Binary, decimal and hex; powers of 2

1. How many bits are in 2 KB?
   Stuck? Peek at 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.

   Read lesson 0.2.1

2. Mesh vs star: links for n nodes?
   Stuck? Peek at 0.1.2 Nodes, links, topologies, LAN/WAN and the boxes in the middle

   • Node = any device; host = node where messages start/end.
   • Point-to-point (2 nodes) vs broadcast (all hear all).
   • Simplex (one way) · half-duplex (two ways, one at a time) · full-duplex (both at once).
   • Links: mesh n(n−1)/2, star n, ring n, tree n−1, bus 1 backbone.
   • LAN (building) ⊂ WAN (cities) ⊂ the Internet (network of networks).
   • Hub repeats to all · switch delivers within a LAN · router picks the next network.

   Read lesson 0.1.2

3. Where does the meaning of a bit pattern come from?
   Stuck? Peek at 0.1.1 Why networks exist: hosts, links and sending data

   • Network = devices + links that exchange data.
   • Host = where a message starts or ends (laptop, phone, server).
   • Link = physical path: copper (voltage), fibre (light), wireless (radio).
   • All content → bits → signals → bits → content; meaning comes from an agreed encoding.
   • One link per pair of n hosts: n(n − 1)/2 links — too many, so links are shared.

   Read lesson 0.1.1

Every address, every length and every counter inside a network is a binary number stored in a fixed number of bits. Read the field wrong by one bit and you are off by a factor of two. Engineers write those numbers in hexadecimal because four bits fit exactly in one hex digit. Fluency here pays off in every phase that follows.

A counter is stored in 8 bits. Guess the largest number it can hold: 8, 128, 255 or 256?

255. 8 bits give 2⁸ = 256 different patterns, from 00000000 (0) to 11111111 (255). The count of patterns is 256; the largest value is one less.

Place value

Decimal: each place is worth 10 times the place to its right (1, 10, 100…). Binary is the same idea with 2: places are worth 1, 2, 4, 8, 16, 32, 64, 128…

Decimal → binary

Method: subtract the largest power of 2 that fits, mark a 1 in that place, repeat. Or divide by 2 repeatedly and read the remainders from bottom to top.

Hexadecimal

Hex is base 16. Digits 0–9 then A=10, B=11, C=12, D=13, E=14, F=15. One hex digit = exactly 4 bits (a "nibble"), so binary ↔ hex is just grouping in fours from the right. Hex is often written with a prefix: 0xC8.

k bits give 2ᵏ values

With k bits there are 2ᵏ patterns, representing 0 to 2ᵏ − 1. To count N different things you need the smallest k with 2ᵏ ≥ N.

Putting it together

• Binary is place value with powers of 2; hex groups 4 bits per digit.
• k bits give 2ᵏ values, from 0 to 2ᵏ − 1.
• Bits needed for N values is the smallest k with 2ᵏ ≥ N.

• k bits hold 2ᵏ values, but the largest value is 2ᵏ − 1.
• "From 0 to 5000" is 5001 values, not 5000.
• Group hex digits from the right (LSB end), not from the left.

"One hex digit is one nibble": four bits, sixteen faces — 0 to F.

Binary numbers are how every field inside the network's wrappers will be stored. Next (0.2.3): networks also combine bit patterns — flipping, adding with a wrap-around, and dividing without carries. Those are the tools that will catch damaged bits in Phase 2.

1. MCQ

   0xA5 in binary is:

   1. 10100101
   2. 10101010
   3. 01011010
   4. 11000101
2. NAT

   Decimal value of 11110000?

   Write your answer before opening the key.

3. NAT

   Minimum bits to give a different number to each of 1000 items?

   Write your answer before opening the key.

4. MSQ

   Which are true for a 16-bit field?

   1. It has 65,536 possible values.
   2. Its largest value is 65,535.
   3. Its largest value in hex is 0xFFFF.
   4. It needs 4 hex digits plus one bit.
5. NAT

   Decimal value of 0x1F4?

   Write your answer before opening the key.

• 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.

• binary
• decimal
• hexadecimal (hex), 0x prefix
• place value
• MSB / LSB
• nibble (4 bits)
• k bits give 2ᵏ values
• powers of 2

1. 0xFF in decimal?
   Reveal answer

   255.
2. Largest value of a k-bit field?
   Reveal answer

   2ᵏ − 1.
3. Bits for 0..5000?
   Reveal answer

   13 (5001 values ≤ 8192).
4. 2³²?
   Reveal answer

   4,294,967,296.

• Binary ↔ hex: group bits in fours from the right; convert each group. — always (pad the leftmost group with zeros).
• Bits needed for N values = smallest k with 2ᵏ ≥ N. — counting values; remember 0..N is N + 1 values.