8.3.1. Addition and Overflow

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 × 161 + 15 × 160. So, in that particular case we will get that this is nothing but 255 in decimal.

In this chunk, we are exploring how numbers are represented in different numeral systems, specifically binary and hexadecimal. The binary system uses base 2 (0s and 1s), while the hexadecimal system uses base 16 (0-9 and A-F). An 8-bit binary number can represent decimal values from 0 to 255. When all bits are '1', it corresponds to 255 in decimal; in hexadecimal, it is represented as 'FF'. This means that 'F' in hexadecimal is equal to 15 in decimal, and 'FF' is calculated as (15 × 16^1 + 15 × 16^0), leading to 255 in decimal.

Think of hexadecimal as a way of compressing information. Just like how a street address can simplify navigation by using numbers and abbreviations instead of long descriptions, hexadecimal allows computers to process binary numbers more efficiently. For instance, instead of writing eight '0's and '1's for a value like 255, you can just write 'FF', making it quicker and easier for programmers.

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.

The hexadecimal system simplifies the representation of binary numbers, which can be cumbersome when they grow long, such as in 32-bit numbers consisting of 32 individual bits (0s and 1s). Instead of trying to manage long binary strings, programmers can represent these numbers in a more concise form, making code easier to read and errors easier to spot. For example, a typical 32-bit binary number can be condensed into 8 hexadecimal digits.

Consider how it's easier to read a long number like '1,000,000' instead of '1000000'. Similarly, hexadecimal condenses long binary representations into a more compact form, allowing programmers to quickly interpret and manage data without getting lost in long strings of bits.

So, if we are going to represent only positive numbers then what will happen I can use all those 256 characters to represent positive numbers from 0 to 255, but if you are going to consider negative number also this is the number line this is 0 and this is your positive side and this is your negative side.

In an 8-bit binary system, you can represent 256 combinations, ranging from 0 to 255 for positive numbers. When negative numbers are included, a portion of these combinations must be reserved for them. Thus, if you reserve some patterns for negative numbers, the effective range of positive numbers narrows down. For instance, with a signed representation, you can depict numbers from -128 to +127 instead, splitting the available combinations evenly around zero.

Imagine you have a scale that weighs from -128 to +127. If you allow weights only on one side of the scale (just positive), you could use all the available space for positive weights up to 255. However, introducing weights on both sides means you have to give up some of the space on the positive side to accommodate negative weights, balancing your scale at zero.