Binary Calculator

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Binary Calculator
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Calculator Use

The binary calculator is a specialized tool designed to perform arithmetic and logic operations on base-2 numbers. While the decimal system (base-10) is what we use in daily life, computers and electronic systems rely exclusively on binary—using only two digits: 0 and 1. This calculator allows you to add, subtract, multiply, and divide binary strings, as well as perform bitwise logic like AND, OR, and XOR.

To use the tool, simply enter two binary sequences and select your desired operator. The calculator will process the inputs and provide the resulting binary value instantly.

Binary Value A
The first sequence of 0s and 1s you wish to calculate. Example: 1101 (which is 13 in decimal).
Binary Value B
The second sequence of 0s and 1s used for the operation.
Choose Operation
Select between standard math (Add, Sub, Mul, Div) or bitwise logic (AND, OR, XOR).

How It Works

Binary calculations follow the same principles as decimal calculations, but instead of carrying over at 10, you carry over at 2. The value of each position in a binary number is a power of 2, increasing from right to left.

Value = (dn × 2n) + … + (d1 × 21) + (d0 × 20)

  • 0 + 0 = 0
  • 0 + 1 = 1
  • 1 + 1 = 0 (carry 1)
  • 1 + 1 + 1 = 1 (carry 1)

Calculation Example

Example: Adding 1011 and 1101

In decimal terms, this is 11 + 13. Let's look at the step-by-step binary addition:

  1. Rightmost column: 1 + 1 = 0, carry 1.
  2. Second column: 1 (original) + 0 (original) + 1 (carry) = 0, carry 1.
  3. Third column: 0 + 1 + 1 (carry) = 0, carry 1.
  4. Fourth column: 1 + 1 + 1 (carry) = 1, carry 1.
  5. Bring down the final carry: 1.
  6. Result = 11000 (which is 24 in decimal).

Common Questions

Why do computers use binary instead of decimal?

Computers use binary because it is easy to represent physically. A transistor or switch can be in two states: On (1) or Off (0). Using base-10 would require circuits that could distinguish between ten different levels of voltage, which is far more susceptible to noise and hardware failure.

How do you handle negative binary numbers?

Negative numbers in binary are typically handled using a system called Two's Complement. In this system, the leftmost bit (the most significant bit) acts as a sign bit. If it is 1, the number is negative. To find the negative equivalent of a number, you flip all the bits (0 to 1 and vice versa) and add 1.

What are AND, OR, and XOR operations?

These are bitwise logic operations. AND results in 1 only if both bits are 1. OR results in 1 if at least one bit is 1. XOR (Exclusive OR) results in 1 only if the bits are different (one is 0 and the other is 1).

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