Understanding Hexadecimal to Binary Conversion

Converting numbers between different bases is a fundamental concept in computer science and digital electronics. This question asks us to convert a hexadecimal number, 4A, into its equivalent binary representation.

Hexadecimal (base 16) uses digits 0-9 and letters A-F to represent values. Binary (base 2) uses only 0 and 1. Each hexadecimal digit can be directly converted into a 4-digit binary number.

Step-by-Step Conversion of Hexadecimal 4A to Binary

To convert the hexadecimal number 4A to binary, we will convert each hexadecimal digit individually into its 4-bit binary equivalent.

Convert the Hexadecimal Digit '4'

The hexadecimal digit '4' represents the decimal value 4.

To convert decimal 4 to binary, we can find the powers of 2 that sum up to 4:

• \(2^2 = 4\)

In 4-bit binary (representing \(2^3\), \(2^2\), \(2^1\), \(2^0\)), this is:

\((4)_{10} = (0 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 0 \times 2^0)_{10} = (0100)_2\)

So, the hexadecimal digit '4' is equivalent to the binary 0100.

Convert the Hexadecimal Digit 'A'

The hexadecimal digit 'A' represents the decimal value 10.

To convert decimal 10 to binary, we find the powers of 2 that sum up to 10:

• \(2^3 = 8\)
• \(2^1 = 2\)
• \(8 + 2 = 10\)

In 4-bit binary (representing \(2^3\), \(2^2\), \(2^1\), \(2^0\)), this is:

\((10)_{10} = (1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0)_{10} = (1010)_2\)

So, the hexadecimal digit 'A' is equivalent to the binary 1010.

Combine the Binary Equivalents

Now we combine the 4-bit binary representations for each hexadecimal digit:

• For '4': 0100
• For 'A': 1010

Putting them together in order:

4A (hex) = 0100 1010 (binary)

The leading zero in the binary representation can be omitted without changing the value.

Thus, 01001010 is the same as 1001010.

Final Binary Representation for 4A Hexadecimal

Combining the binary representations for '4' and 'A', we get 01001010. Removing the leading zero, the binary representation is 1001010.

Therefore, the binary representation for the hexadecimal number 4A is 1001010.

Revision Table: Hexadecimal to Binary Mapping

Additional Information on Number System Conversions

Number system conversions are essential for understanding how computers store and process data. Hexadecimal is often used as a compact way to represent binary data because each hex digit corresponds directly to four binary digits (a nibble).

Other common number system conversions include:

• Decimal to Binary
• Binary to Decimal
• Decimal to Hexadecimal
• Hexadecimal to Decimal
• Binary to Octal
• Octal to Binary

Understanding the place value system in each base is key to performing these conversions accurately.