In the realm of computers and digital systems, binary numbers serve as the fundamental language of communication. Although they may appear mysterious and unfamiliar to the uninitiated, binary numbers are actually quite straightforward and can be easily grasped with a bit of explanation. In this article, we will delve into the world of binary numbers, exploring their origins, significance, and how they are used in modern computing.
At its core, binary is a numeral system that employs only two symbols: 0 and 1. Unlike our familiar decimal system, which uses ten symbols (0–9), binary simplifies things by relying solely on two digits. This seemingly limited range of options forms the foundation of all digital information, as computers represent data using binary digits, commonly known as bits.
The concept of binary numbers can be traced back to ancient civilizations, including the Indian, Chinese, and Mayan cultures. However, it was the German mathematician and philosopher Gottfried Wilhelm Leibniz who formalized the modern binary system in the late 17th century. Leibniz recognized the potential of using binary numbers as the basis for performing calculations and logical operations, leading to the development of binary arithmetic.

Binary Representation

In decimal representation, each digit’s value is determined by its position relative to the rightmost digit, which is multiplied by a power of 10. For example, the number 753 in decimal notation represents (7 * 10²) + (5 * 10¹) + (3 * 10⁰).
Binary representation follows a similar principle, but instead of powers of 10, powers of 2 are used. Each digit in a binary number is called a bit, which can hold one of two values: 0 or 1. The rightmost bit holds a value of 2⁰ (which is 1), the next bit to the left represents 2¹ (which is 2), the next one represents 2² (which is 4), and so on.
To illustrate this, let’s examine the binary number 10110. Breaking it down, we can calculate its decimal equivalent by adding up the values of the bits that are set to 1: (1 * 2⁴) + (0 * 2³) + (1 * 2²) + (1 * 2¹) + (0 * 2⁰) = 16 + 0 + 4 + 2 + 0 = 22. Therefore, in decimal notation, 10110 is equal to 22.

Binary Arithmetic

Let’s delve into some common binary operations that are performed using binary numbers:

1. Binary Addition: Binary addition is performed by adding each pair of corresponding bits, similar to decimal addition. The rules are:

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 0 = 1
• 1 + 1 = 0 (with a carry of 1 to the next column)

Example: Let’s add the binary numbers 101 and 110:

   101
   110
------ +
  1011

2. Binary Subtraction: Binary subtraction follows similar rules to decimal subtraction, including borrowing from the next column if necessary. The rules are:

• 0–0 = 0
• 1–0 = 1
• 1–1 = 0
• Borrowing: When subtracting 0 from 1, borrow 1 from the next column, changing the 1 to 0 and subtracting 1 from the next column.

Example: Let’s subtract the binary number 110 from 1011:

   1011
    110
------- -
    101

3. Binary Multiplication: Binary multiplication is performed using a method similar to decimal multiplication, employing the multiplication table for 0 and 1. The rules are:

• 0 * 0 = 0
• 0 * 1 = 0
• 1 * 0 = 0
• 1 * 1 = 1

Example: Let’s multiply the binary numbers 101 and 110:

    101
    110
------- x
   1010
  10100
------- +
  11110

4. Binary Division: Binary division is akin to decimal division, utilizing long division to find the quotient and remainder. The rules are:

• 0 / 1 = 0 (quotient)
• 0 % 1 = 0 (remainder)
• 1 / 1 = 1 (quotient)
• 1 % 1 = 0 (remainder)

Example: Let’s divide the binary number 1011 by 110:
To divide 1011 by 110 using long division, let’s go through the steps:
Step 1

Step 1

     _________
110 | 1 0 1 1


Step 2

      0
     _________
110 | 1 0 1 1
      0
      -------- -

Step 3

      0 0
     _________
110 | 1 0 1 1
      0
      -------- -
      1 0
      0 0
      -------- -

Step 4

      0 0 0
     _________
110 | 1 0 1 1
      0
      -------- -
      1 0
      0 0
      -------- -
      1 0 1
      0 0 0
      -------- -

Step 5

      0 0 0 1
     _________
110 | 1 0 1 1
      0
      -------- -
      1 0
      0 0
      -------- -
      1 0 1
      0 0 0
      -------- -
      1 0 1 1
        1 1 0  
      -------- -

Step 6

      0 0 0 1 . 0
     ___________
110 | 1 0 1 1
      0
      -------- -
      1 0
      0 0
      -------- -
      1 0 1
      0 0 0
      -------- -
      1 0 1 1
        1 1 0  
      -------- -
        1 0 1
        0 0 0
      -------- -
        1 0 1

If we discard the trailing zeros, we got the quotient as 1 and the remainder as 101, which if we translate back to decimal number system is

the result we got is 1 with remainder 101 in binary system

1011 / 110 <=> (8+0+2+1) / (4+2+0) = 11 / 6 = 1 remainder 5

And turns out, if we check it matches between the decimal
calculation result and the binary calculation results 

Practical Applications

Binary numbers have numerous practical applications across various fields. Here are a few notable examples:
1. Computing and Digital Systems: Binary numbers are the foundation of computing and digital systems. They are used to represent and manipulate data in computers and other digital devices. All digital information, such as text, images, videos, and instructions, is encoded and processed using binary numbers.
2. Data Storage and Transmission: Binary is used in data storage and transmission systems. Digital data is stored in binary format in hard drives, solid-state drives (SSDs), and other storage media. Similarly, binary encoding is used in communication protocols and networks to transmit data reliably and efficiently.
3. Logic and Boolean Algebra: Binary numbers play a crucial role in logic circuits and Boolean algebra. Binary values represent logical states, such as true/false or on/off, in digital logic systems. Logic gates perform operations on binary inputs to produce binary outputs, enabling the design of complex circuits and systems.
4. Cryptography: Binary numbers are used extensively in cryptography, which involves secure communication and data encryption. Encryption algorithms operate on binary data, converting plaintext into ciphertext. Binary operations, such as XOR, are fundamental in cryptographic algorithms for ensuring data confidentiality and integrity.
5. Mathematics and Number Systems: Binary numbers are a fundamental part of the mathematical field of number systems. They provide a basis for understanding other numeral systems, such as octal and hexadecimal. Binary arithmetic, including addition, subtraction, multiplication, and division, is used in various mathematical and computational applications.
6. Electrical Engineering: Binary numbers are widely used in electrical engineering, especially in digital circuits and microprocessor systems. They represent voltage levels, electronic signals, and the states of electronic components. Binary-based control systems are employed in various applications, including robotics, automation, and embedded systems.
7. DNA Sequencing and Genetics: In bioinformatics and genetics, binary numbers are used to represent DNA and genetic information. Genetic sequences are encoded using binary digits (0s and 1s) to store and analyze genetic data, facilitating advancements in genomics and DNA sequencing technologies.
8. Image and Audio Processing: Binary numbers are essential in digital image and audio processing. Images and sound are represented as binary data, allowing for encoding, compression, manipulation, and transmission of multimedia content. Binary representation enables the storage and reproduction of high-quality visual and auditory information.
These are just a few examples of the practical applications of binary numbers. The inherent simplicity and reliability of binary make it a foundational concept in many technological and scientific fields.

Conclusion

Binary numbers form the backbone of modern computing, serving as the basis for representing and manipulating information in digital systems. By delving into binary, one gains insight into the underlying principles that drive the functionality of computers, from the storage and transmission of data to the execution of complex algorithms. This understanding unveils the remarkable efficiency of binary arithmetic, where intricate calculations are executed with simplicity using only two digits. Embracing binary allows us to demystify the inner workings of technology, enabling us to make informed decisions, develop software and hardware solutions, and contribute to the continuous advancement of the digital world.