## 1. The Binary Representation of Positive and Negative Numbers

**1.1 The binary representation of the positive number 5 :**

Assuming we have an `int`

type number with a value of 5, we know that its representation in a computer would typically be a 32-bit binary value.

In this case, the binary representation of the number 5 would be:

`00000000 00000000 00000000 00000101`

Please note that this representation assumes a little-endian byte order, where the least significant bit is stored first.

**1.2 The binary representation of the negative number -5:**

The binary representation of the negative number -5, in a typical 32-bit signed integer representation, would be:

`11111111 11111111 11111111 11111011`

This representation follows the two’s complement method, where the most significant bit (leftmost bit) is set to 1 to indicate a negative value. The remaining bits represent the magnitude of the number in binary form. When converted to hexadecimal, it becomes “`0xFFFFFFFB`

“.

**Why ?**

Negative numbers in computers are typically represented using two’s complement notation.

In two’s complement representation, to express a negative number like -5, you follow these steps:

- Start with the binary representation of the corresponding positive number (5 in this case), which is “
`00000000 00000000 00000000 00000101`

“. - Invert all the bits (change 0s to 1s and 1s to 0s): “
`11111111 11111111 11111111 11111010`

“. - Add 1 to the result obtained in step 2: “
`11111111 11111111 11111111 11111011`

“.

So, the correct binary representation of -5 in a 32-bit signed integer format using two’s complement is “`11111111 11111111 11111111 11111011`

“.

## 2. Tips

- Positive numbers have the same representation in both the one’s complement and two’s complement systems.
- In the one’s complement representation, the negative number’s one’s complement is obtained by flipping all the bits (excluding the sign bit) of its corresponding positive number.
- In the two’s complement representation, the negative number’s two’s complement is obtained by taking the one’s complement of its corresponding positive number and then adding 1 to the least significant bit.