Understanding binary and hexadecimal systems is crucial for working with assembly language, as these number systems are fundamental to how computers operate and process data. This section will cover the basics of these systems, their importance, and how to convert between them.

The binary system is a base-2 numeral system that uses only two digits: 0 and 1. Each digit in a binary number is called a bit. Computers use the binary system because it aligns with their digital nature, where each bit can represent an on (1) or off (0) state.

• Bit: The smallest unit of data in a computer, representing a single binary digit (0 or 1).
• Byte: A group of 8 bits.
• Word: A fixed-sized group of bits handled as a unit by the CPU, typically 16, 32, or 64 bits.

The binary number 1011 can be broken down as follows:

• \(1 \times 2^3 = 8\)
• \(0 \times 2^2 = 0\)
• \(1 \times 2^1 = 2\)
• \(1 \times 2^0 = 1\)

So, 1011 in binary equals \(8 + 0 + 2 + 1 = 11\) in decimal.

The hexadecimal system is a base-16 numeral system that uses sixteen distinct symbols: 0-9 to represent values zero to nine, and A-F to represent values ten to fifteen. Hexadecimal is often used in computing because it can represent large binary numbers more compactly.

• Hex Digit: Each digit in a hexadecimal number, which can be 0-9 or A-F.
• Nibble: A group of 4 bits, which can be represented by a single hexadecimal digit.

The hexadecimal number 2F can be broken down as follows:

• \(2 \times 16^1 = 32\)
• \(F \times 16^0 = 15\) (since F represents 15)

So, 2F in hexadecimal equals \(32 + 15 = 47\) in decimal.

To convert a binary number to hexadecimal, group the binary digits into sets of four (starting from the right). Each group of four bits corresponds to a single hexadecimal digit.

Example:

Convert 11010110 to hexadecimal:

1. Group into sets of four: 1101 0110
2. Convert each group:
   • 1101 = D
   • 0110 = 6

So, 11010110 in binary is D6 in hexadecimal.

To convert a hexadecimal number to binary, replace each hexadecimal digit with its four-bit binary equivalent.

Example:

Convert 3A to binary:

1. Convert each digit:
   • 3 = 0011
   • A = 1010

So, 3A in hexadecimal is 00111010 in binary.

Convert the following binary numbers to decimal:

1. 1010
2. 1111
3. 10001

Solution:

1. 1010 = \(1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 8 + 0 + 2 + 0 = 10\)
2. 1111 = \(1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 8 + 4 + 2 + 1 = 15\)
3. 10001 = \(1 \times 2^4 + 0 \times 2^3 + 0 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 16 + 0 + 0 + 0 + 1 = 17\)

Convert the following hexadecimal numbers to decimal:

1. 1A
2. 7F
3. B4

Solution:

1. 1A = \(1 \times 16^1 + A \times 16^0 = 16 + 10 = 26\)
2. 7F = \(7 \times 16^1 + F \times 16^0 = 112 + 15 = 127\)
3. B4 = \(B \times 16^1 + 4 \times 16^0 = 176 + 4 = 180\) (since B represents 11)

Convert the following binary numbers to hexadecimal:

1. 101110
2. 11100011
3. 10011101

Solution:

1. 101110 = 0010 1110 = 2E
2. 11100011 = 1110 0011 = E3
3. 10011101 = 1001 1101 = 9D

Convert the following hexadecimal numbers to binary:

1. 4C
2. A7
3. 3B

Solution:

1. 4C = 0100 1100
2. A7 = 1010 0111
3. 3B = 0011 1011

In this section, we covered the basics of binary and hexadecimal systems, including their importance in computing, how to convert between them, and practical exercises to reinforce the concepts. Understanding these number systems is essential for working with assembly language and low-level programming. In the next section, we will delve into data types and sizes, building on the foundation we've established here.