How Positional Number Systems Work

Every number system we use daily is positional — the value of a digit depends on its position. In decimal (base 10), the number 347 means 3×10² + 4×10¹ + 7×10⁰ = 300 + 40 + 7. Binary (base 2) works identically but with powers of 2: the binary number 1011 means 1×2³ + 0×2² + 1×2¹ + 1×2⁰ = 8 + 0 + 2 + 1 = 11 in decimal. This same principle applies to octal (base 8) and hexadecimal (base 16). Understanding this pattern makes converting between any two bases straightforward: convert to decimal as an intermediate step, then convert from decimal to the target base.

Binary: The Language of Computers

Computers use binary because transistors — the fundamental building blocks of processors — have two states: on (1) and off (0). Every piece of data in a computer, from text to images to videos, is ultimately stored as sequences of binary digits (bits). A group of 8 bits forms a byte, which can represent values from 0 (00000000) to 255 (11111111). This is why many computing limits relate to powers of 2: a 32-bit integer can hold values up to 2³² − 1 = 4,294,967,295, and a 64-bit system can address 2⁶⁴ bytes of memory (16 exabytes). ASCII encodes each character as a 7-bit binary number, while modern Unicode extends this to 21 bits per code point.

Hexadecimal in Practice

Hexadecimal's power lies in its compact representation of binary data. Since 16 = 2⁴, each hex digit maps to exactly 4 binary digits. This means a full byte (8 bits) is always exactly 2 hex digits, and a 32-bit value is 8 hex digits — far easier to read than 32 binary digits. In web development, CSS colors use hex notation (#RRGGBB) where each pair represents one byte (0–255) of red, green, or blue intensity. In debugging, memory addresses and machine code are displayed in hex. MAC addresses (like A4:CF:12:D0:5B:3E) use hex to represent 6 bytes of network hardware identification. Even Unicode code points are written in hex (U+1F600 is the grinning face emoji).

Frequently Asked Questions