Binary, Hexadecimal, and Number Systems Explained

Every number you encounter in daily life uses the decimal system, but computers, programmers, and engineers constantly work with alternative number bases. Binary powers every digital device, hexadecimal simplifies how developers read machine-level data, and octal still appears in Unix file permissions.

This guide explains how positional number systems work, walks through the four most important bases, and teaches you to convert between them with confidence.

What Are Number Systems?

A number system (or numeral system) is a method of representing numbers using a consistent set of symbols and rules. The key concept behind the number systems used in computing is positional notation, where the value of a digit depends on its position within the number.

In the decimal number 753, the digit 7 represents 7 × 10² = 700, the digit 5 represents 5 × 10¹ = 50, and the digit 3 represents 3 × 10° = 3. Each position corresponds to a power of the base (10 in this case). This same principle applies to every positional number system, whether the base is 2, 8, 16, or any other positive integer.

The base (or radix) of a number system determines how many unique digits it uses and the multiplier for each position. Base 10 uses digits 0 through 9. Base 2 uses only 0 and 1. Base 16 uses 0 through 9 plus the letters A through F.

The Four Major Number Bases

Binary (Base 2)

Binary is the foundation of all digital computing. It uses exactly two digits: 0 and 1, called bits (binary digits). Every piece of data in a computer, from a text document to a streaming video, is ultimately stored and processed as sequences of bits.

In binary, each position represents a power of 2. Reading from right to left, the positions represent 1, 2, 4, 8, 16, 32, 64, 128, and so on.

The binary number 10110101 means: (1 × 128) + (0 × 64) + (1 × 32) + (1 × 16) + (0 × 8) + (1 × 4) + (0 × 2) + (1 × 1) = 181.

Computers use binary because digital circuits operate with two voltage states. A transistor is either on (1) or off (0). This binary state is the most reliable way to store information electronically, since distinguishing between two states is far more noise-resistant than distinguishing between ten states.

Octal (Base 8)

Octal uses digits 0 through 7. Each octal digit corresponds to exactly three binary digits, making it a convenient shorthand for binary when the context calls for it.

The most common modern use of octal is in Unix and Linux file permissions. When you set permissions using a command like chmod 755, each digit is an octal number. The digit 7 (binary 111) grants read, write, and execute permissions, while 5 (binary 101) grants read and execute only.

Decimal (Base 10)

Decimal is the number system you use every day. It uses ten digits (0 through 9) and each position represents a power of 10. The prevalence of base 10 is likely connected to humans having ten fingers, which provided a natural counting tool for early civilizations.

While decimal is intuitive for human calculation, it is not inherently superior to other bases mathematically. It simply became the standard because of our anatomy. Different cultures have used bases 5, 12, 20, and 60 throughout history, each with its own advantages.

Hexadecimal (Base 16)

Hexadecimal (often shortened to "hex") uses sixteen symbols: the digits 0 through 9 and the letters A through F, where A = 10, B = 11, C = 12, D = 13, E = 14, and F = 15.

Each hexadecimal digit represents exactly four binary digits (bits), because 16 = 2&sup4;. This makes hex the preferred shorthand for binary in computing. A single byte (8 bits) can be represented by exactly two hex digits, ranging from 00 to FF (0 to 255 in decimal).

You encounter hex values in web development (HTML color codes like #3B82F6), memory addresses (0x7FFF5FBFF8A0), MAC addresses (00:1A:2B:3C:4D:5E), and debugging output. Wherever raw binary data needs to be displayed in a human-readable format, hexadecimal is the standard choice.

How to Convert Between Number Bases

Converting between number bases is a core skill in computer science and digital electronics. The two most common conversions are decimal to binary and binary to hexadecimal.

Decimal to Binary Conversion

The repeated division method converts any decimal number to binary. Divide the number by 2 repeatedly, recording each remainder. The binary result is the remainders read from bottom to top.

To convert from binary back to decimal, multiply each bit by its positional power of 2 and sum the results. This is the expansion method.

Binary to Hexadecimal Conversion

Converting between binary and hex is straightforward because each hex digit maps to exactly four bits. Group the binary digits into sets of four (starting from the right), then replace each group with its hex equivalent.

For binary numbers whose length is not a multiple of four, pad with leading zeros. For example, 110101 becomes 0011 0101, which converts to 35 in hex.

Practical Examples

Number system conversions arise constantly in technology, design, and networking. Here are three scenarios that demonstrate these skills in action.

Number Base Reference Table

This table provides a quick reference for common values across the four major number systems. Keep it handy when performing conversions or reading technical documentation.

Tips and Complete Guide

Mastering number system conversions requires practice, but these strategies will help you work faster and avoid errors.

• Memorize the 4-bit binary-to-hex table. There are only 16 entries (0000 through 1111 mapping to 0 through F). Once memorized, you can convert between binary and hex instantly without any arithmetic.
• Learn powers of 2 up to 2¹². Knowing that 2¹&sup0; = 1,024 and 2¹&sup6; = 65,536 gives you immediate reference points for evaluating binary numbers and understanding memory sizes.
• Always verify conversions. After converting a number, convert it back to the original base to confirm your result. If 156 in decimal becomes 9C in hex, verify by calculating 9 × 16 + 12 = 156.
• Use prefix notation to avoid ambiguity. In programming and documentation, use 0b for binary (0b10011100), 0o for octal (0o234), and 0x for hexadecimal (0x9C). This prevents confusion about which base a number uses.
• Group binary digits for readability. Long binary strings are hard to read. Group them in fours (for hex conversion) or eights (for bytes): 1001 1100 is much easier to parse than 10011100.
• Practice with real data. Decode hex color codes, read IP addresses from packet dumps, or convert file permissions to binary. Working with real-world examples reinforces the patterns far better than abstract exercises.

Common Mistakes to Avoid

• Reading remainders in the wrong direction. When converting decimal to binary using repeated division, the first remainder is the least significant bit (rightmost). Read the remainders from bottom to top, not top to bottom.
• Forgetting to pad binary groups. When converting binary to hex, if the leftmost group has fewer than four bits, pad it with leading zeros. Otherwise you will map the wrong binary pattern to the hex digit.
• Confusing hex letters with decimal digits. The hex value 10 equals decimal 16, not decimal 10. Always be explicit about which base you are using, especially in written communication.
• Mixing up A-F values. A common error is thinking A = 1 instead of A = 10. Remember: A = 10, B = 11, C = 12, D = 13, E = 14, F = 15.
• Ignoring signed vs unsigned representation. In computing, the same binary pattern can represent different decimal values depending on whether it is interpreted as signed or unsigned. The byte 11111111 is 255 unsigned but -1 in signed two's complement notation.

Last updated: February 23, 2026