Binary Numbers Chapter Notes | AP Computer Science Principles - Grade 9 PDF Download

Introduction

This chapter explores binary numbers, the foundation of how computers store and process data. It covers number bases, how binary and hexadecimal systems work, and their role in computing. The chapter also explains how bits represent various data types, the concept of abstraction, and issues like overflow and rounding errors. Understanding binary numbers is essential for the AP Computer Science Principles exam, particularly for data-related topics.

Binary Numbers

• Data is a collection of facts gathered from sources like experiments, sensors, photos, or videos, used across many professions.
• Computers rely on data, storing it in various forms for processing.
• Data can be stored in spreadsheets or charts for human use, or as variables, lists, or constants in coding.
• At the core, computers store data using bits, which are binary digits.

Number Bases

• A number base is the set of digits or combinations used to represent values in a numeral system.
• The decimal system (base-10) uses digits 0–9 and is commonly used by humans.
• Computers use the binary system (base-2), which only uses 0 and 1 to represent values.
• In the decimal system, place values are powers of 10 (ones, tens, hundreds, thousands, etc.).
  • Example: The number 5,729 in base-10 is 5 thousands (5 × 1000) + 7 hundreds (7 × 100) + 2 tens (2 × 10) + 9 ones (9 × 1).
• In decimal, when a place exceeds 9, it resets to 0, and the next place increases by 1 (e.g., 9 + 1 = 10).
• In the binary system, place values are powers of 2 (ones, twos, fours, eights, etc.), and each place can only hold 0 or 1.
  • Example: The binary number 0101 represents 1 four (1 × 4) + 0 twos (0 × 2) + 0 ones (0 × 1) + 1 ones (1 × 1) = 5 in base-10.
• Binary digits are called bits, and eight bits form a byte.
• Binary strings can be long, so the hexadecimal system (base-16) is used for easier human interaction.
• Hexadecimal uses 0–9 for values 0–9 and A–F for values 10–15, with place values as powers of 16.
  • Example: The number 5,729 in hexadecimal is 1661, which is 1 × 4096 + 6 × 256 + 6 × 16 + 1 × 1 = 5,729 in base-10.

Bit Representations

• Bits represent various data types, such as numbers, colors, and sound waves, depending on the context.
  • Example: In ASCII, the binary string for the letter “A” (01000001) also represents the number 65 in base-10.
• Programs are designed to interpret binary data correctly to avoid confusion.
• Consistent communication standards between programs and computers prevent misinterpretation of binary data.

Abstraction

• Abstraction simplifies complex systems by showing only essential details and hiding unnecessary complexities.
  • Example: Pressing the “start” button on an oven simplifies the process of starting it without needing to know how the oven’s internal components (like gas or fans) work.
• In computers, abstraction hides binary data processing, allowing users to interact with data through variables, lists, spreadsheets, or charts.
• Abstraction makes it easier for humans to use computers or code without understanding binary representations.

Analog Data and Bit Representations

• Analog data is continuous, with values changing smoothly (e.g., volume of a flute solo, time on an analog clock, or temperature on a physical thermometer).
• Digital data is recorded at discrete intervals using sampling techniques, where analog values are measured periodically.
• Each sample is converted into bits for digital storage.
• Digital data uses a finite set of values, unlike the potentially infinite range of analog data.
  • Example: A digital clock updates every minute, while an analog clock’s hands move continuously.
• Digital data approximates analog data, as it cannot capture every detail (e.g., a video is a series of images, not a continuous event).
• This approximation is an example of abstraction, as digital data simplifies real-world analog data for practical use.

Overflow Errors

• In low-level languages like C or C++, numbers are stored in a fixed number of bytes (e.g., 1–8 bytes).
• A byte (8 bits) can store values up to 255 in base-10 (11111111 in binary), calculated as 2⁸ − 1.
• Attempting to store a value larger than the maximum (e.g., 256 in a byte) causes an overflow error.
• Overflow errors may result in incorrect values, such as negative numbers or zero, and are hard to detect because they don’t always crash the program.
• Higher-level languages like Python avoid overflow errors by allowing number sizes limited only by the computer’s memory.
• The AP CSP exam follows the standard of higher-level languages, where overflow errors are less common.

Rounding Errors

• Rounding errors occur when numbers with infinite or high precision (e.g., repeating decimals) are stored with finite bits.
  • Example: In Python, dividing 100 by 3 results in a finite decimal, even though it’s a repeating decimal (33.333…).
• This is an abstraction, as the computer simplifies the number for storage.
• Rounding errors may cause issues in calculations requiring high precision, though they are usually negligible for school assignments.

Key Terms

• Abstraction: Simplifying systems by focusing on essential details and hiding complexity.
• Analog Data: Continuous data representing real-world values, like temperature or sound.
• ASCII Code: A standard assigning numeric values to characters for computer use.
• Constant Value: A fixed, unchanging value in a program.
• Data: Information (numbers, text, images) processed by computers.
• Lists: Ordered collections of items accessed by index in programming.
• Machine Code: Binary instructions directly executed by computer hardware.
• Number Bases: Systems for representing numbers using specific digits (e.g., base-2, base-10).
• Overflow Error: An error when a value exceeds the storage capacity, causing incorrect results.
• Rounding Errors: Inaccuracies from approximating numbers with finite precision.
• Unicode System: A global standard for encoding characters from various languages.
• Variables: Containers for storing changeable data in programs.