Number Base Converter
Convert any number between any bases 2–36: binary, octal, decimal, hex, and beyond. Shows positional expansion and repeated-division step-by-step.
What is a Number Base?
The number base (or radix) is the number of distinct digit symbols used in a positional numeral system. In positional notation, the value of each digit depends on its position: the digit at position i (counting from 0 on the right) represents that digit multiplied by the base raised to the power i.
Common bases:
- Base 2 (binary) — digits 0, 1. The language of digital electronics.
- Base 8 (octal) — digits 0–7. Used in Unix file permissions.
- Base 10 (decimal) — digits 0–9. Our everyday number system.
- Base 16 (hexadecimal) — digits 0–9 and A–F. Standard for memory addresses, color codes, and byte-level data.
Binary, octal, and hex are all powers of 2 (2¹, 2³, 2⁴), which means they interconvert without an intermediate decimal step — 4 binary digits map to one hex digit, and 3 binary digits map to one octal digit.
Formula
Converting to decimal (positional expansion): n₁₀ = d_k × b^k + d_{k−1} × b^{k−1} + … + d_1 × b + d_0
Converting from decimal (repeated division):
- Divide n by the target base b
- Record the remainder (this is the lowest-order digit)
- Replace n with the quotient and repeat until n = 0
- Read remainders from bottom to top
Direct binary ↔ hex: Group 4 binary digits from the right → each group = one hex digit
Variables:
- b — the base (radix)
- d_i — the digit at position i
- n — the integer value in decimal
How to Use
- Enter the number — type in the source number using valid digits for your chosen base (letters A–Z for digits above 9 in bases 11–36).
- Select source base — choose the base your number is currently written in (binary, octal, decimal, hex, or custom 2–36).
- Select target base — choose the base to convert to.
- Click Convert — the result appears in your target base, plus binary, octal, decimal, and hex equivalents shown simultaneously.
- Read the steps — the working panel shows the full positional expansion (from any base to decimal) and the repeated-division steps (decimal to any base).
Example Calculations
Example 1 — Decimal 255 to Binary, Octal, Hex
Convert 255₁₀ to all common bases
1
255 ÷ 2 repeatedly: 255 128 64 32 16 8 4 2 1 → remainders bottom-up: 11111111₂
2
Group binary into 3s from right: 011 111 111 = 377₈
3
Group binary into 4s: 1111 1111 → each group = F → FF₁₆
255₁₀ = 11111111₂ = 377₈ = FF₁₆ (this is the maximum value of one byte)
Try this example →Example 2 — Hex A3 to Decimal and Binary
Convert A3₁₆ to decimal and binary
1
A = 10, so A3₁₆ = 10×16 + 3×1 = 160 + 3 = 163₁₀
2
A₁₆ = 1010₂, 3₁₆ = 0011₂ → 10100011₂
A3₁₆ = 163₁₀ = 10100011₂ = 1010 0011 (nibble groups)
Try this example →Example 3 — Binary 11010 to Decimal
Convert 11010₂ to decimal
1
1×2⁴ + 1×2³ + 0×2² + 1×2¹ + 0×2⁰ = 16 + 8 + 0 + 2 + 0 = 26₁₀
11010₂ = 26₁₀ = 1A₁₆ = 32₈
Try this example →❓ Frequently Asked Questions
What is a number base (radix)?
A number base is the count of distinct digit symbols in a positional numeral system. Base 10 has 10 digits (0–9); base 2 has 2 (0, 1); base 16 has 16 (0–9, A–F). The digit at position i from the right represents digit × base^i. So 1A3₁₆ = 1×256 + 10×16 + 3 = 419₁₀.
How do you convert binary to decimal?
Write out positional values (1, 2, 4, 8, 16, … from right to left). For each 1-bit, add its positional value. Example: 11010₂ = 16+8+0+2+0 = 26₁₀. For fractions: 0.101₂ = 1×(1/2) + 0×(1/4) + 1×(1/8) = 0.5+0+0.125 = 0.625₁₀.
How do you convert decimal to binary?
Repeatedly divide by 2 and collect remainders. Read bottom-to-top. Example: 26: 26÷2=13 R0, 13÷2=6 R1, 6÷2=3 R0, 3÷2=1 R1, 1÷2=0 R1 → read up: 11010₂. The number of divisions equals the number of bits in the result.
What is hexadecimal and why is it used?
Hexadecimal (base 16) uses 0–9 and A–F (A=10 through F=15). One hex digit represents exactly 4 bits (a nibble), so two hex digits = one byte. This makes hex very compact for binary data. Memory addresses, CSS color codes (#RRGGBB), IPv6 addresses, and SHA-256 hashes are all shown in hex.
How do you convert between binary and hex directly?
Group binary digits into nibbles (groups of 4) from the right, padding with leading zeros. Each nibble converts to one hex digit: 0000=0, …, 1001=9, 1010=A, 1011=B, 1100=C, 1101=D, 1110=E, 1111=F. Example: 10110111₂ → 1011 0111 → B7₁₆. The reverse: expand each hex digit to 4 bits.
How do you convert between binary and octal?
Group binary digits into groups of 3 from the right (pad with leading zeros). Each group converts to one octal digit (0–7). Example: 11010110₂ → 011 010 110 → 326₈. Reverse: 5₈ = 101₂, 7₈ = 111₂. This works because 8 = 2³ — octal and binary share the same root.
What is octal used for?
Octal (base 8) was used in early computing because 3-bit groups map naturally to hardware registers. Today its main use is Unix file permissions: “chmod 755” means owner read/write/execute (7=111₂), group read/execute (5=101₂), others read/execute (5=101₂). Octal also appears in some C/C++ literals (prefix 0) and older PDP and IBM mainframe systems.
Can all decimal fractions be represented exactly in binary?
No — most decimal fractions are repeating in binary. Only fractions whose denominator is a power of 2 terminate in binary (1/2, 1/4, 1/8, etc.). For example, 0.1₁₀ = 0.0001100110011…₂ (repeating). This is why floating-point arithmetic has rounding errors: 0.1 + 0.2 ≠ 0.3 in IEEE 754. The calculator shows up to 12 fractional digits in the target base.
What is base 36 used for?
Base 36 uses all alphanumeric characters (0–9, A–Z), making it the most compact case-insensitive single-character-per-digit representation. It’s used in URL shorteners (compactly encoding large IDs), vehicle identification numbers (VINs encode manufacturing info), some database ID schemes, and crockford base32 (a variant). Example: 1,000,000₁₀ = LFLS₃₆.
How many hex digits does it take to represent an n-bit number?
Since each hex digit represents 4 bits: an n-bit number needs ⌈n/4⌉ hex digits. Common sizes: 8-bit (byte) = 2 hex digits (00–FF), 16-bit = 4 hex digits (0000–FFFF), 32-bit = 8 hex digits, 64-bit = 16 hex digits, 128-bit (IPv6 segment, UUID) = 32 hex digits. A SHA-256 hash is 256 bits = 64 hex digits.