8.1. Hexadecimal Representation

So, the similar information I am representing in another number system which is your hexadecimal. Now, you just see what I am saying this is your 8 bit number all 0s to all 1s. So, all 1 in decimal it becomes 255 in hexadecimal it is your FF, FF means 15 F represents 15, 15 × 16^1 + 15 × 16^0. So, in that particular case we will get that this is nothing but 255 in decimal.

In the hexadecimal system, numbers are represented using base 16. This includes the digits from 0 to 9 and the letters A to F, where A represents 10, B represents 11, and so on up to F which represents 15. For an 8-bit binary number that has all bits set to 1 (11111111 in binary), the decimal equivalent is 255. In hexadecimal, this is represented as FF. This conversion is done because F stands for 15, and calculating its value in hexadecimal terms involves understanding its position related to powers of 16:

• F × 16^1 = 15 × 16 = 240
• F × 16^0 = 15 × 1 = 15
• Summing these gives us 240 + 15 = 255.

Think of hexadecimal as a color code in digital graphics. Each color can be represented with 2 hexadecimal digits (one for red, one for green, one for blue). In web design, the color white is represented as #FFFFFF, meaning full intensity of red, green, and blue. This is similar to how numbers are encoded, simplifying complex binary into a more manageable format.

Now, when we are going to discuss that representation of number in computer or in your binary system or binary digit most of the time we are going to take help of hexadecimal number system because it is having one advantage. Say, when I am going to work with 32 bit number I have to write 32 bits ok all 32 bits 0 or maybe combination of 0s and 1 which is slightly difficult. Similarly when we are going to work with 8 bits.

Hexadecimal is particularly beneficial in computing due to its compactness and readability compared to binary. For instance, instead of writing 32 binary bits like 110101101010..., one can write 8 hexadecimal digits which are much easier to read and write. Each hexadecimal digit represents four binary digits (bits), so it's much simpler for programmers to interpret and manage data. This ease of understanding is key when dealing with complex systems.

Consider writing your grocery list in an abbreviated form instead of writing out each item in full. For example, instead of writing '1 dozen eggs,' you could just write '12 eggs.' This saves time and space, making it easier to read.

So, in that particular case for hexadecimal representation this very simple, you just see if I am having a 4 bit number then all 0 if I am having then this is your 0 and if it is all 1 you can say that 8 + 4 = 12, 12 + 2 = 14, 14 + 1= 15.

To convert binary numbers into hexadecimal, you group the bits into sets of four (starting from the right). Each group can be easily translated into a single hexadecimal digit. For instance, the 4-bit binary number 1111 equals 15 in decimal, which is represented as F in hexadecimal. Grouping makes it simple: 0000 (0), 0001 (1), 0010 (2), up to 1111 (F).

Imagine you are counting in groups at a party. Instead of counting everyone one by one, you count groups of four or five. This method makes it quicker and less cumbersome to get an overall count.

So, the range is from 00 to FF, when we are having 12 bit numbers then it will go from 000 to FFF. So, number of bit divided by 4 is going to give me the number of symbols needed for hexadecimal representation.

In hexadecimal, the range of numbers is determined by the number of bits allotted for representation. An 8-bit binary number can represent values from 0 (00000000) to 255 (11111111) in decimal, which corresponds to 00 to FF in hexadecimal. Each additional 4 bits doubles the range of numbers you can represent. For example, a 12-bit hex number ranges from 000 to FFF, indicating it can represent from 0 to 4095 in decimal.

Think of it like a bookshelf. If each shelf can hold 16 books, and you have 8 shelves, you can hold a total of 128 books, but if you add 4 more shelves (making it 12), you can now hold even more books, roughly quadrupling your capacity depending upon the arrangement.

So, in most of the time we are going to represent our information in hexadecimal for understandability and for readability, but when you think about how it is working in the computer basically it working with this particular binary digit only in bit pattern 0s and 1s, but for readability we can write something in our hexadecimal notation.

Practically, hexadecimal is extensively used in programming and network design, especially in contexts like memory addresses and color codes. While computers operate at the binary level (only 0s and 1s), representing data in hexadecimal allows humans to comprehend and work with information more easily. For instance, instead of reading the binary address 1101 0110 1010 1111, one could refer to it simply as D6AF, which is much faster and less error-prone.

Think of a map showing cities with abbreviations instead of full names. Instead of reading 'New York City,' you see 'NYC,' which is much quicker to understand at a glance.

--