Number System

Introduction:

Number systems are the foundation of all mathematical operations and calculations, serving as the basis for how we represent, manipulate, and comprehend numerical information. In the realm of computer science and digital technology, a clear understanding of number systems is paramount. While humans commonly employ the decimal system in their daily lives, computers predominantly utilize the binary system due to their binary-based architecture. Beyond binary and decimal, various other number systems like hexadecimal and octal find application in specific computing contexts. This essay explores the fundamental concepts of number systems, their significance in the digital world, and their diverse applications.

What is Number System?

In computer fundamentals, a number system refers to a mathematical system used to represent and work with numbers in a digital computer. Computers primarily use two main number systems: the binary number system and the decimal number system. Here’s an overview of these number systems:

Binary Number System 

Binary is the fundamental number system used by computers because they work with electrical signals that can be in one of two states, often represented as 0 and 1.

Each digit in the binary system is called a “bit” (short for binary digit).

Binary numbers consist of only two symbols, 0 and 1.

Examples of binary numbers include 1101 (which is equivalent to decimal 13) and 1010 (which is equivalent to decimal 10).

Binary Number: 1101101

In this binary number, each digit represents a power of 2, starting from right to left:

1. The rightmost digit (1) represents 2^0, which equals 1.

2. Moving to the left, the next digit (0) represents 2^1, which equals 2.

3. The next digit (1) represents 2^2, which equals 4.

4. The next digit (1) represents 2^3, which equals 8.

5. The next digit (0) represents 2^4, which equals 16.

6. The next digit (1) represents 2^5, which equals 32.

7. The leftmost digit (1) represents 2^6, which equals 64.

To calculate the decimal equivalent of the binary number 1101101, you can add up these powers of 2:

1 + 0 + 4 + 8 + 0 + 32 + 64 = 109

Decimal Number System

The decimal system is the number system humans typically use in everyday life.

It is a base-10 system, meaning it has ten digits (0 through 9).

Each position in a decimal number represents a power of 10. For example, in the number 123, the “1” represents 1 hundred, the “2” represents 2 tens, and the “3” represents 3 ones.

Decimal numbers are used for most human calculations and expressions.

In addition to binary and decimal, there are other less commonly used number systems, such as hexadecimal (base-16) and octal (base-8), that are used in computer programming and digital systems for their convenient representations of binary data. For example, hexadecimal is often used to represent binary values more concisely because each hexadecimal digit can represent four binary digits (bits).

Understanding different number systems is crucial in computer science and digital electronics because it helps in data representation, conversion, and manipulation, especially when working with binary data and low-level computer operations.

The octal number system has certain characteristics;

 It uses eight digits; 0, 1, 2, 3, 4, 5, 6 and 7.

 It is also known as the base 8 number system.

Each position in an octal number represents a power of the base (8). For example, the highest position would represent an x power of the base (8).

Octal numbers are written using these digits. Here’s an example; 125708.

To calculate the decimal equivalent of an octal number;

Step 1; Take each digit of the octal number and multiply it by the corresponding power of eight. For instance;

The first digit multiplied by (8^4)

The second digit multiplied by (8^3)

The third digit multiplied by (8^2)

The fourth digit multiplied by (8^1)

The fifth digit multiplied by (8^0)

Step 2; Add up all these products to get the decimal equivalent.

For example;

 Octal Number; 125708

 Step by step calculation;

  ((1 x (8^4)) + (2 x (8^3)) + (5 x (8^2)) + (7 x(8^1)) + (0x(80))) = Decimal Number

Step by step calculation for this specific number results in;

((1 x 4096) + (2 x1024) + (5×256) + 

(7×64) + 

(0x1)) = Decimal Number

So, in this case,

Octal Number;125708

Decimal Equivalent;549610

Note that normally “125708” is written as “12570”.

hexadecimal numbers;

The hexadecimal number system has these characteristics;

 It uses ten digits (0 9) and six letters (A F).

 The letters represent numbers starting from 10, where A is equivalent to 10.

That covers the basics of the octal and hexadecimal number systems.In the hexadecimal number system, we use symbols such as B, C, D, E and F to represent the numbers 11, 12, 13, 14 and 15 respectively.

This number system is also referred to as base 16.

Each position in a hexadecimal number represents a power of 16. For example, in the number 160, the last position represents a power of x multiplied by the base (16).

Let’s take an example; the hexadecimal number is given as 19FDE16.

To calculate its decimal equivalent;

Step Binary Number Decimal Number

Step 1 19FDE16 ((1 x (16^4)) + (9 x (16^3)) + (F x (16^2)) + (D x (16^1)) + (E x (16^0)))10

Step 2 19FDE16 ((1 x (16^4)) + (9 x(16^3)) + (15x(162)) + (13x(161)) +  +(14x(160)))10

Step 3 19FDE6   ((65536) +(36864) +(3840) +(208) +(14))10

Step4     The decimal equivalent of this hexadecimal number is calculated to be;10646210.

Note; Normally we write this hexadecimal number as “19FDE”. 

Conclusion:

In conclusion, number systems are the underpinning of digital computation and information storage. While the decimal system is our everyday companion, the binary system reigns supreme in the realm of computers, owing to their binary nature. The versatility of various number systems, such as hexadecimal and octal, has made them indispensable tools in computer programming, networking, and data representation. Understanding these systems not only facilitates efficient data handling but also provides insights into the inner workings of computers. As technology continues to evolve, the importance of number systems in computer fundamentals remains unwavering, serving as a bridge between human thought and digital processing. Whether we are coding software, designing hardware, or troubleshooting network configurations, a solid grasp of number systems is indispensable in the ever-expanding world of computing.