Information Encoding and Binary Representation

Everything Is Bits

A computer stores all information using just two states: on and off, represented as 0 and 1. A single digit is a bit, and eight bits make a byte.

What a bit pattern means depends entirely on its interpretation. The exact same byte $01000001$ can be the decimal number $65$, the letter $A$ in ASCII, or part of an image.

Binary is base-2, where positions are powers of $2$. An $n$-bit value gives $2^n$ combinations; an 8-bit byte holds $2^8 = 256$ values ($0$ to $255$).

Common pitfall: Assuming a bit pattern has an inherent meaning. Always check how data is interpreted.

Hex, Text, and Formulas

Because long binary strings are clumsy, programmers use hexadecimal (base 16, digits $0$–$9$ and $A$–$F$). Each hex digit packs 4 bits, so one byte equals two hex digits.

Text uses numeric mapping: ASCII uses 7 bits for 128 English characters, while Unicode extends this globally to cover every writing system.

The decimal value $d$ of any binary sequence is calculated by summing each bit $b_i$ multiplied by its position's power of 2:

$$d = \sum_{i=0}^{n-1} b_i \cdot 2^i$$

Symbol Guide:

• $d$: The resulting decimal number
• $n$: Total number of bits
• $b_i$: The bit value ($0$ or $1$) at position $i$
• $2^i$: The power of 2 for position $i$