Question

Convert each of the following hexadecimal representations to binary representation and then to its equivalent base 10 representation: a. OxA b. \(0 \times 14\) c. \(0 \times 1 E\) d. \(0 \times 28\) e. \(0 \times 32\) f. \(0 \times 3 \mathrm{C}\) g. \(0 \times 46\) h. \(0 \times 65\) k. \(0 \times 194\) 1\. \(0 \times \mathrm{CA}\)

Short answer

OxA is 10, Ox14 is 20, Ox1E is 30, Ox28 is 40, Ox32 is 50, Ox3C is 60, Ox46 is 70, Ox65 is 101, and OxCA is 202 in decimal.

Step-by-step solution

Understanding Hexadecimal

Hexadecimal is a base-16 number system using digits 0-9 and letters A-F. Each hexadecimal digit corresponds to a 4-bit binary sequence.

Converting Hexadecimal to Binary: OxA

Convert each hex digit to its binary equivalent. The hex 'A' converts to binary '1010'.

Converting Binary to Decimal: OxA

Convert the binary '1010' to decimal by calculating \(1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 10\). Thus, OxA is 10 in decimal.

Convert Hex Ox14 to Binary

Convert hex '14'. '1' is binary '0001', '4' is '0100'. Combined, Ox14 is '0001 0100'.

Convert Binary to Decimal: Ox14

Convert '0001 0100' to decimal, which is \(1 \times 2^4 + 4 \times 2^0 = 20\). Hence, Ox14 is 20 in decimal.

Convert Hex Ox1E to Binary

For hex '1E', '1' is '0001' and 'E' is '1110'. Combined, it becomes '0001 1110'.

Convert Binary to Decimal: Ox1E

Convert '0001 1110' to decimal, which is \(1 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 = 30\). Thus, Ox1E is 30 in decimal.

Convert Hex Ox28 to Binary

Convert '28'. '2' is '0010' and '8' is '1000', which together form '0010 1000'.

Convert Binary to Decimal: Ox28

Convert '0010 1000' to decimal: \(1 \times 2^5 + 1 \times 2^3 = 40\). Hence, Ox28 is 40 in decimal.

Convert Hex Ox32 to Binary

For '32', '3' is '0011' and '2' is '0010', resulting in '0011 0010'.

Convert Binary to Decimal: Ox32

Convert '0011 0010' to decimal: \(3 \times 2^4 + 2 \times 2^0 = 50\). Thus, Ox32 is 50 in decimal.

Convert Hex Ox3C to Binary

The hex '3C': '3' is '0011' and 'C' is '1100', resulting in '0011 1100'.

Convert Binary to Decimal: Ox3C

Convert '0011 1100' to decimal: \(3 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 = 60\). So, Ox3C is 60 in decimal.

Convert Hex Ox46 to Binary

For hex '46': '4' is '0100' and '6' is '0110', making it '0100 0110'.

Convert Binary to Decimal: Ox46

Convert '0100 0110' to decimal: \(4 \times 2^4 + 6 \times 2^0 = 70\). Thus, Ox46 is 70 in decimal.

Convert Hex Ox65 to Binary

Convert '65': '6' is '0110' and '5' is '0101', resulting in '0110 0101'.

Convert Binary to Decimal: Ox65

Convert '0110 0101' to decimal: \(6 \times 2^4 + 5 \times 2^0 = 101\). Hence, Ox65 is 101 in decimal.

Convert Hex OxCA to Binary

For hex 'CA': 'C' is '1100' and 'A' is '1010', resulting in '1100 1010'.

Convert Binary to Decimal: OxCA

Convert '1100 1010' to decimal: \(12 \times 2^4 + 10 \times 2^0 = 202\). Thus, OxCA is 202 in decimal.

Key concepts

Hexadecimal to Binary Conversion

Hexadecimal, or simply hex, is a base 16 number system that includes digits 0 through 9 and letters A through F. This system is a concise way to represent binary numbers, which are in base 2, using fewer digits. Each digit in hexadecimal corresponds exactly to a 4-bit binary sequence. For example:

• 0 in hex is 0000 in binary
• 1 in hex is 0001 in binary
• A in hex is 1010 in binary
• B is 1011, and so on up to F, which is 1111 in binary

To convert a hexadecimal number to binary, you replace each hex digit with its 4-bit binary equivalent. Let's take the hex number '1E' as an example: - '1' in hex converts to '0001' in binary - 'E' in hex converts to '1110' in binary Combine these to get the binary sequence: '0001 1110'. This method also helps in simplifying complex binary calculations as a sequence of lengthy binary digits can be easily presented in a much shorter hex form.

Binary to Decimal Conversion

Binary is a base 2 number system that uses only digits 0 and 1, with each digit representing an increasing power of 2. To convert binary to decimal, we multiply each bit by 2 raised to the power of its position (starting from 0 on the right) and then sum them up. For instance, let's convert the binary number '1010' to decimal:- The leftmost '1' (in the positions 2^3) is worth 8- The '0' in the positions of 2^2 is worth 0- The '1' in the positions of 2^1 is worth 2- And the rightmost '0' in 2^0 is worth 0So, the binary number '1010' equals: \(1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 10\).
This shows the step-by-step breakdown which results in 10 as its base 10 respective decimal value. By following the same method, you can convert any binary sequence to a decimal number.

Base 16 Number System

The Base 16 number system, also known as the hexadecimal system, simplifies the representation of large binary numbers and is widely used in programming and computer science. Unlike the decimal system (Base 10) which has ten digits (0 through 9), Base 16 includes numbers 0 through 9 and extends this by adding the letters A through F, where A stands for 10, B for 11, and so on up to F, which represents 15.
This system is highly efficient in digital computations and systems design. A single hex digit can represent a 4-bit binary number. For example:

• The decimal number 15, which is binary 1111, is represented as F in hexadecimal.
• For larger numbers, each grouped set of four binary bits matches a single hexadecimal digit, significantly reducing the number of characters for complex binary sequences.

Understanding the base 16 system is crucial for decoding memory addresses, color codes in web design, and during multiple stages of programming where binary interactions occur.