# Binary Number System (Definition and Examples)

Binary is a numerical system that employs two numerals, 0 and 1, to represent all real numbers. In contrast to the decimal system, which uses ten numerals, binary relies solely on these two digits.

## Representation and Powers of Two

Each digit in a binary number corresponds to a power of two. The rightmost digit represents the 0th power, the next represents the 1st power, and so on. For instance, the decimal number 1 is also represented as 1 in binary. On the other hand, the decimal number 23 translates to 10111 in binary (16+0+4+2+1).

### Applications of Binary Systems

In a broader context, binary systems encompass any setup offering only two options, not limited to numerical systems. For instance, electronic switches operate on a current-no current basis, while true-false exams and yes-no questions are other examples of binary systems.

### Mathematical Operations of Binary Systems

Mathematical methods exist for converting binary numbers into decimal numbers, and vice versa. Additionally, mathematical tools facilitate operations such as addition, subtraction, multiplication, and division in various base systems, including binary. While conversion to or from decimal may be cumbersome, transitioning between binary and octal or hexadecimal systems (base-eight and base-16 respectively) is simpler. This ease stems from the fact that both eight and 16 are powers of two, aligning well with binary systems. Consequently, octal and hexadecimal are widely used in computer applications.

### Historical of Binary Systems

In 1854, mathematician George Boole published a seminal paper on binary systems, laying the groundwork for what would become Boolean algebra. With the rise of electronics, these systems found practical application. In 1937, Claude Shannon formulated the principles for circuit design using binary arithmetic. Subsequently, in 1940, the era of binary computing commenced with the introduction of Bell Labs Complex Number Computer, capable of executing highly complex mathematical calculations using this system.

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