Convert numbers between binary, octal, decimal, and hexadecimal number systems instantly.
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Number systems are different ways of representing numbers; the decimal system used daily relies on 10 symbols (0-9), the binary system that computers use internally relies on only two symbols (0 and 1), the octal system relies on 8 symbols, and the hexadecimal system relies on 16 symbols (0-9 and A-F) and is used extensively in computer programming and representing colors. This calculator converts any entered number from its original base to the other three bases at once, making it an essential tool for computer science and digital engineering students and anyone working with low-level programming or networking. Every positional number system follows the same underlying logic regardless of its base: each digit's value depends on both the symbol itself and its position, multiplied by the base raised to a power that increases by one moving left — the only thing that changes between binary, octal, decimal, and hexadecimal is how many symbols are available at each position.
It can feel like decimal, binary, octal, and hexadecimal are four unrelated systems that each need to be memorized separately, but they are all built from exactly the same idea: positional value. In decimal, the number present at each position is multiplied by ten raised to an increasing power moving left — the digit 3 in 300 represents 3 × 10², not just '3'. Binary, octal, and hexadecimal work identically, just with 2, 8, and 16 substituted for 10 as the base.
Binary matters because computer hardware, at its most fundamental level, only reliably distinguishes between two physical states — on and off, high voltage and low voltage — which maps naturally onto the two symbols 0 and 1. Every piece of data a computer processes, from a text character to a high-resolution image, is ultimately stored and manipulated as long strings of binary digits, even though almost no software interface shows this to the user directly.
Hexadecimal exists specifically because raw binary is impractical for humans to read or write — a modest binary number can run to dozens of digits. Since 16 is a power of 2 (2⁴), each hexadecimal digit corresponds exactly to a group of four binary digits, making hexadecimal a compact, lossless shorthand for binary that programmers, network engineers, and hardware designers use constantly. Memory addresses, IPv6 addresses, and color codes in web design (like #FF5733) are all conventionally written in hexadecimal for exactly this reason.
Octal is less common today but still appears in specific technical contexts, most notably Unix and Linux file permission codes (a permission setting like 755 is an octal number, where each digit represents a combination of read, write, and execute permissions). Octal's usefulness comes from the same logic as hexadecimal — 8 is 2³, so each octal digit maps cleanly onto three binary digits.
Converting between these systems by hand follows a consistent method: to convert any base to decimal, multiply each digit by its base raised to its positional power and sum the results; to convert decimal to another base, repeatedly divide by the target base and record the remainders in reverse order. Understanding this shared logic — rather than memorizing each conversion as an isolated procedure — is what makes it possible to reason about any positional number system, including ones not covered by name in a typical course.
Because digital circuits naturally represent two states (on/off), which maps directly to the two binary digits, 0 and 1.
In programming (memory addresses, color codes in web design like #FF5733) because it's a compact way to represent binary data.
Yes, just select 'Hexadecimal' as the input base and the result will show the equivalent binary, octal, and decimal values.