Binary & Data Representation

Everything a computer stores — numbers, text, images, sound — is ultimately binary: sequences of 0s and 1s. Understanding this "Data" big idea is central to AP Computer Science Principles (CSP).

Bits and bytes

• A bit is a single 0 or 1.
• A byte is 8 bits and can represent 2⁸ = 256 different values (0–255).
• With n bits you can represent 2ⁿ distinct values.

Binary → decimal

Each place is a power of 2 (…, 8, 4, 2, 1):

1011₂ = 8 + 0 + 2 + 1 = 11₁₀

Decimal → binary

Subtract the largest fitting power of 2:

13 = 8 + 4 + 1 → 1101₂

Representing other data

• Text: each character maps to a number (e.g. ASCII: 'A' = 65), then to binary.
• Images: a grid of pixels; each pixel's colour is stored as RGB numbers (0–255 each).
• Because storage is finite, large or precise values cause overflow (too big to store) or round-off (lost precision) errors.

Worked example

How many values can 4 bits represent, and what is the largest?

2⁴ = 16 values, from 0 to 15 (1111₂).

Exam tip

CSP tests concepts and abstraction, not heavy math. Know that more bits → more values (and more storage), why analog data must be sampled to become digital, and the difference between lossless and lossy compression. You will not need long hand-conversions under time pressure — you need the reasoning.