Converting decimal value...
Enter a decimal number to see its binary representation and place-value breakdown.
How to Convert Decimal to Binary
A Decimal to Binary Converter translates a base-10 number into the base-2 number system used by digital hardware. Binary has only two digits, 0 and 1. For integers, the conversion uses repeated division by 2, recording each remainder from bottom to top.
To convert decimal 13, divide by 2 repeatedly: 13 / 2 = 6 remainder 1, 6 / 2 = 3 remainder 0, 3 / 2 = 1 remainder 1, 1 / 2 = 0 remainder 1. Reading the remainders from bottom to top gives 1101. This calculator performs the same process with exact arbitrary-precision arithmetic and shows each contributing bit.
Decimal to Binary Place Value Chart
Binary place values double from right to left and halve to the right of the radix point. Memorizing the first eight integer powers makes byte-sized conversions faster, while the fractional positions are useful for fixed-point math and embedded systems.
| Position | Power | Decimal Value | Common Meaning |
|---|---|---|---|
| 3 | 2^3 | 8 | Half-byte high bit |
| 2 | 2^2 | 4 | Four-value place |
| 1 | 2^1 | 2 | Second bit |
| 0 | 2^0 | 1 | Single-bit flag |
| -1 | 2^-1 | 0.5 | First fractional bit |
| -2 | 2^-2 | 0.25 | Second fractional bit |
| -3 | 2^-3 | 0.125 | Third fractional bit |
| 7 | 2^7 | 128 | Unsigned byte high bit |
| 8 | 2^8 | 256 | One above byte maximum |
| 15 | 2^15 | 32,768 | Signed 16-bit high bit |
How to Convert a Decimal Fraction to Binary
Decimal fractions convert to binary by repeated multiplication by 2. The integer part of each product becomes the next binary digit to the right of the radix point. The process repeats with the fractional remainder until it becomes exactly zero or reaches the desired precision.
For example, 5.625 has an integer part of 101 and a fractional part that terminates cleanly: 0.625 x 2 = 1.25 (bit 1), 0.25 x 2 = 0.5 (bit 0), 0.5 x 2 = 1.0 (bit 1). The result is 101.101. Not all decimal fractions terminate in binary; 0.1 repeats forever, so the converter truncates at the precision limit and labels the result.
How Two's Complement Converts Signed Decimal to Binary
Most computers store signed integers using two's complement encoding within a fixed bit width. The converter validates that the decimal value fits inside the chosen width, then produces the exact bit pattern.
For -5 in 8-bit two's complement, add 2^8 = 256 to the value: -5 + 256 = 251. Convert 251 to binary and pad to 8 bits: 11111011. The leading 1 indicates a negative value, and the place-value breakdown shows the sign-bit contribution as -128.
Common Decimal to Binary Values
| Decimal | Binary | Octal | Hexadecimal |
|---|---|---|---|
| 0 | 0 | 0 | 0x0 |
| 1 | 1 | 1 | 0x1 |
| 10 | 1010 | 12 | 0xA |
| 13 | 1101 | 15 | 0xD |
| 127 | 1111111 | 177 | 0x7F |
| 128 | 10000000 | 200 | 0x80 |
| 255 | 11111111 | 377 | 0xFF |
| 1024 | 10000000000 | 2000 | 0x400 |
Decimal to Binary FAQ
How do you convert decimal to binary?
For integers, divide by 2 repeatedly and record the remainders from bottom to top. For fractions, multiply by 2 repeatedly and record the integer part of each product as the next binary digit.
What is 10 in binary?
Decimal 10 equals binary 1010. Its set bits contribute 8 and 2, which sum to 10.
How do you convert a decimal fraction to binary?
Multiply the fraction by 2. The integer part of the result is the next binary digit. Repeat with the remaining fraction until it reaches zero or the desired precision limit.
Why is 0.1 not exact in binary?
Decimal 0.1 has no finite binary representation because its denominator contains a factor of 5, which is not a power of 2. The binary representation repeats forever.
How do you convert negative decimal to binary?
Choose a fixed bit width and use two's complement encoding. Add 2 raised to the bit width to the negative value, then convert the result to binary and pad to the chosen width.
Can a decimal number be larger than 64 bits in binary?
Yes. This converter uses arbitrary-precision arithmetic and accepts integer values up to a very large digit count, avoiding the overflow limits of native 64-bit types.
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