## Industrial Computing Numbering and Encoding

**Industrial Computing Numbering and Encoding**

**I. Numbering Systems:**

Humans use ten symbols in their calculations: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Each symbol is called a “digit” in English, hence the use of the terms ‘digital’ and ‘decimal’. Example: N = 1985

1985 = 1 * 10^3 + 9 * 10^2 + 8 * 10^1 + 5 * 10^0 In general, a number N is written as:

N = a_n-1 … a_1 a_0

Where a_i represents the n digits, in a base B:

Notation: The base B is denoted as a subscript in the representation of a number

Note:

The representation of fractional numbers involves the appearance of negative exponents for the digits to the right of the decimal point.

- Octal System:

This system uses 8 symbols: 0, 1, 2, 3, 4, 5, 6, 7. Because certain physical problems can be represented by an octal base (B=8), as in the case of encoding and storing information in first-generation computers.

Remarks:

- To express the same quantity in base (10) in base (8), more places (more memory cells) are required: (1747)8 = (999)10.
- To represent fractional numbers, negative exponents need to be introduced.

Système binaire

Binary Numbering

The binary system only uses two digits (binary digit = bit): ‘0’ and ‘1’. These two symbols are also called ‘states’ and form a binary base (B=2). This base is very convenient for representing information in digital electronic systems (calculators, computers, etc.).

Notations:

N(10) = a_n-1 * 2^(n-1) + a_n-2 * 2^(n-2) + … + a_0 * 2^0

Système binaire

b- Binary System Properties

The binary system, also known as base-2, has the following key properties:

**Simplicity**: It uses only two symbols (0 and 1), making it straightforward for electronic devices to represent and process information.

**Digital Electronics**: Binary is the fundamental language of digital electronic systems like computers, calculators, and other devices. This is because electronic circuits can easily distinguish between two states (on and off).

**Base-2 Representation**: Numbers in the binary system are represented using powers of 2, where each position corresponds to a power of 2.

**Compact Representation**: Binary representation of numbers is very efficient for digital storage and processing, making it ideal for electronic systems.

**Boolean Algebra**: Binary forms the basis of Boolean algebra, which is fundamental in digital logic design and operations.

**Bitwise Operations**: Binary numbers enable bitwise operations like AND, OR, XOR, and NOT, which are fundamental in computer programming and digital signal processing.

**Binary Arithmetic**: Arithmetic operations (addition, subtraction, multiplication, and division) can be performed using binary numbers.

**Error Detection and Correction**: Binary is used in error detection and correction codes, crucial for reliable data transmission and storage.

**Binary Coded Decimal (BCD)**: BCD is a binary-encoded representation of decimal values, commonly used in electronic systems for handling decimal arithmetic.

**Binary Trees**: Binary trees are foundational data structures in computer science, used for organizing and searching data efficiently.

Understanding these properties is essential for working with binary numbers and designing digital systems.