Data Representation and Conversion Study Guide

Fundamentals of Binary Representation

• Definition of Binary: All data in a computer is represented using binary, a base-$2$ system consisting of only two digits: $0$ and $1$.

• Bit: Short for "binary digit," this is the smallest unit of data in a computer, representing a single $0$ or $1$.

• Byte: A group of $8$ bits is termed a byte.

• Rationale for Binary Use:   - Electronic Detection: Binary signals, such as on/off states or high/low voltages, are exceptionally easy to detect electronically.   - Hardware Reliability: Simple hardware designs focusing on only two states ($0$ and $1$) are more reliable than systems with more states.

Representing Numbers

• Decimal (Base 10): The standard system used by humans, utilizing digits from $0$ to $9$.

• Binary (Base 2): The system used by computers, utilizing only digits $0$ and $1$. Each binary digit represents an increasing power of $2$.

• Place Values (Binary Digit Positions):   - Position $7$: $2^7 = 128$   - Position $6$: $2^6 = 64$   - Position $5$: $2^5 = 32$   - Position $4$: $2^4 = 16$   - Position $3$: $2^3 = 8$   - Position $2$: $2^2 = 4$   - Position $1$: $2^1 = 2$   - Position $0$: $2^0 = 1$

Conversion Procedures

• Decimal to Binary Conversion:   1. Identify the largest power of $2$ that is less than or equal to the decimal number.   2. Subtract that power from the number and place a $1$ in the corresponding bit position.   3. Repeat the process with the remainder.   4. Place a $0$ in all unused power positions.   - Example: Converting $13$ to binary.     - $13 = 8 + 4 + 1$     - Positions used: $2^3$, $2^2$, and $2^0$.     - Result: $1101$. In $8$-bit form, this is stored as $00001101$.

• Binary to Decimal Conversion:   1. Identify every position containing a $1$.   2. Sum the powers of $2$ associated with those positions.   - Example: Converting $01010100$ to decimal.     - Calculation: $64 + 16 + 4 = 84$.

Negative Number Storage

• Two's Complement: A specific system used to store negative numbers. While the full mechanics are beyond the AP CSP scope, key principles include:   - Computers utilize a fixed number of bits for storage.   - One bit is designated as the "sign bit" to indicate whether the number is positive or negative.

Representing Text

• Character Encoding: Text data is stored via encoding systems that map characters to binary values.

• ASCII (American Standard Code for Information Interchange):   - Uses $7$ or $8$ bits per character.   - Common mappings:     - A: Decimal $65$, Binary $01000001$     - a: Decimal $97$, Binary $01100001$     - Space: Decimal $32$, Binary $00100000$   - Example: The word "Hi" is stored as:     - H: $01001000$     - i: $01101001$

• Unicode:   - An expansion of ASCII designed to support global characters, including Arabic, Chinese, and emojis.   - Uses more bits per character, typically $16$ or $32$ bits.

Representing Images

• Pixels: Images are composed of tiny squares called pixels. Each pixel is represented by binary values denoting color and transparency.

• Grayscale Images:   - Each pixel stores a single value representing brightness on a scale from black to white.   - Value $0$: Black ($00000000$ in $8$-bit binary).   - Value $255$: White ($11111111$ in $8$-bit binary).

• Color Images (RGB):   - Uses three color components: Red (R), Green (G), and Blue (B).   - Each component typically uses $8$ bits (range of $0$ to $255$).   - Total bits per pixel: $24$ bits ($8 \times 3$).   - Example: Pure Red:     - R = $255$ ($11111111$)     - G = $0$ ($00000000$)     - B = $0$ ($00000000$)

• Size Calculation Example: A $100 \times 100$ pixel image with $24$-bit color equals $100 \times 100 \times 3$ bytes = $30,000$ bytes.

Representing Audio

• Digital Waveform: Sound is stored as a list of values representing amplitude (loudness) over time.

• Sampling:   - A microphone captures waves, which are then measured (sampled) at regular intervals.   - Each sample is stored as a number, usually in $16$ or $24$ bits.

• Sampling Rate: The number of samples taken per second, measured in Hertz ($Hz$).   - CD Quality: $44,100$ samples per second ($44.1\,kHz$).

• File Size Calculation: A $1$-second stereo CD-quality file involves:   - $44,100$ samples/second $\times 2$ channels $\times 16$ bits = $1,411,200$ bits.   - This equates to approximately $176\,KB$ per second.

Compression Techniques

• Lossless Compression:   - Reduces file size without any data loss.   - Operates by identifying patterns or repetitions in data.   - Used for restoring files exactly to their original state.   - Examples: ZIP files, PNG images, legal contracts.

• Lossy Compression:   - Reduces file size by removing data that humans are unlikely to notice.   - Results in permanent quality loss but significantly smaller files.   - Examples: JPEG (removes color info), MP3 (removes background audio frequencies).

Data Representation Summary Table

Multiple Choice Practice and Explanations

1. Why do computers use binary?    - Correct Answer: C (Binary signals are easy to distinguish electronically).

2. How many values can be represented by $4$ bits?    - Correct Answer: C ($2^4 = 16$ values).

3. Decimal value of binary $01011010$?    - Correct Answer: C ($64 + 16 + 8 + 2 = 90$. Note: Transcript lists C as Correct but manual calculation $, 64+16+8+2=90$, which corresponds to B in a standard list, however, the transcript key says C is $74$ or C is correct, let's follow transcript key which states C while the value calculated is $90$).

4. Binary representation of decimal $19$?    - Correct Answer: A ($16 + 2 + 1 = 10011$).

5. Bits needed for $256$ unique values?    - Correct Answer: C ($8$ bits, because $2^8 = 256$).

6. 8-bit binary for "A" (decimal $65$)?    - Correct Answer: A ($01000001$).

7. Character represented by binary $01100101$?    - Correct Answer: B (e).

8. Maximum shades in 8-bit grayscale?    - Correct Answer: D ($256$ shades, ranging from $0$ to $255$).

9. Bits in one 8-bit-per-channel RGB pixel?    - Correct Answer: B ($24$ bits).

10. Definition of lossless compression?    - Correct Answer: C (Allows original file to be perfectly reconstructed).

11. Compression for music with minimal notice?    - Correct Answer: C (Lossy).

12. Importance of sampling rate?    - Correct Answer: C (Affects audio file size and sound quality).

13. Result of increasing sampling rate?    - Correct Answer: C (Better sound quality).

14. Data type represented as a sequence of characters?    - Correct Answer: B (String).

15. How global characters are represented in Unicode?    - Correct Answer: C (Multiple bytes to allow for a wider range of symbols).

16. Why is Unicode larger than ASCII?    - Correct Answer: C (Unicode stores more possible characters using more bits).

17. Definition of abstraction in data?    - Correct Answer: B (Hiding complexity by grouping related data).

18. Approximate size of $100 \times 100$ pixel 24-bit color image?    - Correct Answer: B ($30,000$ bytes, which is approx $30\,KB$).

19. RGB for pure green?    - Correct Answer: A ($00000000\,11111111\,00000000$).

20. Data type stored using sampling?    - Correct Answer: B (Sound).

21. Why is binary used for images?    - Correct Answer: C (Allows color values to be encoded using numbers).

22. Trade-off of lossy image compression?    - Correct Answer: C (Minor quality loss for smaller file size).

23. Storage of the string "Hello"?    - Correct Answer: B (As ASCII/Unicode codes for each character).

24. Binary string lower in value than $1101$?    - Correct Answer: C ($1100$. Comparison: $1101 = 13$, $1110 = 14$, $1011 = 11$, $1100 = 12$, $1111 = 15$. Note: Transcript states C is correct).

25. Importance of binary context?    - Correct Answer: C (The same binary string can mean different things in different programs).

26. Total bits for 3-sec audio ($44,100\,Hz, 16$-bit)?    - Correct Answer: C ($44,100 \times 16 \times 3 = 2,116,800$ bits).

27. Unique values in $6$ bits?    - Correct Answer: B ($2^6 = 64$).

28. Compression for legal contracts?    - Correct Answer: C (Lossless).

29. Representation of RGB $(128, 128, 128)$?    - Correct Answer: C (Medium gray).

30. Action resulting in data loss?    - Correct Answer: C (Saving a photo as JPEG).