Convert numbers between binary, decimal, hexadecimal, and octal. Perform arithmetic operations in any number system.
Binary (base-2) uses only two digits, 0 and 1, because computer hardware fundamentally operates on transistors that are either in an "on" or "off" electrical state - there's no natural way for hardware to reliably represent ten distinct states the way decimal (base-10) requires, which is exactly why binary became the foundation of all digital computing rather than an arbitrary design choice. Every piece of data a computer processes - text, images, video, numbers - is ultimately represented as binary at the hardware level.
Computer science students and programmers convert between binary, decimal, and hexadecimal regularly when working with low-level programming, understanding memory addresses, working with bitwise operations, or debugging network protocols where data is often represented in binary or hex format for compactness.
In binary, each position represents a power of 2 (1, 2, 4, 8, 16, and so on) rather than a power of 10 like decimal - converting binary to decimal means adding up the value of each position where a 1 appears, which is the core mechanic behind manual binary-to-decimal conversion.
Binary (base 2) uses only two digits, 0 and 1, which maps naturally onto the two physical states an electronic circuit can reliably represent — voltage present or absent, current flowing or not. This simplicity makes binary far more reliable for hardware implementation than a system with more possible states, since distinguishing between just two clear states is much less prone to error from electrical noise or component variation than distinguishing between ten states, as decimal would require.
Every piece of digital data — text, images, audio, video, program instructions — is ultimately stored and processed as sequences of binary digits (bits) at the hardware level, even though higher-level programming languages and interfaces abstract this away so most users and even most programmers rarely interact with raw binary directly.
Binary addition follows the same carrying principle as decimal addition, just with a much lower threshold for carrying — 1 + 1 in binary equals 10 (which is 2 in decimal), requiring a carry to the next column, similar to how 9 + 1 carries in decimal. Understanding this carrying behavior is essential for anyone doing manual binary arithmetic, and it's the same underlying principle that computer processors use, implemented in hardware circuits, to perform addition at the electronic level.
Binary is a base-2 number system using only 0 and 1. It's the fundamental language of computers, representing all data as sequences of bits. Each binary digit (bit) represents a power of 2.
Hexadecimal (base-16) is commonly used in computing to represent binary data in a more human-readable format. Each hex digit represents 4 binary digits (bits), making it useful for memory addresses, color codes, and more.
To convert: first convert to decimal, then from decimal to the target base. For example, to convert binary to hex, convert binary to decimal, then decimal to hex. Our calculator does this automatically for you.
Yes! Enter your numbers in their respective bases, select the operation, and the calculator will perform the calculation and show the result in all number systems. This is useful for computer science and digital electronics.
Binary: 0, 1. Octal: 0-7. Decimal: 0-9. Hexadecimal: 0-9 and A-F (or a-f). Hexadecimal letters represent values 10-15 respectively.
Computer hardware operates on transistors that are either on or off, which naturally maps to two states (0 and 1) rather than the ten states decimal would require.
Each binary digit represents a power of 2 based on its position - add up the value of each position where a 1 appears to get the decimal equivalent.
Binary uses only 0 and 1, while hexadecimal uses 16 symbols (0-9, A-F) - hex is often used as a more compact, readable representation of binary data.
Enter your octal number above and convert it to binary by first converting to decimal, then to binary - this calculator handles octal, binary, and decimal conversions together.
