Understanding Hexadecimal to Binary Conversion

Converting a number from one base (or radix) to another is a fundamental concept in digital electronics and computer science. This question asks us to convert a hexadecimal number, C6, into its binary equivalent.

What are Hexadecimal and Binary Numbers?

• Binary Numbers: This is a base-2 number system, using only two digits: 0 and 1. It's the language computers understand.
• Hexadecimal Numbers: This is a base-16 number system. It uses 16 distinct symbols: the digits 0 through 9, and the letters A through F. The letters represent values 10 through 15 (A=10, B=11, C=12, D=13, E=14, F=15). Hexadecimal is often used as a shorthand for binary numbers because each hexadecimal digit can represent exactly four binary digits (bits).

Steps to Convert Hexadecimal C6 to Binary

To convert a hexadecimal number to binary, we convert each hexadecimal digit into its 4-bit binary equivalent. Then, we concatenate (join) these binary groups together in the same order.

Let's break down the hexadecimal number C6:

• The first digit is C.
• The second digit is 6.

Now, let's convert each digit to its 4-bit binary form:

• Convert C to binary:
  • In hexadecimal, C represents the decimal value 12.
  • To convert 12 to 4-bit binary, we find the powers of 2 that sum up to 12: \(2^3=8\), \(2^2=4\), \(2^1=2\), \(2^0=1\).
  • We need 8 and 4 to make 12 (\(8 + 4 = 12\)). So, the 4-bit binary representation for 12 is 1100 (1 for \(2^3\), 1 for \(2^2\), 0 for \(2^1\), 0 for \(2^0\)).
  • So, C in hexadecimal is 1100 in binary.
• Convert 6 to binary:
  • In hexadecimal, 6 represents the decimal value 6.
  • To convert 6 to 4-bit binary, we find the powers of 2 that sum up to 6: \(2^3=8\), \(2^2=4\), \(2^1=2\), \(2^0=1\).
  • We need 4 and 2 to make 6 (\(4 + 2 = 6\)). So, the 4-bit binary representation for 6 is 0110 (0 for \(2^3\), 1 for \(2^2\), 1 for \(2^1\), 0 for \(2^0\)).
  • So, 6 in hexadecimal is 0110 in binary.

Finally, concatenate the binary representations of C and 6 in the correct order:

Binary for C: 1100

Binary for 6: 0110

Concatenated binary: 11000110

Thus, the hexadecimal number C6 is equal to the binary number 11000110.

Comparing with Options

Let's look at the given options:

The calculated binary number 11000110 matches Option 3.

Hexadecimal to Binary Conversion Revision

Additional Information on Number Systems

Number systems are essential for representing data in computers. Here are a few common ones:

• Decimal (Base 10): The system we use daily, with digits 0-9.
• Binary (Base 2): Uses 0 and 1, fundamental for digital circuits.
• Octal (Base 8): Uses digits 0-7. Each octal digit represents three binary digits. Less common than hexadecimal now.
• Hexadecimal (Base 16): Uses 0-9 and A-F. Each hexadecimal digit represents four binary digits, making it convenient for representing byte values.

Conversions between these bases (like hexadecimal to binary, binary to decimal, etc.) are important skills for understanding computer architecture and low-level programming.