Question

What bit patterns are represented by the following hexadecimal notations? a. \(0 \times 8 \mathrm{~A} 9\) b. \(0 \mathrm{xDCB}\) c. \(0 \mathrm{xEF} 3\) d. 0xA01 e. \(0 \mathrm{xC99}\)

Short answer

a. 100010101001, b. 110111001011, c. 111011110011, d. 101000000001, e. 110010011001

Step-by-step solution

Understand Hexadecimal Notation

A hexadecimal number system is base-16, using digits 0-9 and letters A-F. Each hex digit represents a 4-bit binary number.

Convert Each Hex Digit to Binary: 8A9

For the hexadecimal number \(0 \times 8A9\):- 8 is \(1000\)- A is \(1010\)- 9 is \(1001\)Combining these, the binary equivalent is \(100010101001\).

Convert Hex to Binary: DCB

For \(0 \mathrm{xDCB}\):- D is \(1101\)- C is \(1100\)- B is \(1011\)Combining, the binary equivalent is \(110111001011\).

Convert Hex to Binary: EF3

For \(0 \mathrm{xEF} 3\):- E is \(1110\)- F is \(1111\)- 3 is \(0011\)Combining, the binary equivalent is \(111011110011\).

Convert Hex to Binary: A01

For \(0xA01\):- A is \(1010\)- 0 is \(0000\)- 1 is \(0001\)Combining, the binary equivalent is \(101000000001\).

Convert Hex to Binary: C99

For \(0 \mathrm{xC99}\):- C is \(1100\)- 9 is \(1001\)- 9 is \(1001\)Combining, the binary equivalent is \(110010011001\).

Key concepts

Binary Numbers

Binary numbers form the fundamental building blocks of computing. At their core, computer systems process data in binary, which is a base-2 number system. This system uses only two digits: 0 and 1. Each binary digit represents a power of 2, just as each decimal digit represents a power of 10 in our everyday counting system. To understand binary better, we can break it down as follows:

• Each place in a binary number represents an increasing power of 2, starting from the right. For example, in the binary number 1011, the places represent powers of 2 starting with 1 (or 2⁰), 2 (2¹), 4 (2²), and 8 (2³).
• The value of a binary number is the sum of the values represented by places where there are 1s.
• For example, the binary number 1011 equals 1(2³) + 0(2²) + 1(2¹) + 1(2⁰) = 8 + 0 + 2 + 1 = 11 in decimal.

Binary numbers are crucial for digital computing as they align with electrical states, with 0 representing off and 1 representing on, enabling efficient data processing.

Hexadecimal Notation

Hexadecimal notation is a more compact way to represent binary numbers. It's a base-16 number system, using 16 unique symbols: 0-9 for values zero to nine, and A-F for values ten to fifteen. Each hex digit directly corresponds to a 4-bit binary sequence. This makes conversion between hex and binary seamless because each digit in hex represents exactly four bits in binary.

For instance, consider the hexadecimal digit A. In decimal, it represents the number 10, and in binary, it is represented as 1010. Similarly, the hex digit F represents 15 in decimal and 1111 in binary. By converting each hex digit into a 4-bit binary number, you can easily derive the equivalent binary sequence for any hex value."

Hexadecimal is often used in computing and digital electronics because it shortens long binary sequences, making them easier to read and understand. Instead of long strings of binary, concise hex notation is more readable and convenient, especially in programming and debugging.

Bit Patterns

Bit patterns describe the arrangement of binary values—0s and 1s—used to encode data in computing systems. Each pattern of bits (binary digits) holds a unique meaning or function in a computer's memory. These patterns are fundamental to representing various types of data, from numbers and characters to machine instructions.

Here are some key points about bit patterns:

• Each bit in a pattern has a positional value, where its position determines its power of 2.
• Bit patterns are extensively used to perform calculations, execute instructions, and store data efficiently.
• Understanding bit patterns allows better manipulation and interpretation of binary data, which is critical for tasks like error checking in data transmission and encryption.
• In representing numbers, bit patterns can vary in length depending on the required precision and size.

Different systems and languages may denote bit patterns in unique ways, but the essential concept of binary representation remains essential to their functionality.

Number Systems

Number systems are frameworks we use to represent numbers. While humans most commonly use the decimal system (base-10), computers operate more efficiently with binary (base-2), where each digit is a power of 2. Alongside these, hexadecimal is another number system (base-16), widely employed in computing due to its efficiency in representing binary data.

Different number systems have the following characteristics:

• Decimal (Base-10): Uses digits 0-9, aligning with the way we naturally count.
• Binary (Base-2): Utilizes digits 0 and 1, aligning with on/off states in digital circuits.
• Hexadecimal (Base-16): Incorporates sixteen symbols (0-9 and A-F), providing a succinct expression for binary numbers.

The use of hexadecimal simplifies binary string interpretations, especially with longer sequences. Understanding these number systems and how they interconvert is crucial in technology, ranging from programming and database management to network protocols and digital communication. The ease of converting between base-16 and base-2 allows for efficient debugging and system diagnostics, making hexadecimal an essential tool for developers and engineers.