Binary Calculator

Formula

Base-2 Arithmetic

Operations are performed on bits (0s and 1s) carrying values of powers of 2.

Worked Examples

Example 1: Addition

Problem:1010 + 0101

Solution:10 (decimal 10) + 5 (decimal 5) = 15. 15 in binary is 1111.

Result:1111

Example 2: Multiplication

Problem:11 × 10

Solution:3 × 2 = 6. 6 in binary is 110.

Result:110

Frequently Asked Questions

What is Binary?

Binary is a base-2 number system that uses only two digits: 0 and 1. It is the fundamental language of computers, representing on/off states of transistors.

How do you add binary numbers?

0+0=0, 0+1=1, 1+0=1, 1+1=10 (0, carry 1). It is similar to decimal addition but carrying occurs at 2 instead of 10.

How do you convert binary to decimal?

Multiply each bit by 2 raised to the power of its position (from right to left, starting at 0). Example: 101 = 1×2² + 0×2¹ + 1×2⁰ = 4 + 0 + 1 = 5.

How do you subtract binary numbers?

Use borrowing, similar to decimal. 1-0=1, 1-1=0, 0-0=0. For 0-1, borrow from the next column to make it 10-1=1.

Why do computers use binary?

It is efficient and reliable for hardware. Logic gates are easily built to detect two voltage levels (high/low), minimizing errors compared to detecting 10 distinct levels.

How to multiply binary numbers?

It works like decimal long multiplication, but simpler because you only multiply by 0 or 1. 1×1=1, everything else is 0. Then add the shifted rows.

Background & Theory

**Binary Arithmetic:** **Addition Rules:** * 0 + 0 = 0 * 0 + 1 = 1 * 1 + 0 = 1 * 1 + 1 = 10 (0, carry 1) * 1 + 1 + 1 = 11 (1, carry 1) **Subtraction Rules:** * 0 - 0 = 0 * 1 - 0 = 1 * 1 - 1 = 0 * 0 - 1 = 1 (borrow from next column) **Place Value:** Each position represents a power of 2. ... $2^3$ | $2^2$ | $2^1$ | $2^0$ ... 8 | 4 | 2 | 1 Example: 1011 = $1\\times8 + 0\\times4 + 1\\times2 + 1\\times1 = 11$

History

The binary number system was first fully described by Gottfried Wilhelm Leibniz in the early 18th century in his article "Explication de l'Arithmétique Binaire." He showed that all numbers could be represented by 0 and 1. In 1854, British mathematician George Boole published a landmark paper detailing a system of logic that would become known as Boolean algebra. His system used binary variables (true/false) and logic gates (AND, OR, NOT). In 1937, Claude Shannon produced his master's thesis at MIT, demonstrating that electronic relays and switches could implement Boolean algebra. This insight laid the theoretical foundation for digital circuit design and the modern computer age.