In computers, data is stored in bits, the smallest unit of information. A bit can represent two states: 0 or 1. When we group bits, we form bytes, and the sizes of C++ data types are often expressed in bytes.

Bits and Binary Representation:

In the binary system, each bit represents a binary digit, either 1 or 0 The number of bits in a data type directly influences its capacity to represent different values. The formula for calculating the capacity is 2 ^ n, where n is the number of bits. This stems from the face that each bit has two possible values (0 or 1).

Unsigned Representation:

Unsigned integers represent only non-negative values. The range for an n-bit unsigned integer is 0 to 2^(n-1), where n is the number of bits. For example, an 8-bit unsigned integer can represent values from 0 to 255.

• If the value exceeds the maximum limit, it wraps around to 0.

Below is the 16 possible values of a four-bit unsigned int:

bits  value
0000    0
0001    1
0010    2
0011    3
0100    4
0101    5
0110    6
0111    7
1000    8
1001    9
1010   10
1011   11
1100   12
1101   13
1110   14
1111   15

and here are the 16 possible values of a four-bit signed int
bits  value
0000    0
0001    1
0010    2
0011    3
0100    4
0101    5
0110    6
0111    7
1000   -8
1001   -7
1010   -6
1011   -5
1100   -4
1101   -3
1110   -2
1111   -1

1. Char (Character) Data Type:

• Size: 1 byte
• Capacity: 256 possible values (2^8)
• Range: 0 to 255 (unsigned) or -128 to 127 (signed)

2. Integer Data Types:

Short Integer (short):

• Size: Usually 2 bytes.
• Capacity: 65536 possible values (2^16)
• Range: -32,768 to 32,767 (signed) or 0 to 65,535 (unsigned)

Integer (int):

• Size: Typically 4 bytes
• Capacity: 4,294,967,296 possible values (2^32)
• Range: -2,147,483,648 to 2,147,483,647 (signed) or 0 to 4,294,967,295 (unsigned)

Long Integer (long):

• Size: Typically 4 bytes
• Capacity: 

Long Long Integer (long long):

• Size: Usually 8 bytes.
• Capacity: 18,446,744,073,709,551,616 possible values (2^64)
• Range: -9,223,372,036,854,775,808 to 9,223,372,036,854,775,807 (signed) or 0 to 18,446,744,073,709,551,615 (unsigned).

3. Floating-Point Data Types:

Float:

• Size: Typically 4 bytes.

Double:

• Size: Usually 8 bytes.

Long Double:

• Size: The size can vary.

4. Boolean Data Type:

• Size: 1 byte
• Capacity: 2 possible value (2)
• Range: 0 (false) or 1 (true)