Number Conversion   | Decimal (Base 10) → Binary (Base 2)

Supported Numeral Systems

Name	Description	Radix
Binary (Base 2)	The Binary (Base 2) is a numeral system that uses only two digits: 0 and 1. Each digit in a binary number is called a bit (short for binary digit). Features: Only two possible values for each digit (0 or 1). Each position represents a power of 2 (e.g., from right to left: 20, 21, 22, etc.). Example: The binary number 1011 represents: 1 * 23 + 0 * 22 + 1 * 21 + 1 * 20 = 8 + 0 + 2 + 1 = 11 in decimal. Common Uses: Computing The binary system is fundamental to digital computing, where data is processed as binary numbers by computers' hardware (using transistors for 0s and 1s). Storage Binary is used in storage media, like hard drives and memory, to represent data. Logic Circuits Used in digital electronics, including logic gates and circuits that perform mathematical operations.	2
Octal (Base 8)	The Octal (Base 8) numeral system, also known as the octal system, uses eight digits: 0, 1, 2, 3, 4, 5, 6 and 7. Each digit in an octal number represents a power of 8. Features: Only eight possible digits (0 to 7) are used. Each position represents a power of 8 (e.g., from right to left: 80, 81, 82, etc.). Example: The octal number 17 represents: 1 * 81 + 7 * 80 = 8 + 7 = 15 in decimal. Common Uses: Computing Octal was historically used in computing as a shorthand for binary numbers. Each octal digit represents exactly three binary digits (bits). Digital Systems It was used for compact representation of binary data in early computer systems, though hexadecimal (base-16) is more common today. Memory Addressing In some older computer systems, octal was used for addressing and organizing data.	8
Decimal (Base 10)	The Decimal (Base 10) system is the base-10 numeral system, which uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. It is the standard system for denoting integer and non-integer numbers and is the most widely used number system in everyday life. Features: Ten digits are used to represent all numbers. Each position in a decimal number represents a power of 10 (e.g., from right to left: 100, 101, 102, etc.). Example: The decimal number 345 represents: 3 * 102 + 4 * 101 + 5 * 100 = 300 + 40 + 5 = 345. Common Uses: Everyday Life The decimal system is the standard system used for counting, measuring and performing arithmetic in daily activities. Mathematics and Science It is used universally in mathematics, engineering and scientific calculations. Currency Most modern currencies are based on the decimal system.	10
Hexadecimal (Base 16)	The Hexadecimal (Base 16) number system uses sixteen digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, where A through F represent the values 10 to 15. Features: Uses 16 distinct symbols (0 - 9 and A - F) to represent numbers. Each position represents a power of 16 (e.g., from right to left: 160, 161, 162, etc.). Example: The hexadecimal number 1A3 represents: 1 * 162 + A * 161 + 3 * 160 = 1 * 256 + 10 * 16 + 3 = 419 in decimal. Common Uses: Computing Hexadecimal is widely used in computing as a more compact representation of binary numbers. Every 4 bits (a nibble) correspond to 1 hexadecimal digit. Memory Addresses It's used to represent memory addresses in programming, as it's more concise than binary and easier to read. Color Codes In web development and design, hexadecimal codes are used to define colors (e.g., #FF5733).	16

How Positional Notation Works

Every one of these systems works the same way. Each digit position carries a weight that is a power of the base, and the value of the number is the sum of each digit multiplied by its position's weight. Only the base changes.

In decimal, 237 means 2×10² + 3×10¹ + 7×10⁰. In binary, 1101 means 1×2³ + 1×2² + 0×2¹ + 1×2⁰, which comes to 13. The mechanism is identical; the digits available and the multiplier differ.

That is why the same quantity looks so different across bases. The number thirteen is 13 in decimal, 1101 in binary, 15 in octal, and D in hexadecimal — one value, four representations.

Why Each Base Exists

Binary

Two digits, 0 and 1, matching the two states a circuit can reliably distinguish: current or no current. Everything a computer stores is ultimately binary; the other bases here are conveniences for humans reading that data.

Octal

Eight digits, each one standing for exactly three bits. Its most visible survival is Unix file permissions, where chmod 755 encodes three groups of three permission bits — read, write, execute — as three octal digits.

Decimal

Ten digits, near-universal among humans for the obvious anatomical reason. It has no special relationship to computer hardware, which is exactly why conversion is needed so often.

Hexadecimal

Sixteen digits — 0-9 then A-F — each representing exactly four bits, so one byte is always two hex digits. This tidy mapping is why memory addresses, color codes, and byte dumps are written in hex: it is far more compact than binary while still lining up cleanly with the underlying bits.

Reference Table

The first sixteen values in each base:

• 0-3 — binary 0, 1, 10, 11; octal 0, 1, 2, 3; hex 0, 1, 2, 3
• 4-7 — binary 100, 101, 110, 111; octal 4, 5, 6, 7; hex 4, 5, 6, 7
• 8-11 — binary 1000, 1001, 1010, 1011; octal 10, 11, 12, 13; hex 8, 9, A, B
• 12-15 — binary 1100, 1101, 1110, 1111; octal 14, 15, 16, 17; hex C, D, E, F

Notice that four binary digits map onto exactly one hex digit. Converting between binary and hexadecimal is therefore just grouping bits into fours, with no arithmetic required — which is the whole reason programmers prefer hex to binary.

Common Uses

• Web colors — #FF8800 is three bytes: red 255, green 136, blue 0.
• File permissions — reading and writing Unix modes such as 644 or 755.
• Memory addresses and debugging — stack traces, hex dumps, and pointer values.
• Bitmasks and flags — seeing which individual bits a configuration value has set.
• Network addressing — subnet masks make far more sense read as binary.
• Character encodings — Unicode code points are conventionally written in hex, as in U+00E9.

Common Pitfalls

• A leading zero can mean octal. In several programming languages 0755 is not seven hundred fifty-five. This has caused real bugs.
• Hex letters are digits, not text. In 2F the F is the value fifteen, not a letter.
• Watch the prefixes.0b marks binary, 0o octal, and 0x hexadecimal. They label the notation and are not part of the value.
• A digit must be valid for its base. There is no digit 8 in octal and no 2 in binary; such input is not a number in that base at all.
• Case does not matter in hex.ff and FF are the same value, though a given codebase usually settles on one.

Frequently Asked Questions

Why do computers use binary rather than decimal?

Reliability. Distinguishing two voltage states is straightforward and resists noise; distinguishing ten distinct levels in every circuit would be far more error-prone and expensive. Two states also map directly onto Boolean logic.

Why is hexadecimal preferred over binary for display?

Because it is a quarter the length and converts to bits without any arithmetic. A 32-bit address is 32 binary digits, which is hard to read and easy to miscount, or 8 hex digits, which is not.

What does the "0x" in front of a number mean?

It marks the value as hexadecimal. Without it, 10 is ambiguous — sixteen in hex, ten in decimal, two in binary. The prefix removes the ambiguity and is not part of the number itself.

Can I convert negative numbers or fractions?

This tool handles non-negative integers, which covers the great majority of practical cases. Negative values in real systems are usually represented in two's complement, whose meaning depends on the word width in use.

Is octal still used?

Mostly in one place: Unix file permissions, where its three-bits-per-digit grouping fits the permission model exactly. It was more widespread on older hardware whose word sizes were multiples of three bits.