How to Convert Binary — A Complete Guide to Decimal, Hex & Octal
Introduction
Learning how to convert binary to decimal, hexadecimal, and octal is a fundamental skill in computer science, digital electronics, and programming. These number systems, or base systems, are the languages of computers.
Why Learn Binary Conversion?
Understanding binary conversion is key for:
- Computer science and programming fundamentals
- Digital electronics and hardware design
- Network administration and data analysis
- Cybersecurity and systems analysis
- Academic success in technical fields
Essential Technical Skills
These conversions are crucial for:
- Debugging code and understanding data representation
- Analysing network data and packet inspection
- Understanding hardware operations and memory
- Working with assembly language and low-level programming
- Binary arithmetic and logical operations
Manual vs Tool Understanding
While online tools provide instant results, mastering the manual process builds:
- Deep, intuitive understanding of computer systems
- Problem-solving confidence when tools aren't available
- Technical interview preparation and competence
- Foundation knowledge for advanced computing topics
- Troubleshooting abilities in various technical contexts
What You'll Master
This comprehensive guide covers:
- Step-by-step conversion methods for all number systems
- Bit weighting and positional notation
- Two's complement for negative numbers
- Binary arithmetic and practical applications
- Worked examples for confident manual calculation
You'll learn to translate between binary, decimal, hexadecimal, and octal with confidence, building the foundation for advanced computer science concepts.
What are Binary, Decimal, Hexadecimal, and Octal?
At its core, a number system is defined by its base (or radix), which indicates how many digits it uses.
- Decimal (Base-10): The system we use daily. It has ten digits (0-9). Each position represents a power of 10.
- Binary (Base-2): The native language of computers. It has two digits (0 and 1), called bits. Each position represents a power of 2.
- Hexadecimal (Base-16): A compact, human-friendly way to represent binary. It has sixteen digits: 0-9 and A-F (where A=10, B=11, ..., F=15). One hex digit represents four bits (a nibble).
- Octal (Base-8): An older system that uses eight digits (0-7). One octal digit represents three bits.
The Principle of Positional Notation
The value of a number in any system is the sum of each digit multiplied by the base raised to the power of its position, starting from 0 on the right. This is the universal formula for conversion to decimal.
General Formula: Decimal = (dₙ × baseⁿ) + (dₙ₋₁ × baseⁿ⁻¹) + ... + (d₁ × base¹) + (d₀ × base⁰)
Method 1: Binary to Decimal (The Bit Weighting Method)
This is the most direct method, using the positional notation formula.
Step-by-Step:
- Write down the binary number.
- List the powers of 2 above each bit, starting from 2⁰ on the right.
- Multiply each bit by its corresponding power of 2.
- Sum all the products to get the decimal value.
Example: Convert 1101₂ to decimal.
- Binary:
1 1 0 1 - Powers:
2³ 2² 2¹ 2⁰(which are8, 4, 2, 1) - Calculate:
(1×8) + (1×4) + (0×2) + (1×1) - Sum:
8 + 4 + 0 + 1 = 13Result:1101₂ = 13₁₀
Method 2: Decimal to Binary (Repeated Division-by-2)
To convert from decimal to binary, we use repeated division.
Step-by-Step:
- Divide the decimal number by 2.
- Write down the remainder (this will be 0 or 1).
- Use the quotient from step 1 as the new number to divide.
- Repeat steps 1-3 until the quotient is 0.
- The binary number is the remainders read from bottom to top.
Example: Convert 25₁₀ to binary.
Result: 25₁₀ = 11001₂
Method 3: Binary to Hexadecimal (Grouping by Nibbles)
This conversion is easy due to the 4:1 relationship between bits and hex digits.
Step-by-Step:
- Group the binary digits into sets of four, starting from the right. Add leading zeros if necessary.
- Convert each 4-bit group into its corresponding hex digit.
Example: Convert 11010111₂ to hex.
- Group:
1101and0111. - Convert:
1101= D (13),0111= 7. Result:0xD7
Method 4: Binary to Octal (Grouping by Triplets)
This uses a similar grouping method but with sets of three bits.
Step-by-Step:
- Group the binary digits into sets of three, starting from the right. Add leading zeros if necessary.
- Convert each 3-bit group into its corresponding octal digit (0-7).
Example: Convert 10111010₂ to octal.
- Group:
010,111,010. (Note the leading zero added to the first group). - Convert:
010=2,111=7,010=2. Result:0o272
Understanding Signed vs. Unsigned
So far, we've worked with unsigned integers, which are always positive. Computers also need to represent negative numbers using signed integers. The most common method is two's complement.
Key Concept: In an n-bit system (e.g., 8-bit), the leftmost bit (the Most Significant Bit, or MSB) is the sign bit.
0in the MSB means a positive number.1in the MSB means a negative number.
How Two's Complement Works
To find the binary representation of a negative number:
Step-by-Step to Represent a Negative Number:
- Start with the binary representation of the positive equivalent.
- Invert all the bits (change 0s to 1s and 1s to 0s). This is called the ones' complement.
- Add 1 to the result.
Example: Represent -5 as an 8-bit signed integer.
- Positive
5in 8-bit:0000 0101 - Invert all bits:
1111 1010 - Add 1:
1111 1010 + 1 = 1111 1011Result:-5₁₀ = 1111 1011₂in two's complement.
To convert a negative binary number back to decimal, you reverse the process.
How to Use the Binary Converter on calcfort.com
While manual calculation is educational, using an online tool ensures speed and accuracy, especially for signed numbers and large values.
- Navigate to the Tool: Go to the Binary & Hex Converter page.
- Select Input Type: Choose the system of the number you're starting with (Binary, Decimal, Hex, or Octal).
- Enter Your Value: Type or paste your number. Using prefixes like
0b1010(binary) or0xFF(hex) is recommended for clarity. - Configure Settings: This is crucial for binary. Select Signed or Unsigned and a bit-width (e.g., 8-bit, 16-bit) if you are working with negative numbers.
- View Results: The converted values in all other systems are displayed instantly. Use the copy button to easily use the results in your code or notes.
Example: Verifying a Two's Complement Calculation
Scenario: Verify that the 8-bit binary 1111 1011 represents -5 in decimal.
- Select Input: Binary.
- Enter Value:
11111011. - Configure: Select Signed and 8-bit.
- Result: The Decimal output will instantly show -5, confirming your manual calculation.
Conclusion
Mastering how to convert binary manually demystifies the core operations of digital technology. The step-by-step methods for decimal, hex, and octal conversion, along with an understanding of two's complement, provide a solid foundation for any technical field. However, for efficiency and verification in your daily work, a dedicated tool is invaluable. Use our Binary & Hex Converter to check your work, handle large numbers, and explore these concepts with ease.