Decimal to Binary Converter: Convert Decimal Numbers
A decimal to binary converter transforms decimal numbers (base-10 numbers using digits 0-9) into binary numbers (base-2 numbers using only digits 0 and 1) through repeated division by 2, tracking remainders in reverse order to obtain the binary representation essential for computer science, digital electronics, programming, data communication, and understanding how computers internally store and process numerical information. This conversion process uses the division-remainder method where the decimal number is repeatedly divided by 2, recording each remainder (0 or 1), then reading these remainders from bottom to top to form the binary equivalent, enabling translation from human-readable decimal format to machine-readable binary format for computer programming, digital logic design, binary arithmetic, data encoding, and foundational understanding of number systems in computing and electronics.
🔢 Decimal to Binary Converter
Enter a decimal number to convert to binary
Maximum value: 2,147,483,647
Understanding Decimal to Binary Conversion
Decimal is the base-10 number system we use daily, consisting of ten digits (0-9). Binary is the base-2 number system used by computers, consisting of only two digits (0 and 1). Converting decimal to binary reveals how computers represent numbers internally, as digital circuits can only recognize two states: on (1) or off (0).
Decimal to Binary Formula
Division-Remainder Method:
Repeatedly divide the decimal number by 2
Record the remainder at each step (0 or 1)
Read remainders from bottom to top
Mathematical Expression:
\[ \text{Binary} = \sum_{i=0}^{n-1} r_i \times 2^i \]
Where \( r_i \) is the remainder at each division step
Step-by-Step Conversion Method
Division-Remainder Algorithm
- Divide the decimal number by 2
- Record the remainder (0 or 1)
- Divide the quotient by 2 again
- Repeat until quotient becomes 0
- Read remainders from bottom to top for binary result
Detailed Conversion Examples
Example 1: Convert Decimal 13 to Binary
Problem: Convert decimal 13 to binary
Step-by-step division:
Division Quotient Remainder
13 ÷ 2 = 6 1 ← LSB (rightmost)
6 ÷ 2 = 3 0
3 ÷ 2 = 1 1
1 ÷ 2 = 0 1 ← MSB (leftmost)
Read remainders bottom to top: 1101
Answer: 13₁₀ = 1101₂
Verification: (1×8) + (1×4) + (0×2) + (1×1) = 8 + 4 + 0 + 1 = 13 ✓
Example 2: Convert Decimal 42 to Binary
Problem: Convert decimal 42 to binary
Division process:
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 bottom to top: 101010
Answer: 42₁₀ = 101010₂
Verification:
32 + 0 + 8 + 0 + 2 + 0 = 42 ✓
Example 3: Convert Decimal 255 to Binary
Problem: Convert decimal 255 to binary
Quick observation: 255 is the maximum value for 8 bits
Division steps:
255 ÷ 2 = 127 R 1
127 ÷ 2 = 63 R 1
63 ÷ 2 = 31 R 1
31 ÷ 2 = 15 R 1
15 ÷ 2 = 7 R 1
7 ÷ 2 = 3 R 1
3 ÷ 2 = 1 R 1
1 ÷ 2 = 0 R 1
Result: 11111111 (eight 1s)
Answer: 255₁₀ = 11111111₂
💡 Pattern: All 1s = maximum value for n bits = 2ⁿ - 1
Decimal to Binary Conversion Table
| Decimal | Binary | Calculation |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 1 | 2⁰ = 1 |
| 2 | 10 | 2¹ = 2 |
| 3 | odede>11 | 2¹ + 2⁰ = 2 + 1 |
| 4 | 100 | 2² = 4 |
| 5 | odede>101 | 2² + 2⁰ = 4 + 1 |
| 10 | 1010 | 8 + 2 |
| 15 | 1111 | 8 + 4 + 2 + 1 |
| 16 | 10000 | 2⁴ = 16 |
| 32 | 100000 | 2⁵ = 32 |
| 64 | odede>1000000 | 2⁶ = 64 |
| 100 | 1100100 | 64 + 32 + 4 |
| 128 | 10000000 | 2⁷ = 128 |
| 255 | 11111111 | All bits set (8-bit max) |
Powers of 2 Reference (Binary Place Values)
| Position | Power | Decimal Value | Binary |
|---|---|---|---|
| 0 | 2⁰ | 1 | 1 |
| 1 | 2¹ | 2 | 10 |
| 2 | 2² | 4 | 100 |
| 3 | 2³ | 8 | 1000 |
| 4 | 2⁴ | 16 | 10000 |
| 5 | 2⁵ | 32 | 100000 |
| 6 | 2⁶ | 64 | odede>1000000 |
| 7 | 2⁷ | 128 | 10000000 |
| 8 | 2⁸ | 256 | 100000000 |
| 10 | 2¹⁰ | 1,024 | 1KB (approx) |
Alternative Conversion Methods
Method 2: Subtraction Method
Concept: Subtract largest power of 2, mark positions
Example: Convert 45 to binary
Step 1: Find largest power of 2 ≤ 45
32 (2⁵) fits, so position 5 = 1
45 - 32 = 13 remaining
Step 2: Next largest ≤ 13
8 (2³) fits, so position 3 = 1
13 - 8 = 5 remaining
Step 3: Next largest ≤ 5
4 (2²) fits, so position 2 = 1
5 - 4 = 1 remaining
Step 4: Last bit
1 (2⁰) fits, so position 0 = 1
Result: Positions 5,3,2,0 are set = 101101
Real-World Applications
Computer Science
- Data representation: How computers store integers
- IP addressing: IPv4 addresses are 32-bit binary numbers
- File permissions: Unix/Linux file modes (rwx = 111)
- Bit flags: Settings and options in programming
- Network masks: Subnet masks in binary
Digital Electronics
- Logic circuits: Digital circuit design and analysis
- Microcontrollers: Programming embedded systems
- Data encoding: Error detection and correction
- Display systems: Seven-segment displays, LEDs
- Communication: Serial and parallel data transmission
Common Conversion Patterns
Quick Recognition Tips:
- Powers of 2: 1, 2, 4, 8, 16, 32, 64, 128, 256 have single 1 bit
- One less than power of 2: All 1s (7=111, 15=1111, 31=11111)
- Even numbers: Always end in 0 (LSB = 0)
- Odd numbers: Always end in 1 (LSB = 1)
- Multiples of 4: Last two bits are 00
- Max n-bit value: All 1s = 2ⁿ - 1
Tips for Manual Conversion
Conversion Strategies:
- Memorize powers of 2: Up to at least 2¹⁰ = 1024
- Use graph paper: Organize division steps neatly
- Double-check: Convert back to decimal to verify
- Work systematically: Don't skip steps
- Practice regularly: Build speed and accuracy
- Learn patterns: Recognize common decimal-binary pairs
Common Mistakes to Avoid
⚠️ Conversion Errors
- Reading remainders wrong direction: Must read bottom to top, not top to bottom
- Forgetting remainders: Record every remainder (even 0s)
- Stopping too early: Continue until quotient is 0
- Arithmetic mistakes: Verify each division step
- Mixing up MSB/LSB: Most significant bit is leftmost
- Ignoring zeros: Leading zeros can be dropped, but not trailing
- Negative numbers: Basic method only works for positive integers
Binary Representation Ranges
| Bits | Range (Unsigned) | Maximum Value | Binary Max |
|---|---|---|---|
| 4 bits | 0 to 15 | 15 | odede>1111 |
| 8 bits (byte) | 0 to 255 | 255 | 11111111 |
| 16 bits | 0 to 65,535 | 65,535 | 16 ones |
| 32 bits | 0 to 4,294,967,295 | ~4.3 billion | 32 ones |
| 64 bits | 0 to 18,446,744,073,709,551,615 | ~18 quintillion | 64 ones |
Frequently Asked Questions
How do you convert decimal to binary by hand?
Use the division-remainder method: repeatedly divide the decimal number by 2, recording each remainder (0 or 1). Continue dividing the quotient by 2 until it reaches 0. Read all remainders from bottom to top to get the binary result. Example: 13÷2=6 R1, 6÷2=3 R0, 3÷2=1 R1, 1÷2=0 R1. Reading bottom-up gives 1101. Always record remainders in order and read backwards. Practice makes this process quick and intuitive.
Why do we read remainders from bottom to top?
The first remainder represents the least significant bit (rightmost, 2⁰ position), while the last remainder represents the most significant bit (leftmost, highest power of 2). Reading bottom-to-top arranges bits in correct order: MSB to LSB (left to right). This aligns with how we write numbers with most significant digits first. The division order naturally produces bits in reverse, so bottom-up reading corrects this. Standard convention in computer science and mathematics.
What is the binary equivalent of decimal 100?
Decimal 100 equals binary 1100100. Calculation: 100÷2=50 R0, 50÷2=25 R0, 25÷2=12 R1, 12÷2=6 R0, 6÷2=3 R0, 3÷2=1 R1, 1÷2=0 R1. Reading bottom-up: 1100100. Verification: 64+32+4 = 100. This is 7 bits long. Useful to memorize common values like 100 for quick conversions. In hex, 100₁₀ = 64₁₆. Used frequently in programming (percentage values, RGB colors).
How many bits are needed to represent a decimal number?
Use formula: bits needed = ⌈log₂(n+1)⌉ (ceiling of log base 2). Practical: find smallest power of 2 greater than number. Examples: 7 needs 3 bits (2³=8), 15 needs 4 bits (2⁴=16), 255 needs 8 bits (2⁸=256). General rule: n bits can represent 0 to 2ⁿ-1. To represent number N, need at least ⌈log₂(N)⌉ bits. Important for data type selection in programming, memory allocation.
Can decimal fractions be converted to binary?
Yes! Fractional part uses different method: multiply by 2, take integer part as bit, repeat with fractional part. Example: 0.75₁₀. 0.75×2=1.5 (bit=1), 0.5×2=1.0 (bit=1). Result: 0.11₂. Not all decimal fractions have exact binary equivalents (e.g., 0.1₁₀ = 0.0001100110011...₂ repeating). Causes floating-point precision issues in computers. Important for understanding rounding errors in programming, why 0.1+0.2≠0.3 in many languages.
What's the fastest way to convert small decimal numbers?
Memorize common conversions (0-16). Use subtraction method: subtract largest power of 2 that fits, mark that bit position, repeat. Example: 13. Subtract 8 (2³), leaves 5. Subtract 4 (2²), leaves 1. Subtract 1 (2⁰). Positions 3,2,0 set = 1101. Faster than division for small numbers once you know powers of 2. Also recognize patterns: even=ends in 0, odd=ends in 1. With practice, common values become instant recall.
Key Takeaways
Converting decimal to binary is a fundamental skill in computer science and digital electronics. The division-remainder method provides a systematic approach to transform human-readable decimal numbers into machine-readable binary format, essential for understanding how computers store and process data at the most basic level.
Essential principles to remember:
- Binary uses base-2: only digits 0 and 1
- Division-remainder method: divide by 2, record remainders
- Read remainders bottom to top for correct binary
- Each bit position represents a power of 2
- Even numbers end in 0, odd numbers end in 1
- n bits can represent values 0 to 2ⁿ-1
- 8 bits = 1 byte, max value 255
- Always verify by converting back to decimal
- Memorize powers of 2 for faster conversions
- Practice regularly to build speed and accuracy
Getting Started: Use the interactive converter at the top of this page to convert any decimal number to binary instantly. Enter your decimal number, click convert, and receive the binary result with complete step-by-step breakdown showing the division process, remainders at each step, and verification calculation. Perfect for learning, homework, programming, or professional work requiring decimal-to-binary conversions.