81000₂
- Power
2^3
Convert Decimal to Binary in Python
What you’ll learn
- How division by two and remainders produce binary digits from right to left.
- Why you must handle zero and why the first method reverses collected bits.
- A second method using bit scan, plus a live preview.
Overview
Binary uses powers of two. To convert a nonnegative decimal integer into binary, repeatedly divide by 2 and track remainders 0 or 1.
Two methods
Remainder collection and bit-scan.
Live preview
Test quickly with safe integer input.
Inverse
Pair with binary-to-decimal.
Prerequisites
Basic loops, modulo, integer division, and string joining in Python.
//for integer division and%for remainder.- Optional: bit operations like
>>and&.
Decimal to binary
Each binary digit is a coefficient on a power of two. Repeatedly taking n % 2 gives bits from least significant to most significant.
Example: 15 = 8 + 4 + 2 + 1, so binary is 1111.
Decimalbase 10
Binarybase 2
Example15 → 1111
Division algorithm
For n ≥ 0, repeatedly write n = 2q + r with r in {0,1}. The remainders form binary digits from right to left.
1515→7 r1, 7→3 r1, 3→1 r1, 1→0 r1 so binary is 1111.
Intuition
5101₂
- Sum
4 + 1
Takeaway: each division step answers odd/even at the current scale.
Live Preview
Nonnegative integers in JavaScript safe range.
Algorithm (remainder method)
Goal: output binary string of nonnegative n.
Handle n == 0
Return "0".
Collect bits
Append n % 2, then do n //= 2 until n becomes 0.
Reverse
Reverse collected bits and join.
📜 Pseudocode
Pseudocode
function decimal_to_binary(n):
if n == 0:
return "0"
bits = []
while n > 0:
bits.append(n % 2)
n = n // 2
reverse(bits)
return join(bits)1
Divide by two and reverse bits
Classic approach: collect remainder bits, then reverse to print MSB-first.
python
def decimal_to_binary(n: int) -> str:
if n == 0:
return "0"
bits: list[str] = []
while n > 0:
bits.append(str(n % 2))
n //= 2
bits.reverse()
return "".join(bits)
print(decimal_to_binary(15)) # 1111
print(decimal_to_binary(0)) # 0Explanation
Remainders are collected from least significant bit to most significant bit, so reversing is required for normal output.
2
Bit-scan from MSB to LSB
No reversal buffer needed. Start at highest set bit and read downward.
python
def to_binary_bit_scan(n: int) -> str:
if n == 0:
return "0"
highest = n.bit_length() - 1
out: list[str] = []
for b in range(highest, -1, -1):
bit = (n >> b) & 1
out.append("1" if bit else "0")
return "".join(out)
print(to_binary_bit_scan(15)) # 1111
print(to_binary_bit_scan(64)) # 1000000Explanation
bit_length() finds the highest bit position so you can scan MSB to LSB directly.
Optimization
Built-in fast path. For nonnegative integers, format(n, "b") or bin(n)[2:] is concise and reliable.
Fixed width. Use format specifiers like format(n, "08b") when padding is required.
Interview: explain remainder logic first, then mention bit-scan or built-ins.
❓ FAQ
The first remainder from n/2 is the least significant bit (parity). Each later remainder is the next higher bit. Printing from last remainder to first gives MSB to LSB.
The loop while n > 0 will not run, so you must handle n == 0 separately and return '0'.
It checks bits from a high position down to 0 and skips leading zeros. This avoids storing all remainders first.
For learning conversion, many tutorials restrict to nonnegative numbers. If you need signed representation, document whether you use absolute value or fixed-width two's complement.
Either minimal bits (no leading zeros) or fixed width like 8/16/32 bits, depending on problem requirements.
O(b) where b is number of output bits. For positive n, that's O(log2 n) in the remainder method.
🔄 Input / output examples
Use these values to verify both approaches.
| Decimal | Binary (minimal) |
|---|---|
0 | 0 |
1 | 1 |
15 | 1111 |
64 | 1000000 |
Edge cases and pitfalls
Handle zero explicitly and define behavior for negatives before coding.
Zero
n == 0
Return "0"; otherwise a plain while n > 0 loop returns empty output.
Negative
Policy choice
Choose absolute-value conversion or fixed-width two's complement and document it clearly.
Width
Minimal vs padded
Some tasks expect 00001111 instead of 1111; padding rules matter.
Inverse
Binary to decimal
Practice the reverse conversion too for interview completeness.
⏱️ Time and space complexity
| Approach | Time | Extra space |
|---|---|---|
| Remainder + reverse | O(log n) | O(log n) |
| Bit scan using bit_length | O(log n) | O(log n) |
log is base 2 in bit-length terms for positive n.
Summary
- Use remainder and division by 2 to build bits.
- Always handle
n == 0separately. - Bit scan gives a neat MSB-to-LSB alternative.
Did you know?
Repeated division by 2 collects bits from least to most significant, so the usual classroom program prints the array backwards. A bit-scan loop can print from the MSB without an explicit reversal.
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