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Binary Calculator

Perform arithmetic and conversions with binary and decimal numbers, and learn the concepts behind the language of computers.

Binary Calculator

Perform calculations and conversions with binary and decimal numbers.

Understanding Binary

The binary numeral system is a base-2 system that uses only two symbols: 0 and 1. Each digit is referred to as a bit. It is the foundation of all modern computing systems.

Binary to Decimal Conversion

To convert a binary number to decimal, you multiply each bit by 2 raised to the power of its position (starting from 0 on the right) and sum the results.

Example: Convert binary 1101 to decimal.

(1 * 2³) + (1 * 2²) + (0 * 2¹) + (1 * 2⁰)
= (1 * 8) + (1 * 4) + (0 * 2) + (1 * 1)
= 8 + 4 + 0 + 1 = 13

Decimal to Binary Conversion

To convert a decimal number to binary, you repeatedly divide the decimal number by 2 and record the remainder. The binary result is the sequence of remainders read from bottom to top.

Example: Convert decimal 13 to binary.

13 / 2 = 6, remainder 1
6 / 2 = 3, remainder 0
3 / 2 = 1, remainder 1
1 / 2 = 0, remainder 1

Reading the remainders from bottom up gives: 1101.

Binary Arithmetic

Arithmetic in binary is similar to decimal, but simpler because it only involves two digits.

  • Addition: 0+0=0, 0+1=1, 1+0=1, 1+1=10 (0, carry 1).
  • Subtraction: Uses borrowing, similar to decimal. 10 - 1 = 1.
  • Multiplication: Simple rules: 1*1=1, and anything multiplied by 0 is 0.
  • Division: Similar to decimal long division.

This calculator performs these operations by first converting the binary operands to decimal, performing the standard calculation, and then converting the result back to binary.

The Language of Computers: A Guide to Binary Math

As a developer, I work with the binary system every day, even when I don't see it. It's the fundamental language of computers, a base-2 system that uses only two digits: 0 and 1. While it might look intimidating, its simplicity is what makes it so powerful. I built this binary calculator to be a comprehensive tool that not only performs calculations but also helps demystify the world of binary for students, programmers, and the curious alike.

Why Use Binary?

Computers operate using billions of tiny electronic switches that can only be in one of two states: on or off. The binary system perfectly represents this state, with '1' meaning 'on' and '0' meaning 'off'. My calculator provides three main functions to help you work with this system: arithmetic operations, and conversions to and from the decimal system we use every day.

How to Use This Binary Calculator

My tool is split into different tabs for clarity:

  • Arithmetic: Enter two binary numbers and choose an operation (add, subtract, multiply, or divide). My script will show you the result in both binary and decimal format.
  • Binary to Decimal: Enter a binary number, and the calculator will instantly show you its decimal equivalent.
  • Decimal to Binary: Enter a standard decimal number, and it will be converted into its binary representation.

Binary to Decimal Conversion

To convert a binary number to decimal, each digit is multiplied by 2 raised to the power of its position, starting from 0 on the right.

For example, the binary number 1101 is converted as:

(1 * 2³) + (1 * 2²) + (0 * 2¹) + (1 * 2⁰) = 8 + 4 + 0 + 1 = 13

Decimal to Binary Conversion

To convert a decimal number to binary, you repeatedly divide the number by 2 and record the remainders. The binary number is the sequence of these remainders read from bottom to top.

Binary Arithmetic Explained

My calculator handles arithmetic by converting the binary inputs to decimal, performing the calculation, and then converting the result back to binary. Here's a peek at the underlying rules:

  • Addition: 0+0=0, 0+1=1, 1+1=10 (0 and carry over 1).
  • Subtraction, Multiplication, and Division: These follow similar principles to their decimal counterparts, but are simplified by only having two digits to work with.

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