Binary (integer)

• Given: a decimal integer (N)
• Divide repeatedly by 2
• Collect remainders (0 or 1)
• Continue until the quotient becomes 0
• Binary digits = remainders written bottom-to-top
• Check idea: each binary digit corresponds to powers of 2

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B) Binary → Decimal (integer)

• Assign bit positions to the binary number
• For each bit:
  • if bit = 1, add the corresponding power of 2
  • if bit = 0, add nothing

[ \text{Decimal}=\sum (\text{bit}_i \times 2^i) ]

where (i) is the bit position starting from 0 at the rightmost bit.

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C) Binary → Decimal (fractional part, with a point)

• Split into:
  • integer part
  • fractional part
• For fractional bits, use negative powers:

[ \text{Fractional value}=\sum (\text{bit}_{-k}\,/\,2^k) ]

(equivalently (2^{-1}, 2^{-2}, 2^{-3},\dots))

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D) Decimal → Octal (integer)

• Divide repeatedly by 8
• Collect remainders (0–7)
• Read remainders bottom-to-top

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E) Decimal → Octal (with fractional part)

• Integer part: divide by 8
• Fractional part:
  • multiply the fractional part by 8
  • take the integer part of each multiplication as the next octal digit
  • repeat until the fractional part becomes 0 or enough digits are produced
• Final octal result:
  • fractional digits from multiplication steps
  • integer digits from remainders

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F) Octal → Decimal (integer check method)

Use powers of 8:

[ \text{Decimal}=\sum (\text{octal digit}_i \times 8^i) ]

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G) Decimal → Hexadecimal (integer)

• Divide repeatedly by 16
• Collect remainders
• Map remainder:

Remainder Hex 0–9 0–9 10 A 11 B 12 C 13 D 14 E 15 F

• Read remainders bottom-to-top

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H) Decimal ↔ other bases: verification approach

The video suggests verifying conversions by converting back using place-value powers:

• Decimal ↔ Binary using powers of 2
• Decimal ↔ Octal using powers of 8
• Decimal ↔ Hex using powers of 16

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Logic Gates section (digital logic fundamentals)

1) What logic gates are

• Logic gates build digital systems using binary inputs (0 and 1).
• Computers convert instructions into binary.

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2) Basic gates (3 types)

• AND gate
  • Boolean: (a \cdot b) (also written as (ab))
• OR gate
  • Boolean: (a + b)
• NOT gate
  • Output is the complement: (\overline{a})
  • NOT swaps 0 ↔ 1

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3) Universal gates (2 types)

• NAND = NOT(AND)
• NOR = NOT(OR)

Using NAND/NOR, you can implement other gates.

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4) Exclusive gates (XOR / XNOR)

• XOR (exclusive OR)
  • output is 1 when inputs differ, 0 when same
• XNOR
  • opposite of XOR: 1 when same, 0 when differ

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Boolean algebra theorems & laws (as taught)

1) Key identities for binary behavior

• (x + 1 = 1)
• (x \cdot 0 = 0)
• (x + x = x)
• (x \cdot x = x)

Complement rules:

• Complement of 0 is 1
• Complement of 1 is 0
• Complement of complement returns the original value

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2) Commutative / associative / distributive laws

• Commutative
  • (x+y = y+x)
  • (xy = yx)
• Associative
  • ((x+y)+z = x+(y+z))
  • ((xy)z = x(yz))
• Distributive
  • (x(y+z)=xy+xz) (and related forms)

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3) DeMorgan’s Laws (highlighted as important)

• (\overline{x+y}=\overline{x}\cdot\overline{y})
• (\overline{x\cdot y}=\overline{x}+\overline{y})

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4) Truth-table proof method

The video states DeMorgan’s Laws can be proven using truth tables by comparing outputs.

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Canonical / Standard forms (SOP and POS)

1) SOP and POS definitions

• SOP (Sum of Products)
  • denoted as Σ (sigma)
  • product terms are summed
• POS (Product of Sums)
  • denoted as Π (pi)
  • sum terms are multiplied

Emphasis:

• Order matters
  • SOP: do product first, then sum
  • POS: do sum first, then product

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2) Conversion using minterms / maxterms

Canonical representation uses:

• Sum of minterms (SOP form)
• Product of maxterms (POS form)

Rule concept:

• Minterm corresponds to rows where output is 1
• Maxterm corresponds to rows where output is 0

The video teaches a step-by-step logic to write expressions from given indices.

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Implementation using NAND/NOR (construction logic)

NAND-based building strategy

Use NAND to create:

• NOT
• AND
• OR
• XOR/XNOR-style logic (via gate combinations and DeMorgan)

NOR-based building strategy

A similar approach using NOR:

• relies on DeMorgan relationships to convert into the required logic

Overall theme: universal gates can replace all others via correct interconnection.

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Binary arithmetic (addition/subtraction + complements)

1) Binary addition

Emphasized rules:

• (0+0=0)
• (0+1=1)
• (1+1=10) (carry generation)

Carry-forward follows standard binary addition.

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2) Binary subtraction

A borrowing/shift idea is described (e.g., when subtracting a smaller from a larger, borrow affects the base).

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3) One’s complement and two’s complement

• One’s complement: flip bits (0→1, 1→0)
• Two’s complement: one’s complement + 1

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4) Subtraction using complements

To compute (A - B):

• Convert (B) to:
  • one’s complement (for one’s complement subtraction), or
  • two’s complement (for two’s complement subtraction)
• Add to (A)
• Handle carry/discard rules (carry behavior described)
• If no carry occurs in one’s complement subtraction, take complement again (as narrated)

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Codes: BCD and non-weighted codes (Access-3 and Gray)

1) BCD (Binary Coded Decimal)

• Convert each decimal digit (0–9) into a 4-bit binary
• Each group of four bits represents one decimal digit independently

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2) Access-3 (non-weighted code)

Conversion rule:

• take the decimal digit
• add 3
• convert the result into 4-bit binary

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3) Gray code

• Binary → Gray (method taught)
  • keep MSB the same
  • next Gray bits follow XOR-like behavior between adjacent binary bits
• Gray → Binary
  • use a cumulative/zigzag rule:
    • start from MSB and iteratively XOR/accumulate down the sequence (described as a “zigzag” walk)

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Karnaugh Map (K-map)

Purpose

Simplify Boolean expressions graphically, reducing reliance on heavy algebra.

Development

• Mentioned as developed in 1953.

Core procedure

• Create a map with:
  • (2^n) cells for (n) variables
• Fill values:
  • 0/1 from the truth table or given indices
• Grouping:
  • groups of size 2, 4, 8, etc., depending on variable count
  • wrapping/overlap allowed
  • make the largest possible groups first
• Derive the simplified expression:
  • literals come from the grouped cells
  • constant rows/columns correspond to literals

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Speakers / sources featured

• Primary speaker: a single instructor (name not stated in subtitles), associated with Engineerdude-g6m (implied by the video title).