## Number Systems — Decimal, Binary, Octal and Hexadecimal

A number system is characterized as a method of expressing numbers through writing. It is the mathematical notation for consistently representing the numbers in a given set using digits or other symbols. It uniquely reflects each number and the figures’ arithmetic and algebraic structure. Additionally, it enables us to perform basic arithmetic operations such as addition, subtraction, and division.

The value of any digit in a number can be calculated using the following formula:

- The numeric value.
- Its numerical position within the number.
- The fundamental unit of the number system.

**Definition of Number:**

A number is a numerical value that is used to count, measure, or mark objects. Arithmetic equations are carried out using numbers. Normal numbers, whole numbers, rational and irrational numbers are all examples of numbers. Additionally, 0 is a number that represents the null value. Numerous other variants of numbers exist, including even and odd numbers, prime and composite numbers. Even and odd terms indicate whether a number is divisible by two or not, while prime and composite terms indicate whether a number has two or more variables. These numbers are referred to as digits in a number system. The most frequently used digits in the number system are 0 and 1, which are used to represent binary numbers. On the other hand, other number systems also use 0 to 9 digits. Let us first become acquainted with the various forms of number systems.

Mathematics employs a variety of different number systems. The four most prevalent number system forms are as follows:

- The Decimal Number System (Base- 10)
- The Binary Number System (Base- 2)
- The Octal Number System (Base-8)
- The Hexadecimal Number System (Base- 16)

**Decimal Number System (Base 10):**

Since the decimal number system uses ten digits from 0 to 9, it has a base of ten. The locations to the left of the decimal point in the decimal number system denote units, tens, hundreds, and thousands, respectively. This is a decimal-based scheme.

Each location demonstrates the base’s unique strength (10). For instance, the decimal number 1457 is composed of the digits 7 in the units place, 5 in the tens place, 4 in the hundreds place, and 1 in the thousands place.

(1×103) + (4×102) + (5×101) + (7×100)

(1×1000) + (4×100) + (5×10) + (7×1)

1000 + 400 + 50 + 7

=1457

**The Binary Number System (Base 2):**

The base 2 number system is often referred to as the Binary number system, since it contains only two binary digits, namely 0 and 1. More precisely, the standard base-2 is a radix of two. The figures mentioned in this scheme are referred to as binary numbers, which are made up of the digits 0 and 1. 110101, for example, is a binary number.

Every device can be converted to binary and vice versa.

The Octal Number System (Base 8):

The octal number system has an 8-bit base and uses the numbers 0 to 7 to represent numbers. In computer applications, octal numbers are often used. Converting an octal number to decimal is the same as converting a decimal number to an octal number and is demonstrated below with an example.

For instance, convert 2158 to decimal.

**Solution:**

2158 = 2 × 82 + 1 × 81 + 5 × 80

= 2 × 64 + 1 × 8 + 5 × 1

= 128 + 8 + 5

= 14110

In California, the Yuki language uses an octal scheme because speakers count using the spaces between their fingers rather than their actual fingers. Pamean languages spoken in Mexico often use an octal scheme, as their speakers count using their closed fist’s knuckles.

**The Hexadecimal Number System (Base 16):**

The hexadecimal numeral system, sometimes abbreviated as “hex,” is a 16-character numeral system. The decimal (base 10) numeral system is the most widely used and consists of ten symbols: 0,1,2,3,4,5,6,7,8,9. Hexadecimal employs decimal numbers along with six additional symbols. Since there are no numerical symbols for values greater than nine, letters from the English alphabet, specifically A, B, C, D, E, and F, are used. Hexadecimal A is equivalent to decimal 10, and hexadecimal F is equivalent to decimal 15. Hexadecimal is similar to the octal numeral system (base 8) in that both systems can be quickly converted to binary.

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