Math and units toolkit

Units: 1 B = 8 b; rates decimal (k = 10³), sizes binary (K = 2¹⁰) unless the question says otherwise; time to push out = bits ÷ bit/s. Binary/hex: place values 1, 2, 4…; one hex digit = 4 bits; k bits → 2ᵏ values (max 2ᵏ − 1). Bit operations: XOR marks differences and is carry-free addition; 1's-complement addition wraps the carry around; modulo-2 division XORs at each leading 1 and leaves a remainder one bit shorter than the divisor. Logs and probability: bits for N values = ⌈log₂ N⌉; exactly one of n = n·p·(1 − p)ⁿ⁻¹; (1 − 1/n)ⁿ → 1/e.

From 0.2.1 Bits, bytes and prefixes: powers of 10 for rates, powers of 2 for sizes

• 1 B = 8 b. Lower-case b = bit, upper-case B = byte.
• Rates: k = 10³, M = 10⁶, G = 10⁹ (bps).
• Sizes: K = 2¹⁰, M = 2²⁰, G = 2³⁰ (bytes) — unless the question says 1000.
• Time to push out = (size in bits) ÷ (rate in bit/s).
• ms = 10⁻³ s, µs = 10⁻⁶ s, ns = 10⁻⁹ s.

From 0.2.2 Binary, decimal and hex; powers of 2

• Binary places: 1, 2, 4, 8, 16, 32, 64, 128, …; MSB left, LSB right.
• Hex: one digit = 4 bits; A–F = 10–15; group from the right.
• k bits → 2ᵏ values, 0 … 2ᵏ − 1.
• Bits needed for N values = smallest k with 2ᵏ ≥ N.
• 2¹⁰ = 1024 · 2¹⁶ = 65,536 · 2²⁰ ≈ 1.05 M · 2³² ≈ 4.29 × 10⁹.

From 0.2.3 XOR, 1's complement addition and modulo-2 division

• AND (mask), OR, NOT (flip), XOR (1 where bits differ).
• a ⊕ a = 0, a ⊕ 0 = a, (a ⊕ k) ⊕ k = a.
• 1's complement = flip all bits; 1's-complement addition wraps the end carry back in.
• Modulo-2: + = − = XOR, no carries; bits ↔ polynomial.
• Mod-2 division: leading 1 → XOR divisor; leading 0 → XOR zeros; remainder = (divisor length − 1) bits.

From 0.2.4 Exponents, logarithms and basic probability

• 2ᵃ·2ᵇ = 2ᵃ⁺ᵇ; 2¹⁰ ≈ 10³ (estimates only).
• log₂ x = power of 2 giving x; bits for N values = ⌈log₂ N⌉.
• P(none) = (1−p)ⁿ · P(≥ 1) = 1 − (1−p)ⁿ · P(exactly 1) = n·p·(1−p)ⁿ⁻¹.
• (1 − 1/n)ⁿ → 1/e ≈ 0.368; 1/(2e) ≈ 0.184.
• 1 + 2 + … + 2ᵏ⁻¹ = 2ᵏ − 1; expected tries = 1/p.

1. NAT · 2 marks

   A 3 MB file (MB = 2²⁰ B) is pushed out at 24 Mbps. Time in seconds (3 decimals)?

   Write your answer before opening the key.

2. MCQ · 1 mark

   Which value is 0x7F?

   1. 127
   2. 128
   3. 255
   4. 112
3. NAT · 1 mark

   16-bit 1's-complement sum of 0xFFFF and 0x0002, in decimal?

   Write your answer before opening the key.

4. MCQ · 2 marks

   Remainder of 100100000 ÷ 1101 (modulo-2)?

   1. 001
   2. 101
   3. 011
   4. 000
5. MSQ · 1 mark

   Which are equal to 2¹⁶?

   1. 65,536
   2. 0x10000
   3. 64 × 1024
   4. 10⁵ approximately within 1 %
6. NAT · 1 mark

   8 stations each send with p = 0.125. P(none sends)? (4 decimals)

   Write your answer before opening the key.

7. MCQ · 1 mark

   Minimum bits to number 0 to 1023?

   1. 9
   2. 10
   3. 11
   4. 1023
8. MCQ · 1 mark

   100 Mbps expressed in bytes per second is:

   1. 12.5 × 10⁶
   2. 12.5 × 2²⁰
   3. 800 × 10⁶
   4. 100 × 2²⁰
9. NAT · 1 mark

   n = 4, p = 1/4: P(exactly one) to 4 decimals?

   Write your answer before opening the key.