Number Base Converter 2–36
Convert numbers between any bases from binary (2) to base 36 — not just the usual four. Watch every digit’s place value light up, follow the show-your-work steps, and convert fractions too.
100% client-side — your numbers never leave this browserThe converter
Type a number, pick any source and target base from 2 to 36, and convert. Works in both directions.
Digits allowed depend on the source base. Optional prefixes 0x (hex), 0b (binary), 0o (octal) are accepted when they match the source base.
Which converter do you actually need?
Number-base tools look alike but answer different questions. Pick the right one:
- This tool (base-converter): any base 2–36 in both directions, with a digit place-value visualizer and show-your-work steps. The general case.
- Hex ↔ decimal only: use a dedicated hex-to-decimal converter — same math as this tool with the bases locked to 16 and 10.
- Binary ↔ decimal only or binary ↔ hex only: fixed-pair converters for those two bases.
- Text, not numbers: ASCII-to-binary / ASCII-to-hex convert characters (letters, text) into byte values — that is text encoding, not number bases. Binary-to-string goes the other way: bytes back to readable text.
- Just the big four: a hex–dec–oct–bin converter shows all four standard bases at once, but cannot do base 7, base 12, base 36, or show place values.
Worked examples
Three conversions solved by hand, exactly the way the steps panel does it. The same method works for any pair — try 777 from base 8 to base 10 (answer: 511), or decimal 100 to base 6, in the converter above.
Example 1FF16 → decimal
- Each hex digit’s value: F = 15.
- Positional expansion: 15 × 161 + 15 × 160 = 240 + 15.
- Add: 240 + 15 = 255.
Example 225510 → binary
- Repeated division by 2, keeping remainders: 255 ÷ 2 = 127 r1; 127 ÷ 2 = 63 r1; 63 ÷ 2 = 31 r1; 31 ÷ 2 = 15 r1; 15 ÷ 2 = 7 r1; 7 ÷ 2 = 3 r1; 3 ÷ 2 = 1 r1; 1 ÷ 2 = 0 r1.
- Read the remainders from last to first: 11111111.
- Check: 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 255. ✓
Example 31237 → decimal
- Base 7 place values from the right: 70 = 1, 71 = 7, 72 = 49.
- Expansion: 1 × 49 + 2 × 7 + 3 × 1 = 49 + 14 + 3.
- Add: 49 + 14 + 3 = 66.
Methodology: positional notation
The one idea behind every base conversion.
A number’s value is the sum of each digit times its place value. In base b, positions from the right are worth b0, b1, b2, … and positions right of the point are worth b−1, b−2, …
Step 1 — expand to decimal
To leave any base, multiply each digit by its place value and add. That is how FF16 becomes 255 and 1237 becomes 66. The place-value visualizer above draws exactly this expansion for your input.
Step 2 — repeated division (whole part)
To enter a new base, divide the decimal value by the target base repeatedly and read the remainders bottom-up. Example: 10.510 → binary. Whole part: 10 ÷ 2 = 5 r0, 5 ÷ 2 = 2 r1, 2 ÷ 2 = 1 r0, 1 ÷ 2 = 0 r1 → 1010.
Step 3 — repeated multiplication (fraction part)
Multiply the fraction by the target base; the digit left of the point is the next output digit; repeat with what remains. Example: 0.5 × 2 = 1.0 → digit 1, done. Result: 1010.12.
Precision policy (honesty note)
Whole numbers are converted with arbitrary-precision integer math — exact, no matter how long. Fractions are exact only when they terminate in the target base. Otherwise the tool computes up to 12 fraction digits, trims trailing zeros, and labels the result truncated instead of pretending it is exact. Classic case: 0.110 → 0.0001100110012 (truncated), because 1/10 repeats forever in binary just like 1/3 repeats forever in decimal.
Digit alphabet
Digits are 0–9 then A–Z (values 10–35), so base 16 uses A–F and base 36 uses the full alphabet. Input is case-insensitive. A leading − negates the value — negative numbers keep their sign through the conversion, so −25510 becomes −FF16.
Bases at a glance
The tool handles every base from 2 to 36, but a few come up constantly:
| 2 | Binary | 0–1 | computing, logic gates, bit flags |
| 8 | Octal | 0–7 | Unix file permissions (e.g. 755) |
| 10 | Decimal | 0–9 | everyday arithmetic |
| 16 | Hexadecimal | 0–9 A–F | memory addresses, color codes (#FF8800), compact binary |
| 36 | Base 36 | 0–9 A–Z | short IDs, URL shorteners |
One hex digit always equals exactly 4 bits (a nibble) — that fixed 4-to-1 relationship is why programmers read long binary as hex. Octal groups bits in threes instead.
Frequently asked questions
Short answers to the questions people actually ask about base conversion.
How do I convert a number from one base to another?+
Two steps. First, expand the number to decimal using positional notation: multiply each digit by the base raised to its position and add the results (for example, FF in base 16 = 15×16 + 15×1 = 255). Second, convert that decimal value to the target base by repeated division: divide by the new base, keep each remainder, and read the remainders from last to first. The converter above does both steps and shows the work.
How do I convert binary to decimal?+
Write the place values over each binary digit, doubling from the right: 1, 2, 4, 8, 16, and so on. Add the place values where the digit is 1. For example, 1010 in binary = 8 + 2 = 10 in decimal. Type any binary number into the converter and the place-value visualizer draws this out for you.
How do I convert decimal to binary?+
Use repeated division by 2: divide the number by 2, write down each remainder, and read the remainders from bottom to top. For example, 255 ÷ 2 = 127 r1, 127 ÷ 2 = 63 r1, 63 ÷ 2 = 31 r1, 31 ÷ 2 = 15 r1, 15 ÷ 2 = 7 r1, 7 ÷ 2 = 3 r1, 3 ÷ 2 = 1 r1, 1 ÷ 2 = 0 r1, which gives 11111111. The steps panel shows every division.
How do I convert hexadecimal to decimal?+
Multiply each hex digit by 16 raised to its position (A=10, B=11, C=12, D=13, E=14, F=15) and add. For example, FF in hex = 15×16 + 15×1 = 240 + 15 = 255 in decimal. A0 in hex = 10×16 + 0 = 160. The converter accepts lowercase too, and an optional 0x prefix.
How do I convert decimal to hexadecimal?+
Use repeated division by 16 and map remainders 10–15 to A–F. For example, 255 ÷ 16 = 15 r15 (F), 15 ÷ 16 = 0 r15 (F), read bottom-up: FF. For 160: 160 ÷ 16 = 10 r0, 10 ÷ 16 = 0 r10 (A), which gives A0.
Can I convert fractions like 10.5 to binary?+
Yes. Convert the whole part and the fraction part separately. The whole part uses repeated division; the fraction part uses repeated multiplication: multiply the fraction by the new base, take the digit left of the point, and repeat with the remainder. For 10.5 in decimal: 10 becomes 1010, and 0.5 × 2 = 1.0 gives the digit 1, so the answer is 1010.1 in binary. Some fractions never terminate (like 0.1 in decimal to binary), so the tool computes up to 12 fraction digits and labels the result as truncated.
Why does 0.1 become 0.000110011001 in binary?+
Because 1/10 has no exact finite representation in base 2 — just like 1/3 has no exact finite representation in decimal (0.333...). Repeated multiplication by 2 produces the repeating pattern 0011 forever: 0.0001100110011001... The converter shows the first 12 digits and marks the result truncated, which is the honest answer instead of pretending it is exact.
What is a radix, and what does the base of a number mean?+
Radix is another word for base: the number of unique digits a numeral system uses. Base 10 (decimal) uses 0–9, base 2 (binary) uses 0–1, base 16 (hexadecimal) uses 0–9 plus A–F, and base 36 uses 0–9 plus A–Z. The base also sets each digit's place value: in base b, positions from the right are worth b^0, b^1, b^2, and so on.
Which digits are allowed in each base?+
Base b allows digits 0 up to b−1. Base 2 allows 0–1, base 8 allows 0–7, base 10 allows 0–9, base 16 allows 0–9 and A–F (values 10–15), and base 36 allows 0–9 and A–Z (values 10–35). If you type a digit the base does not allow — for example 2 in binary or G in hexadecimal — the converter shows an honest error naming the bad digit and position instead of guessing.
How do I read binary numbers?+
Read each 1 as its place value and add them. From the right, the places are 1, 2, 4, 8, 16, 32, 64, 128, doubling each step. So 11111111 = 128+64+32+16+8+4+2+1 = 255. Grouping bits in fours also helps: 1111 1111 is two nibbles, each worth 15, which is why every 4 bits map neatly to one hex digit.
Why do computers use binary?+
Because a bit has two stable physical states — on/off, high/low voltage — which is the most reliable way to store and transmit information in hardware. Higher bases are still useful as shorthand: one hex digit represents exactly 4 bits, so programmers read long binary strings as compact hex.
Can this converter handle very large numbers exactly?+
Yes. Integer conversion uses arbitrary-precision arithmetic, so there is no 2^53 floating-point ceiling. For example, FFFFFFFFFFFFFFFF in hex converts to exactly 18446744073709551615 in decimal, and a 101-digit binary number converts exactly too. Fraction digits are the only part with a cap (12 digits in the target base), and the tool says so when it truncates.
What is the difference between this converter and a hex-to-decimal converter?+
A hex-to-decimal converter handles one fixed pair of bases. This tool converts between any bases from 2 to 36 in both directions — for example base 7 to base 36 — and additionally shows each digit's place value and the full step-by-step work, which fixed-pair converters do not.
Is my data private when I use this converter?+
Yes. The conversion runs entirely in your browser with JavaScript; nothing you type is sent to any server. You can verify this by disconnecting from the internet — the converter keeps working.