Number Base Converter

Understanding Number Systems

A number system, or numeral system, is a method of representing numbers using a consistent set of symbols and rules. The system we use in everyday life is the decimal (base-10) system, which employs ten digits from 0 to 9. Computers, however, operate on the binary (base-2) system, using only the digits 0 and 1. Two additional systems are widely used in computing: octal (base-8), which uses digits 0 through 7, and hexadecimal (base-16), which extends the digit set with the letters A through F to represent values 10 through 15. Each system serves a different purpose and provides a more convenient way to represent data depending on the context.

Numbers 0-15 in All Four Bases

The table below shows the first 16 values (0 through 15) in binary, octal, decimal, and hexadecimal. These are the building blocks for all larger numbers and are essential to memorize for anyone working with low-level computing.

Notice how each hexadecimal digit corresponds to exactly 4 binary bits, and each octal digit corresponds to exactly 3 binary bits.

How Positional Number Systems Work

All the number systems used in computing are positional, meaning the value of each digit depends on its position within the number. In any base-B system, the rightmost digit has a positional weight of B⁰ (which equals 1), the next digit to the left has weight B¹, then B², and so on. To find the decimal value, multiply each digit by its positional weight and add the results together.

How Base Conversion Works

Converting a number from one base to another involves two fundamental steps. First, the number is interpreted in its original base by multiplying each digit by the base raised to the power of its position, starting from zero on the right. For instance, the binary number 1010 equals 1x8 + 0x4 + 1x2 + 0x1 = 10 in decimal. Second, to express a decimal number in a target base, you repeatedly divide by the target base and collect the remainders in reverse order. Dividing 10 by 2 yields remainders 0, 1, 0, 1, which read backwards gives 1010. For hexadecimal, dividing 255 by 16 gives 15 remainder 15, producing FF. This process works for any base and is the algorithm that our converter uses internally.

Why Hexadecimal Is Used in Computing

Hexadecimal (base 16) is ubiquitous in computing because it provides a compact, human-readable representation of binary data. Since 16 is a power of 2 (2⁴ = 16), each hex digit maps to exactly 4 binary bits. This means an 8-bit byte can always be expressed as exactly two hex digits, a 32-bit value as eight hex digits, and so on, with no wasted digits or ambiguity.

Octal in Unix File Permissions

Octal (base 8) has a special role in Unix and Linux systems for representing file permissions. Each file has three permission groups -- owner, group, and others -- and each group has three permission bits: read (r=4), write (w=2), and execute (x=1). Since three bits produce values 0 through 7, a single octal digit perfectly represents one permission group.

The command chmod 755 translates to: owner gets rwx (7), group gets r-x (5), and others get r-x (5). Similarly, chmod 644 means owner gets rw- (6), while group and others get r-- (4). This compact octal notation is far easier to read and type than the equivalent binary representation of nine individual permission bits.

Binary, Octal, Decimal, and Hexadecimal in Computing

Each number base plays a specific role in software development and computer engineering. Binary is the native language of hardware: processor instructions, memory addresses, and data storage all operate in binary. Octal was historically popular in systems with word sizes that were multiples of three bits. It is still used in Unix file permissions, where chmod 755 grants read, write, and execute permissions using octal notation. Decimal is used for user-facing values, financial calculations, and any context where human readability is paramount. Hexadecimal is the preferred shorthand for binary data because each hex digit maps cleanly to exactly four binary bits, making it far more compact. A single byte (8 bits) is represented by exactly two hex digits, which is why memory addresses, color codes like #FF5733, MAC addresses, and cryptographic hashes are almost always displayed in hexadecimal. Understanding how to move between these bases is an essential skill for programmers, network engineers, and anyone working close to the hardware level.

Common Powers of 2 Reference

Powers of 2 appear constantly in computing -- memory sizes, buffer lengths, hash outputs, and bit widths are all powers of 2. Memorizing the common ones helps you work fluently with binary and hexadecimal values.

Quick Conversion Shortcuts

Experienced developers use mental shortcuts to convert between bases without going through the full division algorithm every time.

• Binary to Hex: Group binary digits into sets of four from the right, then convert each group to its hex digit. For example, 1010 1111 becomes AF (1010=A, 1111=F).
• Hex to Binary: Replace each hex digit with its 4-bit binary equivalent. 3C becomes 0011 1100.
• Binary to Octal: Group binary digits into sets of three from the right. 101 111 becomes 57 in octal.
• Doubling for binary: Each position to the left doubles (1, 2, 4, 8, 16, 32, 64, 128). Mentally add the "on" positions to get decimal. For 11001: 16+8+1 = 25.