8.2.3. Two’s Complement

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.

The two's complement system is a way to represent both positive and negative integers in binary. In an 8-bit binary system, you can use a range from all 0s (00000000) to all 1s (11111111). The maximum positive number you can represent with 8 bits is 255, which, when converted into hexadecimal, is represented as FF (since F = 15, FF = 15 × 16^1 + 15 × 16^0). This highlights how hexadecimal can simplify binary representation.

Think of it like counting in a classroom. You can count students (binary) using hand signals (1s and 0s). When the number of students grows beyond a certain limit (like 255), instead of counting on fingers, you use tokens (hexadecimal) to quickly show how many students are there.

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 we are going to write 8 bits all 0s to all 1.

When working with binary representations, particularly with larger numbers like 32 bits, writing out every single bit can be cumbersome. Instead, hexadecimal provides a more compact form. Each hexadecimal digit corresponds to four binary bits, making it easier and faster to read and write. For instance, a full 32-bit number would require 32 bits written out, but can instead be represented with just 8 hexadecimal characters.

Imagine presenting a long report in a written format. Instead of writing each detail in full, you could summarize the report into a shorter version with bullet points (hexadecimal), making it easier for your audience to grasp the key ideas quickly.

So, in that 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. So, this is your 15 so we need total 16 different symbol 0 to 9 and A to F, F is 16 and A is 10.

In the two's complement format, numbers are represented with a fixed number of bits. In a system that uses 4 bits, the positive numbers (0-15) are represented directly as usual, while the negative numbers are represented by their two's complement forms. Essentially, this means you can go from 0 (0000) to 15 (1111), with A (10) to F (15) filling in the representation for numbers above 9.

Think of a game where scores are tracked from -15 to +15. You can have scores like 0 to 15, but when a player loses, you indicate scores in the negatives. The game has a special way to show negatives using a certain card (two's complement) that has a different color, making it clear when someone is below zero.

So basically what happens? It is basically the maximum number is a 255 or 8 bit representation that 255 is going to represent my -1 now you just count down and go downwards then 554 is going to give me -2, 253 will give me your -3.

In an 8-bit two's complement representation, the range is from -128 to +127. This means that while the maximum positive value is 127 (0111 1111), the minimum negative value is -128 (1000 0000). When counting down from 255, the -1 is represented as the pattern of all 1s (1111 1111) and decreases towards -128.

Imagine a temperature gauge where the midpoint (0 degrees) is balanced. As it extends into negatives (below 0), it counts down further, but once you reach the lowest point (-128), you can't go any lower, just like in the binary system.

So, in two’s complement just I am giving one example just only getting phone numbers now in that particular case what will happen this is the representation if I am having 0 then these are all 0s, + one is simple that magnitude 1, 10, 11 like that if I go for positive 4 then it will be 100 like that and negative number is coming as - 1 is representing as your all 1s then -2 is coming as your all 1 and 0.

To find the two's complement of a binary number, start with the binary representation of the number, invert all bits (change 0s to 1s and vice versa), and then add 1 to the least significant bit (LSB). This method effectively allows representation of negative numbers in binary format.

Consider a clock. When the minute hand points to zero, it means '0' hours. If you want to go back an hour (negative), you could either visually move backwards or calculate the time in reverse using your clock’s mechanism (just like the two's complement method does with binary).

So, just I am calculating it just show it now what will happens say when I use - 7 to - 1 should get - 6 and I am getting it. So, this is also correct result, but when I am using - 7 + - 7 what should be my result my result should be 14, but as a result I am getting 2 only.

In two's complement arithmetic, you may encounter an overflow situation when adding two numbers that produce a result outside of the range representable by the allocated bits. If the result of an addition causes the computation to exceed the maximum (e.g., adding two positive numbers gives a negative result) or the minimum, this indicates overflow.

Imagine trying to put more boxes into a truck than it can hold. If you stack boxes up to the point where they start falling off (like your result going beyond the limits), you've exceeded your truck's capacity (overflow) and can't accurately represent how many boxes there actually are anymore.

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