Binary & Data Representation

💻 Binary & Data Representation

Part 1 of 7 — Binary Numbers, Conversions, and Digital Information

Why Binary?

Computers use binary (base-2) because electronic circuits have two states: ON (1) and OFF (0). ALL data — numbers, text, images, sound — is ultimately stored as sequences of 0s and 1s.

Binary to Decimal Conversion

Each binary digit (bit) represents a power of 2:

Example: 1101 in binary = ?

• 1 x 8 + 1 x 4 + 0 x 2 + 1 x 1 = 8 + 4 + 0 + 1 = 13

Example: 10110 in binary = ?

• 1 x 16 + 0 x 8 + 1 x 4 + 1 x 2 + 0 x 1 = 16 + 4 + 2 = 22

🔑 With n bits, you can represent 2^n different values (0 to 2^n - 1). 8 bits = 1 byte = 256 values.

Concept Check 🎯

Decimal to Binary Conversion

Repeatedly divide by 2 and track remainders (read bottom to top):

Convert 25 to binary:

• 25 / 2 = 12 remainder 1
• 12 / 2 = 6 remainder 0
• 6 / 2 = 3 remainder 0
• 3 / 2 = 1 remainder 1
• 1 / 2 = 0 remainder 1
• Read upward: 11001

Overflow

When a value exceeds the number of bits available:

• 4 bits can store 0-15
• If you try to store 16 in 4 bits, it overflows back to 0
• This is like an odometer rolling over from 999 to 000

Data Units

Representing Other Data Types

Applied Recall ✍️

1. A single binary digit (0 or 1) is called a _______.

2. 8 bits make up one _______.

3. With n bits, you can represent _______ to the power of n different values.

Binary Practice 🔍

AP Exam Strategy: Binary & Data

• Memorize powers of 2: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024
• Know both conversions: binary-to-decimal and decimal-to-binary
• With n bits: 2^n values, range 0 to 2^n - 1
• Overflow = value exceeds bit capacity. AP loves testing this concept
• Know how text (ASCII/Unicode), images (RGB pixels), and sound (samples) are stored in binary
• One additional bit DOUBLES the representable values — this is the key relationship

AP-Style Application 🎯

🔢 Binary & Data Representation

Part 2 of 7 — Key Processes

How Computers Encode Everything in Binary

A computer represents every kind of data — numbers, text, images, sound — using only two digits: 0 and 1. The trick is agreement: a sender and receiver agree on a code, then the same bit pattern can mean a number, a letter, or a pixel color.

Concept Check 🎯

Reading Binary Like a Pro

To convert binary to decimal, line up the bits with their place values from the right and add the place values where the bit is 1.

🔢 Binary & Data Representation

Part 3 of 7 — Patterns & Examples

Patterns of Encoding

Computers use conventions to encode data types. Understanding the convention is half the work.

Concept Check 🎯

Worked Example: Decoding Text

Bytes: 72, 73, 33

• 72 = "H"
• 73 = "I"
• 33 = "!"

Output: HI! Since text is just numbers under an agreed mapping, any file is bytes — the difference is only how the program interprets them.

Worked Example: Building a Color

🔢 Binary & Data Representation

Part 4 of 7 — Connections & Interactions

Why Binary Connects To Everything Else

The "everything is bits" idea makes the rest of the course possible.

Concept Check 🎯

Bits Are Format-Agnostic

A network router doesn't know whether the bits it forwards are an image, a song, or a chess move. It only knows source/destination headers and the payload bits. This is why a single Internet works for so many uses — the binary layer is universal.

Same Bytes, Different Meaning

A 4-byte sequence 01000001 01000010 01000011 01000100 could mean:

• Text (ASCII): "ABCD"
• Four small unsigned integers: 65, 66, 67, 68

🔢 Binary & Data Representation

Part 5 of 7 — Change Over Time

How Storage And Encoding Have Scaled

Storage costs have collapsed and bit budgets have grown. The same data that filled a hard drive in 2000 fits in a phone's photo cache today.

Concept Check 🎯

Lossless vs. Lossy

🔢 Binary & Data Representation

Part 6 of 7 — Problem-Solving Workshop

Binary & Data Workshop

Practice the calculations and design choices the AP exam asks under a time crunch.

Concept Check 🎯

Worked: Bit-Width Sizing

Question: "We have 5 different statuses to encode in each record. Minimum bits per record?"

• 2² = 4 (too small).
• 2³ = 8 ≥ 5 ✓.
• Use 3 bits.

Question: "We have 1,000,000 user IDs. Minimum bits per ID?"

• 2¹⁹ = 524,288 (too small).
• 2²⁰ = 1,048,576 ≥ 1M ✓.
• Use 20 bits.

Worked: Storage Estimation

Question: "Estimate the storage for 10,000 1024×1024 RGB photos, uncompressed."

• One photo: 1024 × 1024 × 3 ≈ 3 MB.
• 10,000 photos: ~30 GB.

With JPEG (~10×): ~3 GB.

Worked: Conversions Both Ways

🔢 Binary & Data Representation

Part 7 of 7 — AP Review

AP Exam Recap — Binary & Data

Final cheat sheet of the binary and data-representation facts most tested on the AP exam.

Concept Check 🎯

Quick-Reference Table

Common Misconceptions to Avoid

• "Binary uses 0 and 1 because computers think in numbers." (No — because hardware reliably represents two voltage levels.)
• "Lossy compression destroys the file." (No — it produces a smaller approximation; original can be saved separately.)
• "More bits is always better." (No — wasted bits = wasted storage / bandwidth. Use the minimum sufficient.)
• "Encryption shrinks data." (No — it transforms data; size stays roughly the same.)

Common Calculations

• Min bits for N values: ⌈log₂ N⌉.
• Image bytes: W × H × bytesPerPixel.
• Audio bytes/sec: sampleRate × bytesPerSample × channels.
• Total storage of M items × k bytes each: M·k.

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