The number system in computers relates to how numbers are represented and used. The binary number system, which is a base-2 system with just the digits 0 and 1, is the most common one used by computers. Computers' representation and processing of digital data are built on binary digits, commonly referred to as bits.

Different sorts of number systems are frequently employed in computer technology and mathematics.

In computer science, the following number systems are mainly used:

Binary Number System

An essential numbering scheme used in computer systems is the binary number system. Because it employs a base-2 system, only the digits 0 and 1 are used. Each digit in binary is referred to as a bit, which is short for binary digit.

The binary system is based on powers of 2 instead of the base-10 decimal system that humans typically employ (using ten digits: 0 to 9). Beginning with the rightmost digit, which represents 2^0 (equal to 1), each digit in a binary number represents a power of two. Leftmost digit stands for 2^1 (equivalent to 2), followed by 2^2 (equal to 4), and so on.

To understand how binary numbers work, let's take an example.

The binary number 1011 can be expanded as follows:

(1 * 2^3) + (0 * 2^2) + (1 * 2^1) + (1 * 2^0)

= 8 + 0 + 2 + 1

= 11 (in decimal)

The widespread usage of binary numbers in computer systems is due to the ease with which digital circuits can encode and manipulate the two states, commonly represented by 0 and 1.

The binary system is the basis for data storage, processing, and communication in computers and other digital devices.

The binary number system is crucial to computer systems and has many uses. The binary number system in computers is used in the following important applications:

Data Representation
Arithmetic Operations
Digital Logic
Memory Addressing
Digital Communication
Machine Language and Programming
Boolean Algebra

1. Decimal Number System

Humans most frequently utilise the decimal number system, usually referred to as the base-10 system, in daily life. Ten numbers are used: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. A fact that these digits are arranged in groups of 10 is what is meant by the name "decimal" means.

Each digit's place value or location in the decimal system establishes its value. The digit on the far right stands for the units place, the digit on the far left for the tens place, the digit after that for the hundreds place, and so on. From 100 (equal to 1) for the rightmost digit to 101 (equivalent to 10) for the following digit, and rising by 10 for each succeeding digit, each place value indicates a power of 10.

Take the following example to see how the decimal system functions:

The decimal number 325 can be expanded as follows:

(3 * 10^2) + (2 * 10^1) + (5 * 10^0)

= 300 + 20 + 5

= 325

The digit 3 in this example stands for 3 multiplied by 10 raised to the power of 2 (100), the digit 2 in the tens place for 2 multiplied by 10 raised to the power of 1 (10), and the digit 5 in the units place for 5 multiplied by 10 raised to the power of 0 (1).

The decimal system is frequently employed because it corresponds to our natural method of counting and numbering. It is utilized in many facets of daily life, including expressing amounts, measuring units, money, and carrying out arithmetic operations.

Even though computers typically use binary internally, the decimal number system has some uses in computer systems.

The decimal number system in computers is used primarily for the following purposes:

Human-Readable Representation
Programming Languages
Financial and Monetary Calculations
Database Systems
Statistical Analysis and Data Processing
Scientific Computing

2. Octal Number System

The base-8 octal number system employs the eight digits 0, 1, 2, 3, 4, 5, and 6 to represent numbers. In computer systems, it is often used, especially in digital and computer programming.

Each digit's value in the octal system is based on its place value inside the number. The numeral at the top right corner stands for the units place, followed by the digit to the left for the eights place, the next for the sixty-fours place, and so on. Every place value corresponds to an 8th power, starting with 8^0 (equal to 1) for the rightmost digit, 8^1 (equivalent to 8) for the following digit, and increasing by an 8th power for each digit after that.

To understand how the octal system works, let's take an example:

The octal number 35 can be expanded as follows:

(3 * 8^1) + (5 * 8^0)

= 24 + 5

= 29 (in decimal)

For a number of reasons, the octal system is frequently employed in digital systems and computer programming.

Compact Representation: Binary numbers are represented succinctly by octal numbers. When working with binary patterns or describing groups of bits in a more condensed manner, the fact that each octal digit represents three binary digits (bits) makes it handy.
Memory Addressing: Computer systems have traditionally employed octal numbers to indicate memory addresses. The efficient encoding and manipulation of memory addresses within a constrained number of digits is made possible by octal representation.
Unix File Permissions: In Unix-like operating systems, file permissions are represented using the octal system. A clear and succinct representation of file permissions is made possible by the fact that each digit in an octal number corresponds to a permission setting for a separate user group (owner, group, and others).

There are several uses for the base-8 octal number system in computer systems. Its use has decreased over time, although it still has certain useful uses in particular niches. The octal number system in computers is used in the following applications:

Representation of binary data
File permissions
Device identification
Digital systems and electronics
Legacy systems and programming

3. Hexadecimal Number System

The base-16 hexadecimal number system, sometimes known as hex, uses numbers to represent data. It represents numbers with 16 digits: 0 to 9 for digits 0 to 9, and A to F for digits 10 to 15. In computer systems, hexadecimal numerals are often used, especially in programming, digital systems, and low-level processes.

Each number in the hexadecimal system corresponds to a power of 16. The digit on the right represents 16^0, which is equal to 1, the digit on the left represents 16^1, which is equal to 16, the digit after that represents 16^2, which is equivalent to 256, and so on. The digits "A" and "B" stand for the values 10, "11," and so on up to "F," which stands for 15.

Computer systems frequently employ hexadecimal integers for the following reasons:

Large binary numbers can be more efficiently represented using hexadecimal numbers, which are more compact. A half-byte, usually referred to as a nibble, or four bits, are represented by each hexadecimal numeral. A byte may therefore be represented by two hexadecimal digits.
Memory addressing: Hexadecimal representations of memory addresses are often used in computer systems. For instance, in programming, memory locations are frequently represented succinctly by hex values.
Hexadecimal numbers are often employed in low-level activities, including bitwise operations, binary file formats, and binary data processing.

To understand hexadecimal numbers better, let's consider an example:

The hex number 3F2 can be expanded as follows:

(3 * 16^2) + (F * 16^1) + (2 * 16^0)

= (3 * 256) + (15 * 16) + (2 * 1)

= 768 + 240 + 2

= 1010 (in decimal)

Numerous areas, notably computer science and digital technology, use the hexadecimal number system, or hex. Following are a few typical uses for the hexadecimal number system:

Memory Addressing
Colour Representation
Binary and Bit Manipulation
Low-Level Programming
Network Addressing

4. Floating-Point Number System

A technique for representing and computing with real numbers that have a fractional component is the floating-point number system. Wherever great accuracy and a large range of integers are needed, it is extensively utilised in computer science, scientific computations, and computer graphics.

A number is stated using a combination of the sign, significand (also known as the mantissa), and exponent in the floating-point system. The sign denotes the number's positive or negative value, the significand stands for the number's significant digits, and the exponent establishes the size of the number.

By moving the decimal point, the floating-point format enables a dynamic range of values. This is comparable to the mathematical scientific notation where a number is shown as a decimal fraction times a power of ten.

The most popular format for displaying floating-point integers in contemporary computer systems is IEEE 754. Single-precision and double-precision formats are defined.

Single-precision floating-point format:

32 bits in length

1 bit for the sign (positive or negative)

8 bits for the exponent

23 bits for the significand

Double-precision floating-point format:

64 bits in length

1 bit for the sign

11 bits for the exponent

52 bits for the significand

Typically, the significand is given in normalised form, with the most important bit considered to be 1. This enables a wider range of values and greater accuracy.

With varying degrees of accuracy depending on the number of bits assigned to the exponent and significand, floating-point numbers may represent a broad range of values, from very tiny to extremely big. Because there are only a finite number of bits available, floating-point values can occasionally be off and have a limited degree of precision.

In computer systems, real numbers are represented using the floating-point number system. It can handle numbers with a broad range of magnitudes and fractional parts. The floating-point format is often used in many different applications, particularly those that call for a high degree of precision and accuracy.

The floating-point number system is used in the following major applications:

Scientific Computing
Engineering and Computer Graphics
Financial and Economic Modelling
Physical Simulations
Machine Learning and Data Science

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