Binary to Decimal Converter
Binary to Decimal Converter. Enter your binary string and the Binary to Decimal Converter instantly calculates its Decimal Output. Use the hex to text converter to reveal the original message hidden behind a string of hex character codes.
Type a string of 0s and 1s into the binary to decimal converter and you get back the decimal equivalent in one click, so you never have to add up eight or sixteen place values by hand. Every number you enter is read as a row of bits, and the result shows both the plain positive reading and the signed reading, so you can see exactly what those digits mean to a computer.
How the Binary to Decimal Converter Works
The tool takes your binary string, treats each digit as a switch that is either on or off, and adds up the value of every switch that is on. You do not need to pad the input or add a prefix. Paste your bits, press the button to convert it, and the decimal value appears next to the original string, ready to copy.
- Enter your binary number using only the digits 0 and 1 (spaces between groups of four are ignored).
- Choose whether to read the bits as an unsigned value or as a signed two's complement value.
- Click the convert button and read the decimal result, along with the working shown step by step.
- Copy the answer, or clear the field and try another binary to decimal lookup.
Behind the scenes, this binary to decimal converter tool applies the same sum you would write on paper. Every bit is multiplied by the power of two that belongs to its position, and the products are added together:
$$\text{decimal} = \sum_{i=0}^{n-1} d_i \times 2^{i}$$Here di is the bit at position i counted from the right, starting at zero, and n is the number of bits. A bit that is 0 contributes nothing, and a bit that is 1 contributes its full place value.
Binary System and Decimal System Basics
The binary system is a base 2 numeral system: it has only two symbols, 0 and 1, and each position is worth twice as much as the one to its right. That simplicity is why digital electronics and computers use it. A wire carries an electric signal that is either off or on, and a single binary digit records which state it is in. Text, images, sound and every other kind of data that you see on screen is stored as long strings of these digits, which is why people call it binary code. Reading those strings back as ordinary base 10 values is exactly the job of this converter.
The decimal system is the base 10 system you use every day. It has ten symbols, 0 through 9, and each position is worth ten times the one to its right. The Hindu-Arabic notation that made this positional layout standard is the reason a 3 in the hundreds place means 300. Both systems work the same way and differ only in their radix, which is the number of symbols available before you carry into the next position.
How to Convert Binary to Decimal by Hand
Doing a calculation yourself once or twice is the best way to trust the converter afterwards, since you can convert a short string by hand and check its accuracy against the tool. All the examples below use the same 8-bit string, 10110111, so you can follow one number through each method and compare it with the tool's output.
Positional System and Powers of 2
In a positional system, a digit's value depends on where it sits. Write the place values under the bits, starting on the right with 20, and keep doubling as you move left. The rightmost bit is the least significant, and the leftmost bit is the most significant.
| Position | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
|---|---|---|---|---|---|---|---|---|
| Power of 2 | 27 | 26 | 25 | 24 | 23 | 22 | 21 | 20 |
| Place value | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
| Bit | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 |
| Contribution | 128 | 0 | 32 | 16 | 0 | 4 | 2 | 1 |
Multiply each digit by its place value, then add the results: \(128 + 32 + 16 + 4 + 2 + 1 = 183\). So 101101112 = 18310. The zeros in positions 6 and 3 skip the values 64 and 8, which is why the sum stops short of 255, the largest value eight bits can hold.
Double Dabble Method
The double dabble method is a short algorithm for longer strings that you can run in your head: start at 0, then for each bit from the left, double the running total and add the bit. It needs no powers of 2 at all, only repeated doubling.
| Bit read | Doubling step | Running total |
|---|---|---|
| 1 | 0 × 2 + 1 | 1 |
| 0 | 1 × 2 + 0 | 2 |
| 1 | 2 × 2 + 1 | 5 |
| 1 | 5 × 2 + 1 | 11 |
| 0 | 11 × 2 + 0 | 22 |
| 1 | 22 × 2 + 1 | 45 |
| 1 | 45 × 2 + 1 | 91 |
| 1 | 91 × 2 + 1 | 183 |
The final total, 183, matches the positional sum exactly. Doubling is the same operation as shifting every earlier bit one position to the left, which is why the method works for any binary string you throw at it.
Fractional Binary Digits
A binary string can also hold a fraction. Digits after the point are worth 2-1, 2-2, 2-3 and so on, which means one half, one quarter, one eighth. Take 1011.101: the whole part is 8 + 2 + 1 = 11, and the fractional values are 0.5 + 0.125 = 0.625, giving 11.625. Some decimals, such as 0.3, have no finite binary form, so a tool with arbitrary-precision arithmetic still has to stop at a chosen number of bits and show an approximation.
Checking a Sensor Register with the Binary-Decimal Converter
Dana is logging temperatures from a small I2C sensor in a cold-storage unit. The datasheet says the temperature register is an 8-bit signed two's complement value, one degree Celsius per step, with a rated range of -40 to 125 °C. The logger dumps the raw byte as 11010010, and Dana needs to know whether the freezer really dropped that far.
Dana pastes the byte into the converter and first sees the unsigned reading: 210. That cannot be a temperature, because the sensor tops out at 125 °C. The sign bit is 1, so Dana switches to the signed reading. The tool inverts the bits to 00101101 (45), adds 1 and reports -46. The decimal equivalent of that byte is therefore -46 °C, which is 6 degrees below the sensor's rated minimum of -40 °C.
Dana does not trust a reading outside the part's specification, so the next step is a re-read after the compressor cycle ends. The new byte is 11011000. Converting it gives 216 unsigned and 216 - 256 = -40 signed, exactly at the rated floor. The first sample was a transient dip, not a valid measurement, so Dana adds a rule: the signed decimal result from the converter is checked against the -40 limit, and anything lower is flagged for review.
Unsigned vs Signed Results With Two's Complement
A string of bits does not say by itself whether it stands for a positive-only integer or a signed one, so the converter shows both readings. An unsigned reading uses every bit as a value bit, so the result is always zero or positive. A signed reading uses two's complement, where the leftmost bit decides whether the number is negative.
Sign Bit and Bit Width
The sign bit is the leftmost bit. If it is 0, the signed and unsigned results agree. If it is 1, the number is negative: invert all bits, add 1, and put a minus sign in front. For 10110111, the sign bit is 1. Inverting gives 01001000, which is 72, and adding 1 gives 73, so the signed result is -73. You get the same answer by subtracting 256 from 183.
The bit width matters just as much as the digits. The same string padded with zeros on the left to 16 bits becomes 0000000010110111, whose sign bit is 0, so the signed value is now +183. The table lists the ranges that common widths can store.
| Data type | Bits | Minimum | Maximum |
|---|---|---|---|
| Unsigned integer | 8-bit | 0 | 255 |
| Signed integer | 8-bit | -128 | 127 |
| Unsigned integer | 16-bit | 0 | 65,535 |
| Signed integer | 16-bit | -32,768 | 32,767 |
| Unsigned integer | 32-bit | 0 | 4,294,967,295 |
| Signed integer | 32-bit | -2,147,483,648 | 2,147,483,647 |
| Floating point (IEEE754) | 32 bits | about -3.4 × 1038 | about 3.4 × 1038 |
Programming languages build their data types on these word sizes, so choosing the wrong reading is a classic source of bugs: a byte that stores 183 in one place can be read as -73 in another. Floating point values follow the IEEE754 layout instead of plain two's complement, so they need a different decoder; this converter reads integers only.
Binary to Decimal Conversion Table for Common Values
These landmark values show up constantly in memory sizes, colour channels and network masks, and the hex column helps when you jump between bases.
| Binary | Decimal | Hexadecimal |
|---|---|---|
| 0111 1111 | 127 | 7F |
| 1000 0000 | 128 | 80 |
| 1011 0111 | 183 | B7 |
| 1111 1111 | 255 | FF |
You can convert in either direction with this table. Every group of four bits maps to exactly one hexadecimal digit, so 1011 0111 reads as B and 7 without any arithmetic. That shortcut is why programmers write hex whenever the raw bits would be too long to read.
Related Number Conversion: Hex, Octal and Decimal to Binary
Once binary to decimal feels natural, the same positional logic carries across every other base. Octal groups bits in threes, hex groups them in fours, and going from decimal to binary simply reverses the process: divide by 2 repeatedly and read the remainders from bottom to top. A good binary converter handles the trip in both directions, and a decimal/binary converter lets you paste an answer back in to confirm that you land on your starting number. Text connects back to the same skill: each ASCII character is stored as one 8-bit value, so every byte you convert to a decimal code maps to one letter. For any kind of number conversion, a quick sanity check is to estimate the size first. A string with n bits can never exceed 2n - 1 when read as unsigned.
Whenever you want a quick, private answer, an online binary-decimal converter does the validation for you by rejecting any input that contains a digit other than 0 or 1, and it shows the working so you can learn the pattern while you use it.