What is Decimal to Binary Number Converter?
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Decimal to Binary Converter
Decimal to binary conversion via repeated division by 2. Foundation of digital computing.
About decimal to binary conversion
Decimal to binary conversion rewrites a base-10 integer as a string of 0s and 1s in base-2, the native representation used inside every digital computer. Each bit is a power of 2, switched on or off, and the sum of the active powers reconstructs the original number.
How it works: positional notation and division by 2
Binary uses positional notation just like decimal, but each position is a power of 2 instead of 10. The rightmost bit is 2^0 = 1, then 2^1 = 2, 2^2 = 4, 2^3 = 8, and so on. A binary number 1011 means 1 times 8, plus 0 times 4, plus 1 times 2, plus 1 times 1 = 11 decimal.
N = b_k * 2^k + b_(k-1) * 2^(k-1) + ... + b_1 * 2 + b_0
algorithm:
while N > 0:
bit = N mod 2 # remainder is 0 or 1
N = N // 2 # integer division
push bit to output
read output in reverse
- N = the decimal integer you want to convert.
- b_i = the bit (0 or 1) at position i, where position 0 is the rightmost (least significant) bit.
- mod 2 = the remainder after dividing by 2, which is the next bit.
- // 2 = integer division by 2, dropping the remainder, which shifts the number right by one bit.
- Bits are produced low-to-high but written high-to-low, so the output stack must be reversed.
Worked example: convert 156 to binary
Apply the division-by-2 algorithm step by step. Each row records the quotient and the remainder, and the remainders read bottom-up form the answer.
- 156 / 2 = 78 remainder 0
- 78 / 2 = 39 remainder 0
- 39 / 2 = 19 remainder 1
- 19 / 2 = 9 remainder 1
- 9 / 2 = 4 remainder 1
- 4 / 2 = 2 remainder 0
- 2 / 2 = 1 remainder 0
- 1 / 2 = 0 remainder 1
- Read remainders bottom to top: 1 0 0 1 1 1 0 0.
Reference: powers of 2 and small decimal-to-binary table
Memorising the first 16 powers of 2 and the first 16 binary numbers lets you convert most everyday values in your head.
| Power | Decimal value | Common use |
|---|---|---|
| 2^0 | 1 | 1 bit |
| 2^1 | 2 | flag pair |
| 2^2 | 4 | 2-bit range |
| 2^3 | 8 | 1 byte halved |
| 2^4 | 16 | 1 nibble (hex digit) |
| 2^7 | 128 | signed byte limit |
| 2^8 | 256 | 1 byte total range |
| 2^10 | 1,024 | 1 KiB (kibibyte) |
| 2^16 | 65,536 | 1 word (Unicode BMP) |
| Decimal | Binary (4 bits) | Decimal | Binary (4 bits) |
|---|---|---|---|
| 0 | 0000 | 8 | 1000 |
| 1 | 0001 | 9 | 1001 |
| 2 | 0010 | 10 | 1010 |
| 3 | 0011 | 11 | 1011 |
| 4 | 0100 | 12 | 1100 |
| 5 | 0101 | 13 | 1101 |
| 6 | 0110 | 14 | 1110 |
| 7 | 0111 | 15 | 1111 |
Common pitfalls
- Reading remainders the wrong way. The first remainder you compute is the least significant bit. Read remainders bottom-up, not top-down, or your answer will be bit-reversed.
- Forgetting to pad with leading zeros. A byte is 8 bits. The number 5 is 101 in pure binary but 00000101 as a byte. APIs and file formats almost always expect the padded form.
- Confusing signed and unsigned ranges. 11111111 means 255 unsigned but minus 1 in two's complement signed. Always confirm which encoding the system expects.
- Floating-point rounding. Decimal fractions like 0.1 have no finite binary expansion, so 0.1 + 0.2 returns 0.30000000000000004 in most languages. Use integer cents for currency.
- Off-by-one bit count. floor(log2(N)) + 1 gives the bit count for unsigned N greater than zero. The number 0 still needs one bit (the bit value 0).
Related tools and glossary
Frequently asked questions
How do I convert 156 to binary by hand?
Use division by 2 and read remainders bottom-up. 156/2 = 78 r 0; 78/2 = 39 r 0; 39/2 = 19 r 1; 19/2 = 9 r 1; 9/2 = 4 r 1; 4/2 = 2 r 0; 2/2 = 1 r 0; 1/2 = 0 r 1. Reading bottom to top: 10011100. Verify by adding the powers of 2: 128 + 16 + 8 + 4 = 156.
How many bits do I need to store a decimal number?
floor(log2(N)) + 1 bits for an unsigned integer N. Examples: 100 needs 7 bits (max 127), 1,000 needs 10 bits (max 1023), 1,000,000 needs 20 bits (max 1,048,575). For signed two's complement, add one bit for the sign.
What is the largest 8-bit unsigned number?
11111111 binary = 255 decimal (2^8 minus 1). For signed two's complement, the range is -128 to 127 (10000000 to 01111111). 16-bit unsigned reaches 65,535, 32-bit unsigned reaches 4,294,967,295.
What is two's complement and why use it?
Two's complement is the standard way computers store negative integers. To negate a number, invert all bits and add 1. Example: -5 in 8 bits is 00000101 inverted = 11111010, plus 1 = 11111011. It works because addition of a number and its negation wraps around to zero with one shared zero representation, which simplifies CPU arithmetic circuits.
Sources
- Knuth, Donald (1997) The Art of Computer Programming, Vol 2: Seminumerical Algorithms, Section 4.1 - positional number systems and radix conversion.
- IEEE 754-2019 standard for floating-point arithmetic - binary fraction representation.
- Intel 64 and IA-32 Architectures Software Developer's Manual, Volume 1, Section 4.2 - signed integer encoding and two's complement.