Binary to Decimal Converter
Convert binary to decimal — and decimal back to binary — with a tap-to-flip bead abacus that visualizes every place value, plus step-by-step working you can copy into homework or documentation.
Converter
Digits 0 and 1 only — anything else gets an honest error.
The bead abacus — tap a bead to flip its bit
Tap any bead to flip that bit and watch the decimal update.
Place-value breakdown
Show your work
Which converter do you need?
This page is deliberately binary ↔ decimal only. Pick the right tool for your job:
Binary to Decimal Converter
Dedicated binary ↔ decimal with the interactive bead abacus, place-value breakdown, binary fractions, and copyable steps. Deepest on exactly these two bases.
Binary Converter
Multi-base generalist: binary, octal, decimal, and hexadecimal in one place. Use it when you juggle more than two bases at once.
Base Converter
Arbitrary bases 2–36 with a place-value visualizer for the general case. Use it when your conversion goes beyond binary and decimal.
Number Base Converter Pro
Arbitrary bases 2–36 plus bitwise operations and two's complement. Use it for programmer-level base work beyond plain conversion.
Worked examples — step by step
Three binary-to-decimal conversions worked out by hand, plus the reverse direction. Every number below is verified by the converter's own engine.
Example 1: 1010₂ → 10₁₀
The classic textbook case — four bits, two of them set:
- 1 × 2³ = 8
- 0 × 2² = 0
- 1 × 2¹ = 2
- 0 × 2⁰ = 0
10102 = 8 + 0 + 2 + 0 = 1010
Example 2: 11111111₂ → 255₁₀
All eight bits of a byte set — the largest value one byte can hold:
- 1 × 2⁷ = 128
- 1 × 2⁶ = 64
- 1 × 2⁵ = 32
- 1 × 2⁴ = 16
- 1 × 2³ = 8
- 1 × 2² = 4
- 1 × 2¹ = 2
- 1 × 2⁰ = 1
111111112 = 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 25510
Example 3: 101.1₂ → 5.5₁₀
A binary fraction: digits after the point use negative powers of two:
- 1 × 2² = 4
- 0 × 2¹ = 0
- 1 × 2⁰ = 1
- 1 × 2⁻¹ = 0.5
101.12 = 4 + 0 + 1 + 0.5 = 5.510
Example 4 (reverse): 42₁₀ → 101010₂
Repeated division by 2, then read the remainders bottom to top:
- 42 ÷ 2 = 21 remainder 0
- 21 ÷ 2 = 10 remainder 1
- 10 ÷ 2 = 5 remainder 0
- 5 ÷ 2 = 2 remainder 1
- 2 ÷ 2 = 1 remainder 0
- 1 ÷ 2 = 0 remainder 1
Reading upward: 1 0 1 0 1 02 = 4210
How binary-to-decimal conversion works
Positional notation: every position doubles
Binary is a base-2 positional system: each digit (bit) is worth twice the digit to its right. The rightmost bit is the 1s place, then 2s, 4s, 8s, and so on — the powers of 2. A 1 means "take this place value", a 0 means "skip it". Add up the taken values and you have the decimal number. The leftmost bit is the most significant bit (MSB); the rightmost is the least significant bit (LSB).
The powers of two (your cheat sheet)
Memorize the first few and every conversion gets easier:
| Power | Decimal value | Binary |
|---|---|---|
| 20 | 1 | 1 |
| 21 | 2 | 10 |
| 22 | 4 | 100 |
| 23 | 8 | 1000 |
| 24 | 16 | 10000 |
| 25 | 32 | 100000 |
| 26 | 64 | 1000000 |
| 27 | 128 | 10000000 |
| 28 | 256 | 100000000 |
| 29 | 512 | 1000000000 |
| 210 | 1024 | 10000000000 |
| 211 | 2048 | 100000000000 |
| 212 | 4096 | 1000000000000 |
| 213 | 8192 | 10000000000000 |
| 214 | 16384 | 100000000000000 |
| 215 | 32768 | 1000000000000000 |
| 216 | 65536 | 10000000000000000 |
Binary → decimal by hand (the method)
- Write the bits in a row, rightmost first.
- Label each bit with its power of two: 1, 2, 4, 8, 16, …
- Multiply each bit by its label: 1-bits contribute their label, 0-bits contribute 0.
- Add everything up. That sum is the decimal value.
Decimal → binary by hand (repeated division)
- Divide the decimal number by 2; write down the remainder (0 or 1).
- Repeat with the quotient until the quotient is 0.
- Read the remainders bottom to top — that is your binary number.
Binary fractions
Digits after the binary point use negative powers of two: the first is ½ (0.5), then ¼ (0.25), ⅛ (0.125), and so on. So 101.1₂ = 4 + 1 + 0.5 = 5.5. An important honesty note: while binary fractions always convert to exact decimals, the reverse is not true — many decimal fractions (like 0.1) repeat forever in binary, so the converter caps output at 24 fraction bits and labels such results approximate instead of silently rounding.
Frequently asked questions
How do I convert binary to decimal?
Write down each binary digit with its place value: powers of 2 starting from 1 on the right (1, 2, 4, 8, 16, …). Multiply each bit by its place value and add the results. For example, 1010₂ = 1×8 + 0×4 + 1×2 + 0×1 = 10. The converter above does this automatically and shows every step in the place-value breakdown.
What is 1010 in binary in decimal?
1010 in binary equals 10 in decimal. Reading the bits from right to left: 0×1 + 1×2 + 0×4 + 1×8 = 0 + 2 + 0 + 8 = 10. It is the classic textbook example of binary-to-decimal conversion.
What is 11111111 in decimal?
11111111 in binary equals 255 in decimal: 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 255. It is the largest value you can store in a single 8-bit byte, which is why 255 appears everywhere in computing — from IPv4 octets to RGB color channels.
How do you convert a binary fraction to decimal?
Use negative powers of 2 after the binary point: the first bit right of the point is worth 1/2, the next 1/4, then 1/8, 1/16, and so on. For example, 101.1₂ = 1×4 + 0×2 + 1×1 + 1×1/2 = 5 + 0.5 = 5.5. Unlike decimal fractions converted to binary, binary fractions always have an exact decimal value.
How do I convert decimal to binary?
Use repeated division by 2: divide the number by 2, write down the remainder (0 or 1), and repeat with the quotient until it reaches 0. Then read the remainders from bottom to top. For 42: 42÷2=21 r0, 21÷2=10 r1, 10÷2=5 r0, 5÷2=2 r1, 2÷2=1 r0, 1÷2=0 r1 — reading upward gives 101010₂.
What is 42 in binary?
42 in decimal is 101010 in binary: 32 + 8 + 2 = 42. It is also the largest power-of-two pattern trick worth memorizing: any decimal number can be broken into a sum of powers of 2, and each power you use becomes a 1-bit in that position.
Why is 102 not a valid binary number?
Because binary uses only two digits — 0 and 1. The digit 2 never appears in a binary number, so "102" is rejected by this converter with an honest error that points at the exact invalid digit. If you meant the decimal number 102, switch to the Decimal → Binary tab: 102₁₀ = 1100110₂.
Why do computers use binary?
Because hardware is built from switches — transistors that are either on or off. Two stable states map perfectly onto the digits 0 and 1, making circuits simple, fast, and resistant to noise. Every number, letter, image, and video in a computer is ultimately encoded as long strings of bits.
How many bits are in a byte?
There are 8 bits in a byte. Half a byte (4 bits) is called a nibble. With 8 bits you can represent 256 different values, from 00000000₂ (0) to 11111111₂ (255) — the familiar 0–255 range used in IP addresses and color codes.
What is the binary to decimal formula?
Decimal = Σ (bitᵢ × 2ⁱ), where i is the bit's position counting from 0 at the rightmost digit. For 10110₂: 0×2⁰ + 1×2¹ + 1×2² + 0×2³ + 1×2⁴ = 0 + 2 + 4 + 0 + 16 = 22. For digits after the binary point, use negative positions: 2⁻¹ = 0.5, 2⁻² = 0.25, and so on.
Can binary numbers have a decimal point?
Yes — it is called the binary point, and digits after it have fractional place values (1/2, 1/4, 1/8, …). So 101.1₂ = 5.5₁₀ exactly. Note the asymmetry: many decimal fractions cannot be written exactly in binary. The decimal 0.1, for instance, repeats forever in binary (0.0001100110011…₂), which is why this converter flags such results as approximate.
What is the largest decimal number you can write with 8 bits?
255 — that is 11111111₂. In general, n bits can represent every integer from 0 up to 2ⁿ − 1: 4 bits give 0–15, 8 bits give 0–255, 16 bits give 0–65,535, and 32 bits give 0–4,294,967,295.
What is the difference between this tool and a general base converter?
This page is a dedicated binary ↔ decimal converter: it focuses on exactly two bases and goes deep — an interactive bead abacus you can tap to flip bits, place-value breakdown, binary fractions, and copyable step-by-step working. Our binary-converter is a multi-base generalist (binary, octal, decimal, hex), base-converter handles arbitrary bases 2–36 with its own place-value visualizer, and number-base-converter-pro adds bitwise operations and two's complement. If you only ever convert between binary and decimal, this page is the faster, clearer tool.
Is this binary converter private? Does my input leave my browser?
Yes, it is fully private. The conversion runs entirely in your browser with JavaScript — your numbers are never sent to any server, stored, or logged. You can even disconnect from the internet after the page loads and it keeps working.