Binary Numbers

Computers store everything — numbers, text, images, sound — as a sequence of bits. A bit (binary digit) is either 0 or 1. A group of 8 bits is a byte.

• Binary (base 2) uses two digits; each position is a power of 2 (…, 16, 8, 4, 2, 1), unlike decimal (base 10).
• Abstraction — the same bits can mean different things (a number, a letter, a color) depending on how the program interprets them.
• Analog data varies smoothly and continuously (sound waves). Computers capture it by sampling — measuring at regular intervals — to get digital data. More samples means a closer match to the original.

• n bits can represent 2ⁿ different values, from 0 up to 2ⁿ − 1. Adding one bit doubles the number of values.
• To represent N distinct items, you need enough bits so that 2ⁿ ≥ N.
• Overflow error — a result is too large for the fixed number of bits available.
• Round-off error — some real numbers (like 1/3 or 0.1) can't be stored exactly in binary, so the computer stores a close approximation.
• Examples of representation: text via ASCII/Unicode codes, images as pixels with red-green-blue values, sound as sampled amplitudes.

Data Compression

Data compression reduces the number of bits needed to store or transmit data. Smaller files save storage space and send faster over a network. There are two types:

• Trade-off — lossy gives smaller files but lower quality; lossless keeps quality but saves less space.
• Choose based on need: lossless for text, code, or medical images where every bit matters; lossy for streaming video or music where small quality loss is acceptable.
• Lossless compression often finds repeated patterns. Example — run-length encoding:

Original :  AAAABBBCC     (9 characters)
Compressed: 4A3B2C        (6 characters)
// Decompressing gives back AAAABBBCC exactly, so this is lossless.

Extracting Information from Data

Data is raw values; information is the meaning and patterns drawn from it. Computing lets us find trends in datasets far too big for people to analyze by hand.

• Metadata — data about data (e.g. a photo's date, location, size, or author). It helps organize, find, and group data, and changing it doesn't change the main data.
• Big data — extremely large datasets that need computers (often many working together) to process.
• Correlation is not causation — two things changing together doesn't prove one causes the other.
• Data cleaning — fixing problems such as incomplete, duplicate, invalid, or inconsistent entries so results are accurate.
• Bias in data — a dataset that leaves out groups or is collected unfairly can lead to misleading conclusions.
• Combining data sources can reveal new insights (e.g. weather data plus sales data), but can also raise privacy concerns.
• Scalability — a solution that still works well as the amount of data grows.

Using Programs with Data

Programs help us collect, store, process, and visualize data. Data can come from files, databases, sensors, websites, or user input, and programs let us work with it at scale.

passing <- []
FOR EACH score IN scores
{
    IF (score >= 70)
    {
        APPEND(passing, score)     // filtering
    }
}
DISPLAY(LENGTH(passing))           // counting

• Tools — spreadsheets, databases, and programs can all be used to analyze data; programs are best for large or repeated tasks.
• Visualizations (charts, graphs, tables) make patterns easier to see and communicate.
• Challenges — data can be messy, huge, biased, or in different formats; cleaning and checking it is a major part of the work.
• Privacy and security — collecting and combining personal data must be done responsibly.