Binary Calculator

How to Calculate with Binary Numbers

Binary arithmetic involves performing calculations using only two digits: 0 and 1. Here's a comprehensive guide on how to perform binary calculations:

Binary Conversion Formula

To convert a binary number to decimal, use the following formula:

\[ (b_n b_{n-1} ... b_1 b_0)_2 = b_n \times 2^n + b_{n-1} \times 2^{n-1} + ... + b_1 \times 2^1 + b_0 \times 2^0 \]

Where:

• \(b_i\) represents each digit of the binary number (0 or 1)
• \(n\) is the position of the digit (starting from 0 for the rightmost digit)

Calculation Steps

1. Convert both binary numbers to decimal using the formula above.
2. Perform the desired arithmetic operation (addition, subtraction, multiplication, or division) on the decimal numbers.
3. Convert the result back to binary using repeated division by 2.
4. If the result is negative, convert the absolute value to binary and add a negative sign.

Example Calculation

Let's subtract the binary numbers 1010 from 0110.

Step 1: Convert to Decimal

1010 to decimal:

\[ 1010_2 = 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 8 + 0 + 2 + 0 = 10_{10} \]

0110 to decimal:

\[ 0110_2 = 0 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 0 + 4 + 2 + 0 = 6_{10} \]

Step 2: Perform the Operation

Subtract the decimal numbers:

\[ 6 - 10 = -4_{10} \]

Step 3: Convert to Binary

To convert 4 to binary, divide by 2 repeatedly and read the remainders from bottom to top:

4 ÷ 2 = 2 remainder 0
2 ÷ 2 = 1 remainder 0
1 ÷ 2 = 0 remainder 1
        

Therefore, 4 in binary is 100.

Since the result is negative, we add a negative sign: -100

Final Result

0110 - 1010 = -100 (in binary)

Visual Representation

This diagram illustrates the process of subtracting two binary numbers (0110 and 1010) and obtaining the result in binary (-100).