What are the most common number bases used in computing?

Binary (base-2): the fundamental language of all digital electronics. Octal (base-8): Unix file permissions and legacy systems. Decimal (base-10): human-readable numbers. Hexadecimal (base-16): memory addresses, color codes, machine code, and network protocols. Base-64: encoding binary data as text. Base-36: compact alphanumeric identifiers. Each has a specific niche where its digit grouping matches the underlying data structure.

Can I convert floating-point numbers between bases?

Floating-point conversion is significantly more complex than integer conversion. A decimal fraction (e.g., 0.1₁₀) often produces a repeating binary fraction (0.0001100110011...₂), which is why 0.1 + 0.2 ≠ 0.3 in JavaScript. Our base converter handles integers; for floating-point understanding, the converter explains IEEE 754 representation concepts in the detailed breakdown section.

What is endianness and does it affect base conversion?

Endianness refers to byte ordering in memory: big-endian stores the most significant byte first (like we write numbers — "123" has the most significant digit first), while little-endian stores the least significant byte first (common in x86 and ARM architectures). Endianness affects how numbers are stored as bytes, but base conversion operates on the mathematical value, which is endian-agnostic. When you convert a number between bases, you're working with the abstract value, not its memory representation.

How do I mentally convert binary to hex and back?

Binary to hex: split the binary string into groups of 4 bits starting from the right (pad with leading zeros if needed), then replace each group with its hex digit. 0000=0, 0001=1, 0010=2, ..., 1001=9, 1010=A, ..., 1111=F. Example: 10111100 → 1011 1100 → B C → BC₁₆. Hex to binary: replace each hex digit with its 4-bit binary equivalent. This shortcut works because 16 = 2⁴, so every hex digit maps to exactly 4 bits.

Why do programmers use hexadecimal instead of decimal?

Hexadecimal directly mirrors binary in a human-readable format. One hex digit = 4 bits = a nibble. Two hex digits = 8 bits = a byte (values 00 to FF / 0 to 255). This makes hex ideal for viewing memory dumps, defining colors (RGB each fits in one byte), debugging network packets, and reading machine code. A 32-bit memory address like 0x7FFF_FFFF is far easier to parse in hex than as the decimal 2,147,483,647 — the structure (7FFF region) jumps out immediately.

What custom bases are practically useful beyond 16?

Base-36 (digits 0-9, A-Z): compact, case-insensitive, URL-safe identifiers used by URL shorteners, database keys, and JavaScript's toString(36). Base-58 (Bitcoin's choice): eliminates confusing characters (0, O, I, l) for human-readable cryptocurrency addresses. Base-62: similar to base-58 but includes all alphanumerics, used by YouTube video IDs. Base-64: the standard for encoding binary data as text in email (MIME), data URIs, JSON Web Tokens, and certificate files (PEM format). Each custom base reflects a specific tradeoff between compactness, readability, and character set constraints.

Calculators2026-05-10

Base Converter: How Number Systems Work and Why Every Programmer Needs One

Every number you've ever written is expressed in a base — you're just so used to base-10 that you don't think about it. But computers speak binary (base-2), programmers debug with hexadecimal (base-16), and Unix file permissions use octal (base-8). A base converter bridges these worlds, translating a value from one radix to another in milliseconds. Our free online base converter handles binary, octal, decimal, hexadecimal, and custom bases from 2 through 64, so whether you're decoding a hex color like #FF5733, verifying a memory address like 0x7FFF_FFFF, or making sense of Unix permission bits like 755, you'll have exactly the tool you need. All conversion runs locally in your browser — no data sent anywhere, no account required, and results appear the moment you type.

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Number Base Converter

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Step-by-Step Guide

Enter Your Number and Select the Source Base

Type or paste your number into the input field, then select the base it's currently in. Our converter supports bases 2 through 64 — binary (2), octal (8), decimal (10), hexadecimal (16), and beyond. The input is validated against the selected base: if you select binary, you can only type 0 and 1; if you select hex, digits 0-9 and letters A-F are accepted. For bases above 16, the converter uses uppercase letters A-Z followed by lowercase a-z and symbols to represent digit values beyond 9.

Choose One or More Target Bases

By default, the converter shows results in binary, octal, decimal, and hexadecimal simultaneously — the four number systems you need 95% of the time. You can add custom target bases (like base-36 for database keys or base-64 for data encoding) and remove ones you don't need. Each result appears in its own copyable card below the input, so you can grab the hex value for your CSS and the binary for your bitmask at the same time.

Read the Mathematical Breakdown

Below the results, the converter shows step-by-step division for the decimal conversion — the repeated division-by-base method where you read remainders from bottom to top. For example, converting decimal 202 to binary: 202÷2=101 R0, 101÷2=50 R1, 50÷2=25 R0, 25÷2=12 R1, 12÷2=6 R0, 6÷2=3 R0, 3÷2=1 R1, 1÷2=0 R1 — reading remainders upward gives 11001010₂. This is invaluable for students learning the concept and professionals verifying results.

Tips & Best Practices

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The division method (repeatedly divide by the target base, collect remainders) works for any base. For base-16, remainders 10-15 map to A-F. For base-2, the only possible remainders are 0 and 1. Practice it manually on small numbers — it builds intuition for how positional notation actually works.

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Hexadecimal color codes are just 3 bytes in base-16. #FF5733 means: red = FF (255 decimal), green = 57 (87 decimal), blue = 33 (51 decimal). A base converter lets you extract and manipulate these channels. Darken a color by reducing each hex pair proportionally.

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Unix file permissions use octal digits where each bit controls read (4), write (2), and execute (1). chmod 755 means: owner=4+2+1=7 (rwx), group=4+0+1=5 (r-x), others=4+0+1=5 (r-x). Understanding binary-to-octal mapping makes permission math second nature.

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Base-64 encoding (used in data URIs, email attachments, and JWTs) packs 6 bits per character — that's why the alphabet has 64 symbols (A-Z, a-z, 0-9, +, /). Every 3 bytes of binary data expand to 4 base-64 characters. Our converter handles bases up to 64 so you can experiment with this mapping directly.

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Base-36 (0-9 + A-Z) is the largest case-insensitive alphanumeric base. It's used in URL shorteners, database IDs, and JavaScript's toString(36). Converting a large integer to base-36 produces a compact, URL-safe string — 1,000,000,000 becomes "gjdgxs" in base-36.

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When converting between power-of-2 bases (binary, octal, hex), you can skip decimal entirely. Group binary digits: 3 bits per octal digit, 4 bits per hex digit. 1101 0110₂ = D6₁₆ in one step without ever converting to decimal first. This is how experienced engineers do it mentally.

Frequently Asked Questions

To convert a decimal number to base-N, repeatedly divide the number by N and collect the remainders. The first remainder is the least significant digit (rightmost). Read remainders from last to first to get the final number. Example: decimal 29 to base-16 — 29÷16=1 R13 (D), 1÷16=0 R1. Reading upward: 1D₁₆. This works for any positive integer and any base from 2 to 64.

A base converter is one of those tools that turns a 30-second mental grind into an instant, error-free result. Whether you're a programmer reading hex dumps, a student learning positional notation for the first time, or a sysadmin deciphering octal permissions, our free online base converter handles bases 2 through 64 with step-by-step breakdowns. Try it now and never count on your fingers in binary again.

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