Binary Calculator
Perform arithmetic operations with binary numbers.
Binary Arithmetic
Table of Contents
Complete Binary System Guide
What is the Binary System?
The binary number system is a base-2 numeral system that uses only two symbols: 0 and 1. Unlike our everyday decimal system (base-10) which uses digits 0-9, binary represents all numbers using combinations of just these two digits. Each digit in a binary number represents a power of 2, making binary arithmetic the foundation of all modern computing and digital electronics.
In binary, each position represents a power of 2:
- Rightmost position: 20 = 1
- Second from right: 21 = 2
- Third from right: 22 = 4
- Fourth from right: 23 = 8
- And so on...
Binary to Decimal Conversion
Converting binary to decimal involves multiplying each binary digit by its corresponding power of 2 and summing the results:
| Binary | Calculation | Decimal |
|---|---|---|
| 1010 | (1 × 23) + (0 × 22) + (1 × 21) + (0 × 20) | 10 |
| 1101 | (1 × 23) + (1 × 22) + (0 × 21) + (1 × 20) | 13 |
Decimal to Binary Conversion
To convert decimal to binary, use the "successive division by 2" method:
- Divide the decimal number by 2
- Record the remainder (0 or 1)
- Divide the quotient again by 2
- Repeat until the quotient becomes 0
- Read the remainders from bottom to top
Example: Convert 13 to binary
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
Reading bottom to top: 1101
Example: Convert 25 to binary
12 ÷ 2 = 6 remainder 0
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
Reading bottom to top: 11001
Importance in Computing
Binary is the foundation of modern computing for several key reasons:
- Electronic Implementation: Digital circuits operate using electrical signals that can be in one of two states: on/off, high/low, or true/false.
- Simplicity: Binary systems are simpler to design and less prone to errors compared to systems with more states.
- Data Storage: All data in computers, including text, images, videos, and programs, are ultimately stored as sequences of binary digits (bits).
- Boolean Logic: Binary enables the implementation of Boolean logic (AND, OR, NOT operations), which forms the basis of digital circuit design and computer programming.
Binary Number Properties
Binary Patterns
- All 1s: 2n - 1 (e.g., 1111 = 15)
- Powers of 2: Single 1 followed by 0s (e.g., 1000 = 8)
- Even numbers: Always end with 0
- Odd numbers: Always end with 1
Common Binary Values
| 0 | 0 |
| 1 | 1 |
| 10 | 1010 |
| 100 | 1100100 |
| 255 | 11111111 |
Advanced Binary Applications
Binary Coding Systems
Binary forms the foundation for various coding systems like ASCII, Unicode, UTF-8, and other character encoding schemes that represent text in computers.
Digital Signal Processing
Binary representations enable efficient processing of digital signals for applications like audio, video, and image processing.
Cryptography
Many encryption algorithms rely on binary operations like XOR for data security and privacy protection.
Binary Logic Operations
Binary logic operations form the foundation of digital circuit design and computer programming. These operations work on individual bits and are essential for implementing all computational tasks.
Basic Logic Operations
| Operation | Symbol | Description |
|---|---|---|
| AND | & | 1 only when both bits are 1 |
| OR | | | 1 when at least one bit is 1 |
| XOR | ^ | 1 when bits are different |
| NOT | ~ | Inverts bits (0→1, 1→0) |
Bit Shift Operations
| Operation | Symbol | Description |
|---|---|---|
| Left Shift | << | Shifts bits left, filling with 0s |
| Right Shift | >> | Shifts bits right, filling with 0s |
Binary Number Systems in Computing
Binary in Memory Organization
In computing, memory is organized in hierarchical units based on binary:
- Bit: A single binary digit (0 or 1)
- Byte: 8 bits, can represent 256 different values (28)
- Word: Typically 16, 32, or 64 bits, depending on computer architecture
- Kilobyte (KB): 210 bytes = 1,024 bytes
- Megabyte (MB): 220 bytes = 1,048,576 bytes
- Gigabyte (GB): 230 bytes = 1,073,741,824 bytes
Binary-Based Number Systems
Several number systems related to binary are commonly used in computing:
| System | Base | Digits | Usage |
|---|---|---|---|
| Binary | 2 | 0-1 | Machine code, low-level operations |
| Octal | 8 | 0-7 | File permissions in Unix systems |
| Decimal | 10 | 0-9 | Human-readable values, calculations |
| Hexadecimal | 16 | 0-9, A-F | Memory addresses, color codes, debugging |
Binary in Modern Technologies
Digital Communications
Binary coding schemes enable efficient data transmission through various communication channels, including the internet, wireless networks, and satellite communications.
Machine Learning
Binary is fundamental to neural networks and machine learning algorithms, which often use binary weights or activation functions in their computational models.
Quantum Computing
While traditional computing uses bits, quantum computing uses quantum bits or "qubits" that can exist in multiple states simultaneously, exponentially increasing computational power.
When working with binary numbers, group them in sets of four bits to make them easier to read and convert to hexadecimal. For example, the binary number 10110110 can be grouped as 1011 0110.
Binary Arithmetic
Binary arithmetic is the foundation of computer operations. It uses only two digits (0 and 1) and follows specific rules for addition, subtraction, multiplication, and division.
Binary Operations
Here are the basic binary operations:
-
1Addition: 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, 1 + 1 = 10 (carry 1)
-
2Subtraction: 0 - 0 = 0, 1 - 0 = 1, 1 - 1 = 0, 0 - 1 = 1 (borrow 1)
-
3Multiplication: 0 × 0 = 0, 0 × 1 = 0, 1 × 0 = 0, 1 × 1 = 1
-
4Division: Similar to decimal division, but using binary digits
Binary Arithmetic Examples
Example 1 Binary Addition
Adding binary numbers 1010 and 1100:
1010 + 1100 = 10110
Example 2 Binary Multiplication
Multiplying binary numbers 101 and 11:
101 × 11 = 1111
Example 3 Binary Division
Dividing binary numbers 1100 by 11:
1100 ÷ 11 = 100