CMS 704 · Everything Left in the Notes and the Outline

A digit is a symbol; a number is a quantity. A digit is one of the individual symbols a number system provides for writing values · base 10 has ten digits (0–9), base 8 has eight (0–7), base 2 has two (0 and 1) and base 16 has sixteen (0–9 and A–F). A number is the actual value or quantity being represented, which is built by placing digits in positions, each position carrying a weight equal to the base raised to that position.

So the same number can be written with completely different digits depending on the base, while the quantity itself never changes:

A neat closing line: digits are the alphabet of a number system; numbers are the words written with them.

Three inputs, so 2³ = 8 rows. Fill the input columns by counting up in binary from 000 to 111 so that no combination is missed.

If the question also asks for the gate symbols, draw the two three-input gates · the same shapes as in Volume I §3, with three input lines instead of two · and give the expressions A·B·C and A+B+C.

Watch the follow-up: examiners often extend this to (A·B·C)' and (A+B+C)', which are simply those two columns inverted, and which De Morgan turns into A'+B'+C' and A'·B'·C' respectively.

Why hexadecimal and computer storage fit together

Computer storage is organised in bits grouped into bytes of 8 bits. Because 16 = 2⁴, one hexadecimal digit represents exactly four bits (a nibble), so two hexadecimal digits represent exactly one byte. The mapping is exact and needs no arithmetic, which is why memory addresses, memory dumps, machine code, colour values and MAC addresses are all written in hex.

1 hex digit  = 4 bits  = 1 nibble
2 hex digits = 8 bits  = 1 byte      e.g. FF = 11111111 = 255
4 hex digits = 16 bits = 1 word

11010011₂  →  1101 0011  →  D3₁₆

The point to state: binary is what the hardware stores, but it is long and error-prone for a human to read; hexadecimal is a compact, lossless shorthand for binary · four times shorter, with every digit still mapping cleanly onto a fixed group of bits. Octal does the same job in groups of three, but hex matches the 8-bit byte exactly, which is why it dominates.

The three encoding schemes

Classwork 1 · as recorded

F = A'BC' + AB'C' + AB'C + ABC
      ①       ②       ③      ④

① and ② ⇒ A'BC' + ABC'  ⇒ BC'(A'+A) = BC'
② and ③ ⇒ AB'C' + AB'C  ⇒ AB'(C'+C) = AB'
③ and ④ ⇒ AB'C  + ABC   ⇒ AC(B'+B)  = AC

∴ F = BC' + AB' + AC

Note how each pair is chosen: the two terms are identical except that one variable appears once plain and once complemented. That variable is the one that disappears.

Classwork 2 · as recorded

X = A'B'C' + A'B'C + AB'C + ABC
      ①        ②       ③      ④

① and ② ⇒ A'B'(C'+C) = A'B'
② and ③ ⇒ B'C(A'+A)  = B'C
③ and ④ ⇒ AC(B'+B)   = AC

∴ X = A'B' + B'C + AC

The recorded answer line reads AB' + A'C + B'C, which does not follow from these four terms · evidence that some minterms in the scan are misread. Trust the method, not the copied line.

Why the PC is incremented during T₁ rather than later: the increment uses the ALU and the memory read uses the bus, so the two are independent and can be done in the same cycle. This is a small but real example of the parallelism mechanisms in the organization list · and it is exactly the kind of point that separates a full-mark answer from an average one.

Fixed point

A fixed-point number reserves a fixed number of bits for the fractional part, so the binary point never moves. An 8-bit number with 4 fractional bits stores values in steps of 1/16.

0101.1100₂ = 4 + 1 + 0.5 + 0.25 = 5.75

• Advantages · simple hardware, fast, exact within its step size; ordinary integer circuits can be reused.
• Disadvantages · a narrow range; very large and very small values cannot both be represented.
• Used in embedded systems, digital signal processing and financial arithmetic.

Floating point

A floating-point number stores a number in the form ± mantissa × base^exponent, so the binary point floats · a far wider range for the same number of bits, at the cost of precision.

Normalisation: the mantissa is shifted so there is exactly one non-zero digit before the point; in binary that digit is always 1, so it is not stored · the "hidden bit", which buys one extra bit of precision free.

Memory addressing

16 address lines address 64 K locations; 32 lines address 4 G. This is the same 2ⁿ counting rule as truth-table rows and unsigned ranges · one idea, three appearances on this syllabus.

Volatile vs non-volatile · a likely one-liner

The link back to the hierarchy: the top three levels are volatile and electronic; the bottom levels are non-volatile and (historically) mechanical. That is exactly why the pyramid's lower levels are so much slower · and why a computer must load programs upward from disk into RAM before it can run them.

Virtual memory is a memory-management technique in which the operating system gives each program the illusion of a large, contiguous private address space, while the actual data is spread across physical RAM and secondary storage. It extends the memory-hierarchy idea downward: cache makes RAM look faster; virtual memory makes disk look like RAM.

How it works

1. The virtual address space is divided into fixed-size pages; physical memory is divided into frames of the same size.
2. A page table maps each virtual page to the physical frame holding it.
3. The MMU (memory management unit) performs this translation on every access, in hardware.
4. If the page is not in RAM, a page fault occurs: the OS fetches the page from disk, evicts another page if necessary, updates the page table and restarts the instruction.
5. The TLB (translation lookaside buffer) caches recent translations so the page table need not be consulted every time · the TLB is named in the organization list in Volume I §2.

Why it is worth having

• Programs larger than physical RAM can run · only the active pages need be resident.
• Protection and isolation · one process cannot address another's memory, because the page tables differ.
• Simpler programming · every program sees the same flat address space, whatever the machine has installed.
• Efficient sharing · two processes can map the same physical frame, e.g. for shared libraries.

The system bus · three groups of lines

This is the hardware behind the bus notation in the notes: Bus ← R1 means R1 drives the data bus, and R2 ← Bus means R2 latches from it. Only one device may drive the bus at a time, which is why bus arbitration exists · a scheme deciding which unit becomes bus master when several request it at once.

The two I/O addressing models

These are the two options listed as item 7 of the architecture list · "input/output model: memory-mapped I/O or special I/O instructions".