INTRODUCTION
Computers store and process data using only binary digits—0s and 1s—because their hardware is built on electrical circuits that have two states: on and off. While representing positive whole numbers in binary is straightforward, handling negative numbers is more complex since binary doesn’t include a minus sign.

To solve this, computer systems use a method called two’s complement. This system allows both positive and negative numbers to be stored and used efficiently in binary form. It simplifies the design of arithmetic circuits, making it possible to perform addition, subtraction, and other operations using the same logic for both positive and negative values.

Understanding how negative numbers are represented in computers is essential in areas like programming, hardware design, and digital electronics, because it affects how data is stored, manipulated, and interpreted by the machine.

Let's quickly look at some of the methods asocciated with storing data in Binary.

SIGN AND MAGNITUDE

Is one of the earliest and simplest ways to represent negative numbers in binary. It separates a number into two parts:

How It Works
The most significant bit (MSB) (the leftmost bit) is the sign bit:

Where;
0 = Positive number

1 = Negative number

The remaining bits represent the magnitude (absolute value) of the number in binary.

Example (8-bit representation)
Let’s represent +11 and -11:

+11:
where;
Sign bit = 0 (for positive numbers),
Magnitude = 0001011 (representation of 11 in Binary)

The 8-bit representation for Sign bit (0) and Magnitude (0001011) will give us the result → 00001011

-11:
where;
Sign bit = 1 (for negative numbers),
Magnitude = 0001011 (representation of 11 in Binary)

The 8-bit representation for Sign bit (1) and Magnitude (0001011) will give us the result → 10001011

ONE'S COMPLEMENT (1's COMPLEMENT)

In simple words, one's complement is toggling or exchanging all the 0's into 1 and all the 1's into 0 of any number.

For example, there is a binary number 11001001, then exchanging all the 0's into 1 and all the 1's into 0, the result for one's complement will be 00110110.

Let's get the one's complement for 11
First, we convert 11 (Base 10) to Binary (Base 2)

Where;
11 (Base 10) = 00001011 (Base 2)
exchanging all the 0's into 1 and all the 1's into 0 for 0001011 (base 2) will give = One's complement 1110100

TWO'S COMPLEMENT (2's COMPLEMENT)

Two’s complement is a mathematical operation on binary numbers that is used to signify positive or negative integers in binary. One method is to complement each bit and then add one, i.e. 6 in binary (0110) becomes 1010. Another method is to go right to left complementing each bit after the first 1, i.e. 10 in binary (01010) becomes 10110.

Computers store negative numbers using a method called two's complement. Let's break it down using -11 as an example.

Steps to Represent -11 in Two's Complement:

Convert to Binary: First, convert the positive version of the number (11) to binary.

This is achieved using the below method;

Where Conversion to Binary will be;

11 Devided by 2 = 5 R 1
5 Devided by 2 = 2 R 1
2 Devided by 2 = 1 R 0
1 Devided by 2 = 0 R 1

We read the remainders 'R' as the result starting from the bottom.

Therefore, 11 in binary is 1011.

Determine Bit Length: Decide how many bits you want to use. For this example, let's use 8 bits.

So, 11 in 8 bits is 00001011. This is simply achieved by adding 0's to the corresponding binary number for 1011.

Invert the Bits: Change all 0s to 1s and all 1s to 0s.

Inverting 00001011 gives you 11110100.
Add 1: Finally, add 1 to the inverted binary number.

11110100 + 00000001 = 11110101.
Result:
The two's complement representation of -11 in 8 bits is 11110101.

A STEP-BY-STEP CONVERSION OF -11 INTO CORRESPONDING HEXADECIMAL VALUE

First, convert positive 11 to binary:

Where Conversion to Binary will be;

11 Devided by 2 = 5 R 1
5 Devided by 2 = 2 R 1
2 Devided by 2 = 1 R 0
1 Devided by 2 = 0 R 1

We read the remainders 'R' as the result starting from the bottom. Therefore, 11 in binary is 1011.

Using 8 bits length +11 = 00001011

Invert the Bits (One’s Complement)
Flip every bit:
00001011 → 11110100

Add 1 (to get Two’s Complement)
Add 1 to the inverted result:
11110100 + 1 = 11110101

So, −11 in 8-bit two’s complement binary is:
11110101

Convert Binary to Hexadecimal
Now, split the binary into 4-bit (Nibble) groups:
1111 0101

Convert each group to hex:

1111 = F

0101 = 5

Final Hexadecimal Value: 0xF5