Base Converter

A base converter calculator is a tool that converts numbers between different numeral systems (binary, octal, decimal, hexadecimal, and other bases from 2 to 36). It takes a number in one base and translates it to an equivalent representation in another base, showing the mathematical process behind each conversion.

The decimal system (base 10) uses digits 0-9. It is the standard system humans use for everyday counting and mathematics. Each position represents a power of 10.

The binary system (base 2) uses only digits 0 and 1. It is the fundamental language of computers, where each bit represents an on/off state in digital circuits.

The hexadecimal system (base 16) uses digits 0-9 and letters A-F (where A=10, B=11, ..., F=15). It provides a compact representation of binary data, making it easier for programmers to read and work with.

The octal system (base 8) uses digits 0-7. It was commonly used in early computing and is still used in Unix file permissions (e.g., chmod 755).

Multiply each bit by 2 raised to its position power (starting from 0 on the right), then sum all the results. For example, 1011 = (1×2³) + (0×2²) + (1×2¹) + (1×2⁰) = 11.

Repeatedly divide the decimal number by 2 and record the remainders. Read the remainders from bottom to top to get the binary result. For example, 13 ÷ 2 = 6 remainder 1, 6 ÷ 2 = 3 remainder 0, 3 ÷ 2 = 1 remainder 1, 1 ÷ 2 = 0 remainder 1 → 1101.

Multiply each hex digit by 16 raised to its position power, using A=10, B=11, C=12, D=13, E=14, F=15 for letters. Sum all results. For example, 2A = (2×16¹) + (10×16⁰) = 42.

Repeatedly divide by 16 and record remainders. Remainders 10-15 become A-F. Read remainders from bottom to top. For example, 42 ÷ 16 = 2 remainder 10 (A), 2 ÷ 16 = 0 remainder 2 → 2A.

Multiply each octal digit by 8 raised to its position power, then sum all results. For example, 157 = (1×8²) + (5×8¹) + (7×8⁰) = 111.

Repeatedly divide by 8 and record remainders. Read remainders from bottom to top. For example, 111 ÷ 8 = 13 remainder 7, 13 ÷ 8 = 1 remainder 5, 1 ÷ 8 = 0 remainder 1 → 157.

Group binary digits into sets of 4 (from right to left), then convert each group to its hex equivalent. For example, 11010110 → 1101 0110 → D6.

Convert each hex digit to its 4-bit binary equivalent. For example, 3F → 3 = 0011, F = 1111 → 00111111.

Group binary digits into sets of 3 (from right to left), then convert each group to its octal equivalent. For example, 101110 → 101 110 → 56.

Convert each octal digit to its 3-bit binary equivalent. For example, 56 → 5 = 101, 6 = 110 → 101110.

First convert hex to binary, then convert the binary result to octal. Alternatively, convert hex to decimal first, then decimal to octal.

First convert octal to binary, then convert the binary result to hex. Alternatively, convert octal to decimal first, then decimal to hex.

The calculator supports arbitrary precision, so you can convert very large numbers. However, extremely large values may take longer to process.

Yes, the calculator handles both integer and fractional parts. For fractions, multiply the fractional part by the target base repeatedly and collect the integer parts.

For binary fractions, multiply each bit after the binary point by 2 raised to its negative position power. For example, 0.101 = (1×2⁻¹) + (0×2⁻²) + (1×2⁻³) = 0.625.

Repeatedly multiply the fractional part by 2 and record the integer result (0 or 1). Continue until the fraction becomes 0 or you reach desired precision. For example, 0.625 × 2 = 1.25 (1), 0.25 × 2 = 0.5 (0), 0.5 × 2 = 1.0 (1) → 0.101.

Yes, the calculator handles signed numbers. In binary, negative numbers are typically represented using two's complement notation.

Two's complement is a method of representing signed integers in binary. To get the negative of a number, invert all bits and add 1. For example, +5 = 0101, -5 = 1011.

One's complement represents negative numbers by inverting all bits of the positive value. For example, +5 = 0101, -5 = 1010.

Sign-magnitude uses the leftmost bit as the sign bit (0=positive, 1=negative) and remaining bits for the magnitude. For example, +5 = 0101, -5 = 1101.

Enter your number in the input field, select the source base (e.g., binary, decimal), select the target base, and click "Convert" to see the result with step-by-step explanation.

Bases 2-36 are supported because they can be represented using digits 0-9 and letters A-Z. Higher bases would require additional symbols that are not standard.

Binary (base 2), octal (base 8), decimal (base 10), and hexadecimal (base 16) are the most commonly used bases in computing.

Binary is the native language of computers. All data—text, images, programs—is ultimately stored and processed as binary sequences of 0s and 1s.

Hex provides a human-readable representation of binary data. Each hex digit corresponds to exactly 4 bits, making it easy to read and interpret binary values.

Octal is used in Unix file permissions (chmod), some programming languages, and certain legacy systems. It provides a compact representation using 3-bit groups.

Yes, you can convert between any bases from 2 to 36. Simply select your source base and target base.

Place value determines a digit's contribution based on its position. In base B, the digit at position n (from right, starting at 0) contributes digit × B^n.

Read from left to right. Each position represents a power of 2, starting from 2⁰ on the right. Add the values where the bit is 1.

Read from left to right. Each position represents a power of 16, starting from 16⁰ on the right. Letters A-F represent values 10-15.

Each hex digit corresponds to exactly 4 bits. This makes hex a convenient shorthand for binary—1 hex digit = 1 nibble = 4 bits.

A nibble is a group of 4 bits, which can represent values from 0 to 15. It corresponds to one hexadecimal digit.

A byte is a group of 8 bits, which can represent values from 0 to 255. It corresponds to 2 hex digits.

IP addresses are typically represented in decimal dotted notation (e.g., 192.168.1.1). Convert each octet (0-255) to binary for subnet calculations.

RGB colors are often represented in hexadecimal (e.g., #FF5733). Each pair of hex digits represents one color channel (0-255 in decimal).

Unix file permissions use octal notation (e.g., 755). Each digit represents read (4), write (2), and execute (1) permissions for owner, group, and others.

Memory addresses are often displayed in hexadecimal for readability. Convert to decimal for offset calculations or to binary for bit-level operations.

Big-endian stores the most significant byte first; little-endian stores it last. Reverse the byte order when converting between the two representations.

Scientific notation represents numbers as a × 10^b. It is useful for very large or very small numbers, but base conversion typically works with standard notation.

The calculator supports arbitrary precision, so you can convert numbers of any size. For extremely large values, the conversion may take slightly longer.

For very small decimal numbers, use scientific notation or ensure sufficient precision in the fractional part during conversion.

Yes. Convert each pair of hex digits to decimal. For example, #FF5733 → FF=255 (red), 57=87 (green), 33=51 (blue).

Yes. Convert each decimal color value (0-255) to hexadecimal and concatenate. For example, RGB(255, 87, 51) → #FF5733.

Group binary into 8-bit bytes, convert each byte to decimal, then use the ASCII table to find the corresponding character.

Look up each character in the ASCII table to get its decimal value, then convert each decimal to 8-bit binary.

A base converter changes number representation between bases. A calculator performs arithmetic operations (add, subtract, multiply, divide).

Unsigned conversion handles only positive numbers. Signed conversion also handles negative numbers using representations like two's complement.

Integer conversion handles whole numbers only. Fractional conversion also handles the decimal part of a number.

Instant conversion shows only the final result. Step-by-step shows the complete conversion process, useful for learning.

Use binary conversion when working with digital logic, bitwise operations, computer architecture, or low-level programming.

Use hex conversion when working with memory addresses, color codes, hash values, or reading binary dumps.

Use octal conversion when working with Unix file permissions, certain programming languages, or legacy systems.

The calculator uses standard mathematical algorithms and provides exact results for integer conversions. Fractional conversions may have rounding for some bases.

Yes, the base converter calculator is completely free to use with no registration or payment required.

Yes, all conversions are performed in your browser. No data is transmitted to any server.

Yes, the calculator is mobile-friendly and works on smartphones, tablets, and desktop computers.

Once loaded, the calculator performs all conversions locally in your browser, so it works offline.

Convert between binary, hex, and decimal for debugging, writing code, or understanding how data is represented in memory.

Convert IP addresses between decimal and binary for subnet calculations and network configuration.

Convert between binary and other bases when designing or analyzing digital circuits and logic gates.

Use step-by-step mode to learn how number systems work and understand the conversion process.

Convert between bases when working with encryption keys, hash values, or encoding schemes.

Convert between number bases for various encoding schemes used in data transmission and storage.

Convert between binary and hex when working with hash values (MD5, SHA-1, SHA-256).

Convert UUID hex components to binary for processing or comparison.

For integer conversion: repeatedly divide by target base, collect remainders. For fraction: repeatedly multiply and collect integer parts.

Value = Σ(digit_i × base^i) for each position i, starting from 0 on the right.

For negative numbers, convert the absolute value first, then apply the appropriate sign representation (two's complement, one's complement, or sign-magnitude).

The calculator uses big integer arithmetic to handle numbers of any size without rounding.

The calculator shows each division or multiplication step with intermediate results, explaining the process.

Yes, you can convert between any base from 2 to 36.

Yes, enable the step-by-step option to see the complete conversion process.

Yes, convert both integer and fractional parts.

Yes, convert binary to decimal, octal, hex, or any other base.

Yes, convert hexadecimal to binary, decimal, octal, or any other base.

Yes, convert decimal to binary, octal, hex, or any other base.

Yes, convert octal to binary, decimal, hex, or any other base.

Yes, enter any base from 2 to 36.

Yes, handle scientific notation, engineering notation, and standard form.

Binary (base 2) is the most fundamental, but hexadecimal (base 16) is most commonly used for human-readable representation.

Hex provides a compact, human-readable representation of binary data. Each hex digit maps to exactly 4 bits.

Convert between hex and binary when examining memory dumps, register values, or error codes.

Convert octal file permissions (chmod) to understand or set access rights.

Convert hex color codes to RGB values or vice versa for styling.

Convert between bases when working with binary data types or hex-encoded values.

Convert between bases when analyzing encryption keys, certificates, or hash values.

Convert between hex and binary when programming microcontrollers or working with hardware registers.

Convert between bases when analyzing compressed data formats or implementing compression algorithms.

Convert between binary patterns and other representations for barcode encoding and decoding.

Convert between binary data and hex for QR code generation and reading.

Convert between binary data and hex for NFC programming and data exchange.

Convert between binary data and hex for RFID tag encoding and reading.

Convert between bases for app development, data encoding, and compatibility testing.

Convert between bases for device communication, sensor data, and protocol implementation.

The Base Converter uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Base Converter accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.