Binary Calculator - Number System Converter

Convert between binary, decimal, hexadecimal, and octal number systems. Perform bitwise operations and understand computer number representation with visual explanations.

Number System Conversion

Enter any integer (positive or negative)

Enter values and click Calculate

Common Binary Examples

The answer to life, the universe, and everything

Dec:42
Bin:101010
Hex:0x2A
Oct:0o52

Maximum 8-bit unsigned value

Dec:255
Bin:11111111
Hex:0xFF
Oct:0o377

Power of 2 (2^7)

Dec:128
Bin:10000000
Hex:0x80
Oct:0o200

1 Kilobyte (in binary)

Dec:1024
Bin:10000000000
Hex:0x400
Oct:0o2000

Maximum 16-bit unsigned value

Dec:65535
Bin:1111111111111111
Hex:0xFFFF
Oct:0o177777

Understanding Binary Number Systems

Binary is the fundamental number system used by all digital computers and electronic systems. Understanding binary, hexadecimal, and octal number systems is essential for computer science, programming, digital electronics, and cybersecurity.

Number System Bases

Binary (Base 2)

Uses only 0 and 1. Each digit represents a power of 2.

101010₂ = 1×2⁵ + 0×2⁴ + 1×2³ + 0×2² + 1×2¹ + 0×2⁰ = 42₁₀

Decimal (Base 10)

Uses 0-9. The standard number system we use daily.

42₁₀ = 4×10¹ + 2×10⁰

Hexadecimal (Base 16)

Uses 0-9 and A-F. Compact representation of binary.

2A₁₆ = 2×16¹ + 10×16⁰ = 42₁₀

Octal (Base 8)

Uses 0-7. Often used in Unix file permissions.

52₈ = 5×8¹ + 2×8⁰ = 42₁₀

Real-World Applications

1.Network IP addresses and subnet masks (IPv4)
2.File permissions in Unix/Linux systems
3.Color representation (RGB values)
4.Cryptography and hash functions
5.Compression algorithms
6.Embedded systems and hardware programming
7.Flags and configuration bits in software
8.Graphics and image processing

Bitwise Operations Explained

AND (&):

Returns 1 only if both bits are 1. Used for masking specific bits.

1100 & 1010 = 1000
OR (|):

Returns 1 if at least one bit is 1. Used for setting bits.

1100 | 1010 = 1110
XOR (^):

Returns 1 if bits are different. Used for toggling and encryption.

1100 ^ 1010 = 0110
NOT (~):

Inverts all bits. Creates ones complement.

~1100 = 0011
Left Shift (<<):

Shifts bits left, multiplies by 2ⁿ. Fast multiplication.

0101 << 2 = 10100 (5 × 4 = 20)
Right Shift (>>):

Shifts bits right, divides by 2ⁿ. Fast division.

10100 >> 2 = 0101 (20 ÷ 4 = 5)

Two's Complement for Negative Numbers

Computers represent negative numbers using two's complement. This allows the same hardware to perform addition and subtraction without special cases for negative numbers.

How to Calculate Two's Complement:

  1. Write the positive number in binary
  2. Flip all the bits (ones complement)
  3. Add 1 to the result
5 in binary: 00000101
Flip bits: 11111010
Add 1: 11111011 = -5

Standards and Specifications

IEEE 754

Standard for Binary Floating-Point Arithmetic

Authority: IEEE (Institute of Electrical and Electronics Engineers)
Learn more →

ISO/IEC 10646

Universal Coded Character Set (Unicode)

Authority: ISO/IEC
Learn more →

Educational Resources

MIT OpenCourseWare - Introduction to Computer Science

Binary numbers and computer representation

Visit resource →

Khan Academy - Computer Science

Binary numbers and bitwise operations

Visit resource →

Stanford CS Education Library

Binary and bit manipulation techniques

Visit resource →

Recommended Reading

Computer Systems: A Programmer's Perspective

by Bryant & O'Hallaron

Comprehensive coverage of binary representation and computer systems

The Art of Computer Programming

by Donald Knuth

Classic computer science text covering number systems

Frequently Asked Questions

Why do computers use binary?

Computers use binary because digital circuits have two stable states: on (1) and off (0). This corresponds to high and low voltage levels in transistors. Binary is the most reliable and efficient way to represent information in electronic circuits. All data - numbers, text, images, videos - is ultimately stored as sequences of 0s and 1s.

What is hexadecimal used for?

Hexadecimal (base 16) is a compact way to represent binary numbers. Each hex digit represents exactly 4 bits, making it much easier to read and write than long binary strings. It's widely used for memory addresses, color codes in web design (RGB values), MAC addresses, cryptographic hashes, and assembly language programming. For example, the color white is #FFFFFF in hex, which is 111111111111111111111111 in binary.

How do I convert binary to decimal quickly?

Start from the rightmost bit (least significant bit). Each position represents a power of 2: 1, 2, 4, 8, 16, 32, etc. Add up the decimal values of positions where the bit is 1. For example, binary 1011: (1×8) + (0×4) + (1×2) + (1×1) = 8 + 0 + 2 + 1 = 11. With practice, you'll recognize common patterns like 11111111 = 255 or 10000000 = 128.

What are bitwise operations used for?

Bitwise operations are fundamental in systems programming, embedded systems, graphics, cryptography, and optimization. Common uses include: setting/clearing flags in configuration registers, implementing permissions systems (like Unix file permissions), fast multiplication/division by powers of 2, color manipulation in graphics, network subnet masks, data compression, encryption algorithms, and optimizing code performance. They're much faster than arithmetic operations.

What is the difference between signed and unsigned integers?

Unsigned integers can only represent positive numbers (including zero). An 8-bit unsigned integer can represent 0 to 255. Signed integers use two's complement to represent both positive and negative numbers. An 8-bit signed integer can represent -128 to +127. The most significant bit (leftmost) indicates the sign in signed integers: 0 for positive, 1 for negative. This is why the same bit pattern 11111111 means 255 unsigned but -1 signed.

Why is octal (base 8) used?

Octal was historically popular because each octal digit represents exactly 3 bits, making conversion from binary straightforward. It's still widely used in Unix/Linux file permissions (e.g., chmod 755), where each digit represents read (4), write (2), and execute (1) permissions. While hexadecimal is more common today, octal remains important in systems programming and legacy systems.

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Last updated: January 2025. This disclaimer applies to all calculators on Top Calculator.