Binary Calculator

Understanding Binary Numbers

In binary, each digit position represents a power of 2, starting from 2^0 on the right. For example, the binary number 1101 represents: (1×2³) + (1×2²) + (0×2¹) + (1×2⁰) = 8 + 4 + 0 + 1 = 13 in decimal. The rightmost digit is called the Least Significant Bit (LSB), and the leftmost is the Most Significant Bit (MSB).

1101_2 = 1 × 2^3 + 1 × 2^2 + 0 × 2^1 + 1 × 2^0 = 13_10

Binary Addition Rules

Binary addition follows four fundamental rules: 0+0=0, 0+1=1, 1+0=1, and 1+1=10 (which means 0 with a carry of 1). When adding multiple binary numbers, carries propagate to the left just like in decimal addition. This process forms the basis of all arithmetic operations in digital computers.

\beginalign 0 + 0 &= 0 \\ 0 + 1 &= 1 \\ 1 + 0 &= 1 \\ 1 + 1 &= 10 \text (carry 1) \endalign

Binary Subtraction Rules

Binary subtraction uses four basic rules: 0-0=0, 1-0=1, 1-1=0, and 0-1=1 (with a borrow of 1 from the next higher bit). When borrowing is needed, we borrow from the next significant bit, converting it from 1 to 0 and adding 2 to the current position (since we're in base-2).

\beginalign 0 - 0 &= 0 \\ 1 - 0 &= 1 \\ 1 - 1 &= 0 \\ 0 - 1 &= 1 \text (borrow 1) \endalign

Binary Multiplication

Binary multiplication is simpler than decimal multiplication because you only multiply by 0 or 1. Multiplying by 0 gives 0, and multiplying by 1 gives the original number. The process involves shifting and adding: for each 1 in the multiplier, add the multiplicand shifted left by the appropriate number of positions.

\beginalign 0 × 0 &= 0 \\ 0 × 1 &= 0 \\ 1 × 0 &= 0 \\ 1 × 1 &= 1 \endalign

Binary Division

Binary division follows the same long division process as decimal division, but uses binary subtraction. At each step, determine if the divisor fits into the current portion of the dividend. If yes, write 1 in the quotient and subtract; if no, write 0 and bring down the next bit.

\textQuotient = \frac\textDividend\textDivisor, \text Remainder = \textDividend \bmod \textDivisor

Binary to Decimal Conversion

To convert a binary number to decimal, multiply each digit by its corresponding power of 2 and sum the results. Start from the rightmost digit (2⁰) and move left, increasing the power by 1 for each position. This method is called positional notation.

\textDecimal = Σ_i=0^n-1 b_i × 2^i \text where b_i \text is the bit at position i

Decimal to Binary Conversion

Convert decimal to binary using the division method: repeatedly divide the decimal number by 2 and record the remainders. The binary number is formed by reading the remainders from bottom to top (last remainder first). Alternatively, use the subtraction method by subtracting the largest possible power of 2 repeatedly.

\textDecimal ÷ 2 = \textQuotient + \textRemainder, \text continue until quotient = 0

Powers of 2 Table

Understanding powers of 2 is crucial for binary arithmetic. Common powers include: 2⁰=1, 2¹=2, 2²=4, 2³=8, 2⁴=16, 2⁵=32, 2⁶=64, 2⁷=128, 2⁸=256, 2⁹=512, 2¹⁰=1024. These values help in quick binary-decimal conversions and understanding bit positions.

2^n \text where n = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...

Applications in Computing

Binary arithmetic is fundamental to computer operations: CPUs perform all calculations in binary, memory addresses are binary numbers, and data storage uses binary encoding. Understanding binary helps in programming (bit manipulation, bitwise operations), computer architecture, and digital system design.

\textComputer Operations arrow \textBinary Arithmetic arrow \textDigital Logic

Worked Example

Inputs

• operation: add
• binary1: 1010
• binary2: 1101

Result: 10111

Explanation

1010 (10) + 1101 (13) = 10111 (23)

Second Scenario

Inputs

• operation: add
• binary1: 1212
• binary2: 1101

Result: 10111

Explanation

This scenario uses different inputs (operation = add, binary1 = 1212, binary2 = 1101) to show how changing one variable affects the binary result. Run the calculator above with these values to get the exact updated output with step-by-step work.