Decimal to Binary Converter - Convert Numbers to Binary Code
Decimal to Binary Converter - Free Online Tool
Convert Any Decimal Number to Binary Instantly
Our free decimal to binary converter helps you change regular numbers (decimal) into computer language (binary). Just type any number from 0 to 65,535 and get the binary result immediately. Perfect for students, programmers, and anyone learning about computer numbers.
This tool shows you step-by-step how the conversion works, so you can learn while you convert. No registration needed - completely free to use!
What You Get:
- Instant decimal to binary conversion
- Step-by-step conversion process
- Binary, octal, and hexadecimal results
- Verification calculations
- Educational examples and explanations
- Mobile-friendly design
- Print and download options
Why Use Our Decimal to Binary Converter?
Easy to Use
Simple interface that anyone can understand. Just enter a number and see the result.
Educational
Learn how binary conversion works with detailed step-by-step explanations.
Accurate
Get precise results every time with built-in verification to check your answers.
Enter any positive integer
Conversion Results:
Decimal (Base 10)
42
Binary (Base 2)
101010
Octal (Base 8)
52
Hexadecimal (Base 16)
2A
Division Method Steps:
1. 42 ÷ 2 = 21 remainder 0
2. 21 ÷ 2 = 10 remainder 1
3. 10 ÷ 2 = 5 remainder 0
4. 5 ÷ 2 = 2 remainder 1
5. 2 ÷ 2 = 1 remainder 0
6. 1 ÷ 2 = 0 remainder 1
Binary result: 101010 (read remainders bottom to top)
Verification:
Position values:
Position 5: 1 × 2^5 = 32
Position 4: 0 × 2^4 = 0
Position 3: 1 × 2^3 = 8
Position 2: 0 × 2^2 = 0
Position 1: 1 × 2^1 = 2
Position 0: 0 × 2^0 = 0
Sum: 0×2^0(1) + 1×2^1(2) + 0×2^2(4) + 1×2^3(8) + 0×2^4(16) + 1×2^5(32) = 42
How Decimal to Binary Conversion Works
Input Number
Enter decimal value
Divide by 2
Record remainders
Repeat
Until quotient is 0
Conversion Formulas and Methods
Division Method Formula
Decimal ÷ 2 = Quotient + Remainder
Repeat until quotient = 0
Example: 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
Result: 101010₂
Position Value Formula
Decimal = Σ(bit × 2^position)
Sum of powers of 2
Example: 101010₂
= 1×2⁵ + 0×2⁴ + 1×2³ + 0×2² + 1×2¹ + 0×2⁰
= 1×32 + 0×16 + 1×8 + 0×4 + 1×2 + 0×1
= 32 + 0 + 8 + 0 + 2 + 0
= 42₁₀
Base Conversion Rules
Base₁₀ → Base₂
Successive division by 2
Binary: Uses digits 0, 1
Octal: Uses digits 0-7
Hexadecimal: Uses 0-9, A-F
Powers of 2: 1, 2, 4, 8, 16, 32, 64...
Each position = base^power
Decimal to Binary Conversion Table
| Decimal | Binary | Octal | Hexadecimal | Bit Length |
|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 bits |
| 2 | 10 | 2 | 2 | 2 bits |
| 4 | 100 | 4 | 4 | 3 bits |
| 8 | 1000 | 10 | 8 | 4 bits |
| 16 | 10000 | 20 | 10 | 5 bits |
| 32 | 100000 | 40 | 20 | 6 bits |
| 64 | 1000000 | 100 | 40 | 7 bits |
| 128 | 10000000 | 200 | 80 | 8 bits |
| 256 | 100000000 | 400 | 100 | 9 bits |
| 512 | 1000000000 | 1000 | 200 | 10 bits |
| 1,024 | 10000000000 | 2000 | 400 | 11 bits |
| 2,048 | 100000000000 | 4000 | 800 | 12 bits |
| 4,096 | 1000000000000 | 10000 | 1000 | 13 bits |
| 8,192 | 10000000000000 | 20000 | 2000 | 14 bits |
| 16,384 | 100000000000000 | 40000 | 4000 | 15 bits |
Note: This table shows powers of 2 and their binary representations. Each row represents 2^n where n increases by 1.
Binary Number System Progression Chart
2⁰
2¹
2²
2³
2⁴
2⁵
Practice Problems with Step-by-Step Solutions
Problem 1: Convert 75 to binary
75 ÷ 2 = 37 remainder 1
37 ÷ 2 = 18 remainder 1
18 ÷ 2 = 9 remainder 0
9 ÷ 2 = 4 remainder 1
4 ÷ 2 = 2 remainder 0
2 ÷ 2 = 1 remainder 0
1 ÷ 2 = 0 remainder 1
Solution: 75₁₀ = 1001011₂
Problem 2: Convert 156 to binary
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
Solution: 156₁₀ = 10011100₂
Problem 3: Convert 11010₂ to decimal
Position values: 1×2⁴ + 1×2³ + 0×2² + 1×2¹ + 0×2⁰
= 1×16 + 1×8 + 0×4 + 1×2 + 0×1
= 16 + 8 + 0 + 2 + 0
Solution: 11010₂ = 26₁₀
Problem 4: Convert 200 to hexadecimal
200 ÷ 16 = 12 remainder 8
12 ÷ 16 = 0 remainder 12 (C in hex)
Read remainders bottom to top: C8
Solution: 200₁₀ = C8₁₆
Problem 5: Convert 127 to octal
127 ÷ 8 = 15 remainder 7
15 ÷ 8 = 1 remainder 7
1 ÷ 8 = 0 remainder 1
Read remainders bottom to top: 177
Solution: 127₁₀ = 177₈
Problem 6: Convert 1111₂ to all bases
Binary to Decimal: 1×8 + 1×4 + 1×2 + 1×1 = 15
Decimal to Octal: 15 ÷ 8 = 1 remainder 7 → 17₈
Decimal to Hex: 15 = F₁₆
Solution: 1111₂ = 15₁₀ = 17₈ = F₁₆
5 Daily Uses of Binary Conversion
Programming code uses binary for computer instructions
Digital cameras store photos in binary format
Internet routers use binary for network addresses
Mobile apps convert data to binary for storage
Gaming consoles process graphics using binary numbers
What is Decimal to Binary Conversion?
Understanding Decimal Numbers
Decimal numbers are the numbers we use every day. They use ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. This is called base-10 because it has ten different symbols. When we count, we go from 0 to 9, then start over with 10.
Each position in a decimal number represents a power of 10. For example, in the number 123, the 3 is in the ones place (10⁰), the 2 is in the tens place (10¹), and the 1 is in the hundreds place (10²).
Understanding Binary Numbers
Binary numbers use only two digits: 0 and 1. This is called base-2 because it has only two symbols. Binary is the language that computers speak. Every piece of information in a computer is stored as binary numbers.
Each position in a binary number represents a power of 2. For example, in the binary number 101, reading from right to left: the first 1 is 2⁰ (which equals 1), the 0 is 2¹ (which equals 0), and the last 1 is 2² (which equals 4). So 101 in binary equals 1+0+4 = 5 in decimal.
Why Do We Need Binary Conversion?
Converting decimal to binary is important because computers only understand binary. When you type a number on your keyboard, the computer converts it to binary to process it. When programmers write code, they often need to work with binary numbers directly.
Binary conversion helps in computer programming, digital electronics, network settings, and understanding how computers store data. It's also useful for students learning computer science and anyone working with technology.
How Decimal to Binary Conversion Works
The Simple Division Method
The easiest way to convert decimal to binary is using the division method. Here's how it works in simple steps:
Step-by-Step Process:
- Take your decimal number and divide it by 2
- Write down the remainder (it will be either 0 or 1)
- Take the result (quotient) and divide it by 2 again
- Keep writing down the remainders
- Continue until the quotient becomes 0
- Read all the remainders from bottom to top - that's your binary number!
Example: Convert 13 to Binary
13 ÷ 2 = 6 remainder 1
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
Reading remainders from bottom to top: 1101
So 13 in decimal = 1101 in binary
Alternative Method: Powers of 2
Another way to convert is using powers of 2. This method helps you understand what each binary digit means.
Powers of 2 Table:
2⁰ = 1
2¹ = 2
2² = 4
2³ = 8
2⁴ = 16
2⁵ = 32
2⁶ = 64
2⁷ = 128
Example: Convert 13 using Powers of 2
13 is less than 16 (2⁴) but greater than 8 (2³)
So we use: 8 + 4 + 1 = 13
In powers of 2: 2³ + 2² + 2⁰
Binary positions: 1101
Result: 13₁₀ = 1101₂
Common Decimal to Binary Examples
Everyday Numbers in Binary
Here are some common decimal numbers and their binary equivalents. These examples show how different types of numbers look in binary format. Notice the patterns!
The smallest positive number - just one bit needed
1 = 2⁰ = 1
Five uses three bits with a pattern
5 = 4 + 1 = 2² + 2⁰
Ten shows alternating pattern in binary
10 = 8 + 2 = 2³ + 2¹
Perfect power of 2 - only one bit is 1
16 = 2⁴
Twenty-five combines multiple powers of 2
25 = 16 + 8 + 1 = 2⁴ + 2³ + 2⁰
Fifty is an even number ending in 0
50 = 32 + 16 + 2 = 2⁵ + 2⁴ + 2¹
One hundred needs seven bits to represent
100 = 64 + 32 + 4 = 2⁶ + 2⁵ + 2²
Another perfect power of 2 with 8 bits
128 = 2⁷
Maximum 8-bit number - all bits are 1
255 = 128+64+32+16+8+4+2+1
Easy Patterns to Remember
- Powers of 2: Numbers like 1, 2, 4, 8, 16, 32 have only one '1' bit in binary
- Even numbers: Always end with '0' in binary (like 10, 50, 100)
- Odd numbers: Always end with '1' in binary (like 1, 5, 25)
- All 1's: Numbers like 7 (111), 15 (1111), 31 (11111) are 2ⁿ - 1
- Zero: The only number that is just '0' in binary
Quick Recognition Tips
- Count the bits: More bits mean bigger numbers
- First bit: If it starts with 1, the number is at least that power of 2
- Last bit: Tells you if the number is odd (1) or even (0)
- All zeros: Only the number 0 has all zeros
- Mixed pattern: Most numbers have a mix of 1s and 0s
Real-World Examples
Age 25: 11001 in binary (5 bits needed)
Days in year (365): 101101101 in binary (9 bits)
Hours in day (24): 11000 in binary (5 bits)
Minutes in hour (60): 111100 in binary (6 bits)
Seconds in minute (60): 111100 in binary (6 bits)
Months in year (12): 1100 in binary (4 bits)
Frequently Asked Questions About Decimal to Binary Conversion
What is the difference between decimal and binary numbers?
Decimal numbers use 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and are based on powers of 10. This is the number system we use in daily life for counting money, telling time, and measuring things.
Binary numbers use only 2 digits (0 and 1) and are based on powers of 2. Computers use binary because electronic switches can only be in two states: on (1) or off (0). This makes binary perfect for digital devices like computers, phones, and calculators.
How do I convert large decimal numbers to binary?
For large numbers, use the same division method but it will take more steps. Keep dividing by 2 and write down each remainder until you reach 0. Our online calculator can handle numbers up to 65,535 instantly.
For numbers bigger than 65,535, you can use programming languages like Python, JavaScript, or online calculators that support larger numbers. The method stays the same - just more divisions!
Why do computers use binary instead of decimal?
Computers are made of millions of tiny electronic switches called transistors. Each switch can only be in two states: on or off. It's much easier and more reliable to represent these two states as 1 (on) and 0 (off) than to try to create 10 different states for decimal digits.
Think of it like a light switch - it's either on or off. It would be very complicated and unreliable to make a switch that could be in 10 different positions accurately.
What is the maximum number I can convert with this tool?
Our converter works with numbers from 0 to 65,535. This range covers most everyday needs and educational purposes. The number 65,535 is special because it's the largest number you can represent with 16 binary digits (bits).
If you need to convert larger numbers, you can use the same division method by hand or use programming tools. The process is exactly the same, just with more steps.
How can I check if my binary conversion is correct?
The best way to verify is to convert the binary number back to decimal. Take each binary digit, multiply it by its corresponding power of 2, and add them all up.
For example, to check if 1010 in binary equals 10 in decimal: (1×8) + (0×4) + (1×2) + (0×1) = 8 + 0 + 2 + 0 = 10. Our calculator shows this verification step automatically.
Can I convert negative numbers to binary?
Yes, but negative numbers in binary are more complex. Computers use a special method called "two's complement" to represent negative numbers. This involves flipping all the bits and adding 1.
Our basic converter focuses on positive integers, which are easier to understand and more commonly used in learning. For negative numbers, you'll need specialized tools or programming knowledge.
Where is binary conversion used in real life?
Binary conversion is everywhere in technology! Programmers use it when writing software, network engineers use it for IP addresses, and digital artists use it for color codes. Even your smartphone uses binary to store your photos, messages, and apps.
Students learning computer science, electronics technicians, and anyone working with digital systems need to understand binary conversion. It's also useful for understanding how file sizes work (like KB, MB, GB).
How many bits do I need for a specific decimal number?
The number of bits needed depends on how big your decimal number is. Here's a simple guide: Numbers 0-1 need 1 bit, 0-3 need 2 bits, 0-7 need 3 bits, 0-15 need 4 bits, and so on.
A quick rule: if your number is less than 2^n, then you need n bits. For example, numbers less than 256 (which is 2^8) need 8 bits. Our calculator shows you exactly how many bits your number needs.
Is there a pattern to binary numbers?
Yes! Binary numbers follow clear patterns. Even numbers always end in 0, odd numbers always end in 1. Powers of 2 (like 1, 2, 4, 8, 16) have only one '1' bit. Numbers that are one less than a power of 2 (like 3, 7, 15, 31) have all '1' bits.
Learning these patterns makes binary conversion much easier and helps you spot mistakes quickly.
What's the easiest way to learn binary conversion?
Start with small numbers like 1-15 and practice the division method. Use our calculator to check your work. Try to memorize common numbers like powers of 2 (1, 2, 4, 8, 16, 32, 64, 128).
Practice converting your age, the current year, or other meaningful numbers. The more you practice, the easier it becomes. Don't worry about making mistakes - that's how you learn!