Binary Calculator
Perform binary arithmetic operations (addition, subtraction, multiplication, division) with this free online calculator. Get accurate results in both binary and decimal formats.
Binary Arithmetic Operations
Binary arithmetic follows the same fundamental principles as decimal arithmetic, but uses only two digits (0 and 1). Here's how each operation works:
Binary Addition
Binary addition follows these rules:
- 0 + 0 = 0
- 0 + 1 = 1
- 1 + 0 = 1
- 1 + 1 = 0 (with a carry of 1)
Example: 1011 + 0101
1011
+ 0101
------
10000
= 16 in decimal (11 + 5 = 16)
Binary Subtraction
Binary subtraction follows these rules:
- 0 - 0 = 0
- 1 - 0 = 1
- 1 - 1 = 0
- 0 - 1 = 1 (with a borrow of 1)
Example: 1101 - 0011
1101
- 0011
------
1010
= 10 in decimal (13 - 3 = 10)
Binary Multiplication
Binary multiplication follows these rules:
- 0 × 0 = 0
- 0 × 1 = 0
- 1 × 0 = 0
- 1 × 1 = 1
Example: 101 × 11
101
× 11
-----
101
101.
-----
1111
= 15 in decimal (5 × 3 = 15)
Binary Division
Binary division works similarly to decimal division, but using only 0 and 1 as digits. Our calculator performs integer division, truncating any remainder.
Example: 1110 ÷ 10
111
10 ) 1110
10
--
11
10
--
1
= 7 in decimal (14 ÷ 2 = 7)
Common Applications of Binary Arithmetic
Binary arithmetic forms the foundation of all computer operations. CPUs perform calculations using binary arithmetic logic units (ALUs) that implement these basic operations.
Engineers use binary arithmetic when designing digital circuits, including adders, subtractors, multipliers, and other components that perform mathematical operations in hardware.
Low-level programming and embedded systems often require understanding of binary arithmetic for tasks like bit manipulation, efficient algorithm design, and hardware control.
Binary arithmetic is a fundamental topic in computer science education, helping students understand how computers process and manipulate data at the most basic level.
Frequently Asked Questions
Computers use binary arithmetic because electronic circuits are most reliable and efficient when representing just two states (on/off, high/low voltage). Binary arithmetic's simplicity makes it ideal for implementation in digital hardware, requiring fewer components and less power compared to other number systems.
Bit width refers to the number of binary digits (bits) used to represent a number. It determines the range of values that can be represented. For example, an 8-bit binary number can represent values from 0 to 255. If a calculation results in a value outside this range, overflow occurs. Computers typically use standard bit widths like 8, 16, 32, or 64 bits for arithmetic operations.
In computing, negative numbers are typically represented using two's complement notation. This system allows for addition and subtraction operations to work the same way for both positive and negative numbers. To represent a negative number in two's complement, invert all the bits and add 1. You can use our Two's Complement Calculator to see this in action.
Overflow occurs when the result of a binary operation exceeds the maximum value that can be represented with the specified bit width. For example, if you're using 4-bit numbers, the maximum representable value is 1111 (15 in decimal). If an addition results in a value greater than 15, overflow occurs. Our calculator warns you about potential overflow issues and recommends increasing the bit width when needed.
| Operation | Result | Carry |
|---|---|---|
| 0 + 0 | 0 | 0 |
| 0 + 1 | 1 | 0 |
| 1 + 0 | 1 | 0 |
| 1 + 1 | 0 | 1 |
| 1 + 1 + 1 | 1 | 1 |
This table shows the basic rules of binary addition, including the carry bit which is added to the next position to the left.