Unit 1 — Number Systems, Evolution, Block Diagram and Generations

Topics Covered

Syllabus topics: Number systems — binary, decimal, octal, hexadecimal; conversions between number systems. Evolution of computers — history from early mechanical devices to modern-day systems. Block diagram of a computer — components like input unit, output unit, memory, CPU (ALU + CU). Generations of computers — first to fifth generation technologies, characteristics, examples.

1.1 Number systems

Why computers use binary

A transistor is either conducting or not. Two states — so the natural number system for electronics has two digits, 0 and 1. Every number, letter, image and instruction inside a computer is ultimately a pattern of these.

Conversion: any base → decimal

Multiply each digit by its positional weight and add.

Binary 1011 → decimal

8 + 0 + 2 + 1 = 11

Hexadecimal 2AF → decimal

• 2 × 16² = 2 × 256 = 512
• A(10) × 16¹ = 10 × 16 = 160
• F(15) × 16⁰ = 15 × 1 = 15
• Total = 512 + 160 + 15 = 687

Octal 745 → decimal

7 × 64 + 4 × 8 + 5 × 1 = 448 + 32 + 5 = 485

Conversion: decimal → any base

Divide repeatedly by the base and read the remainders bottom-up.

Decimal 45 → binary

Reading the remainders upwards: 101101

Check: 32 + 8 + 4 + 1 = 45 ✓ — always verify by converting back.

Decimal 255 → hexadecimal

• 255 ÷ 16 = 15 remainder 15 (F)
• 15 ÷ 16 = 0 remainder 15 (F)

Reading upwards: FF

Conversion: binary ↔ octal (group in 3s)

Binary 110101 → octal

Group from the right in threes: 110 | 101 → 6 | 5 → 65

Octal 47 → binary

4 → 100, 7 → 111 → 100111

Conversion: binary ↔ hexadecimal (group in 4s)

Binary 11010110 → hexadecimal

Group from the right in fours: 1101 | 0110 → 13(D) | 6 → D6

Hexadecimal 3E → binary

3 → 0011, E → 1110 → 00111110

Group from the right, always. 1101101 splits as 110 | 1101 → 6, D → 6D. Group it from the left instead and you get 1101 | 101 → D, 5 → D5, which is 213 rather than 109. Padding the short leftmost group with zeros — 0110 | 1101 — changes nothing about the answer; it is a tidiness habit that makes the direction obvious at a glance, and that is its whole value.

Octal ↔ hexadecimal

There is no direct route. Go via binary: octal → binary → regroup in fours → hex.

Octal 725 → binary 111 010 101 → regroup as 0001 1101 0101 → 1D5

Handy reference table

Memorise the 0–15 row. Every conversion becomes lookup rather than arithmetic.

Fractional conversions

For the part after the point, multiply by the base and read the integer parts downwards.

Decimal 0.625 → binary

Reading downwards: 0.101

Check: ½ + 0 + ⅛ = 0.625 ✓

Some fractions never terminate in binary — 0.1 decimal is infinite in binary, which is exactly why 0.1 + 0.2 != 0.3 in Python and in every other language.

Binary arithmetic

Addition rules: 0+0=0 · 0+1=1 · 1+0=1 · 1+1=10 (0 carry 1) · 1+1+1=11 (1 carry 1)

   1011   (11)
 + 1101   (13)
 ------
  11000   (24)  ✓

Subtraction rules: 0−0=0 · 1−0=1 · 1−1=0 · 0−1=1 with a borrow

Representing negative numbers

2's complement is what real computers use, because addition and subtraction then use the same circuit, and there is only one representation of zero (sign–magnitude and 1's complement both have +0 and −0).

Finding the 2's complement of 5 (00000101):

1. Flip every bit → 11111010
2. Add 1 → 11111011

Check: 5 + (−5) should be 0. 00000101 + 11111011 = 100000000; the ninth bit overflows out of 8 bits, leaving 00000000 ✓

Codes

ASCII values worth knowing: A = 65, a = 97, 0 = 48, space = 32.

The 32 difference between A and a is a single bit, which is why case conversion in C can be done with ch + 32 or a bitwise OR.

1.2 Evolution of computers

Charles Babbage is the "Father of the Computer" for the Analytical Engine — which was never built in his lifetime but contained every logical element of a modern machine: a store (memory), a mill (processor), input and output.

Ada Lovelace wrote an algorithm for it, making her the first programmer. She also observed that such a machine could manipulate symbols in general, not only numbers — an insight a century ahead of its time.

The von Neumann architecture

Proposed by John von Neumann in 1945. Its central idea is the stored-program concept: instructions and data live in the same memory.

Before this, reprogramming ENIAC meant physically rewiring it — a job that took days. Storing the program as data made a computer general-purpose in the modern sense.

Almost every computer today is a von Neumann machine. Its known weakness is the von Neumann bottleneck — instructions and data share one path to memory, so the CPU waits. The Harvard architecture, which separates them, is used in some embedded processors.

1.3 Block diagram of a computer

        ┌───────────────────────────────────────────────┐
        │                    CPU                        │
        │   ┌───────────────┐   ┌───────────────────┐   │
  INPUT─┼──▶│ Control Unit  │   │ Arithmetic Logic  │───┼──▶ OUTPUT
  UNIT  │   │     (CU)      │◀─▶│    Unit (ALU)     │   │    UNIT
        │   └───────┬───────┘   └─────────┬─────────┘   │
        │           │      ┌──────────┐   │             │
        │           └─────▶│ Registers│◀──┘             │
        │                  └──────────┘                 │
        └───────────────────────┬───────────────────────┘
                                │
                    ┌───────────▼────────────┐
                    │     MEMORY UNIT        │
                    │  Primary + Secondary   │
                    └────────────────────────┘

      ────▶  data flow        ◀──▶  control signals

CPU = ALU + CU + Registers. The CU is often called the "nerve centre" — it processes no data itself, but tells every other unit what to do and when.

Registers

Small, extremely fast storage inside the CPU:

The machine cycle

Fetch → Decode → Execute → Store, repeated billions of times per second.

1.4 Generations of computers

Characteristics by generation

First (vacuum tubes) — enormous, consumed huge power, generated great heat, very unreliable (tubes burned out constantly), used punched cards, magnetic drum memory. ENIAC filled a room and weighed 30 tonnes.

Second (transistors) — the transistor, invented at Bell Labs in 1947, replaced the vacuum tube: smaller, cheaper, far more reliable, much less heat. Magnetic core memory. Assembly and the first high-level languages appeared.

Third (integrated circuits) — many transistors on a single silicon chip. Keyboards and monitors replaced punched cards. Operating systems appeared, enabling multiprogramming and time-sharing.

Fourth (microprocessors) — VLSI put an entire CPU on one chip. The Intel 4004 (1971) was the first. This generation brought the personal computer, the GUI, networks and the Internet.

Fifth (AI) — parallel processing, ULSI, natural language processing, machine learning, quantum computing. Defined by capability rather than by component technology, which is why its boundary is fuzzy.

The trend across all five generations: smaller, faster, cheaper, more reliable, and using less power. That single sentence answers "compare the generations" questions.

📝 Worked practice

   10110      (22)
 + 01101      (13)
 -------
  100011      (35)

Exam questions from this unit

Two marks

1. Why do computers use the binary system?
2. Convert (1010)₂ to decimal.
3. What is the 2's complement, and why is it preferred?
4. Who is the Father of the Computer, and why?
5. Expand ALU and CU.

Five marks

1. Convert a given number between all four bases, showing every step.
2. Explain the stored-program concept and the von Neumann architecture.
3. Explain the block diagram of a computer with a diagram.
4. Explain binary addition and subtraction with examples.

Ten marks

1. Explain the generations of computers with technology, characteristics, languages and examples for each.

2. Explain number systems and all the conversions between them, with examples.

Mistakes that cost marks