Converting Binary to Hexadecimal

Understanding how to convert between different number systems is a fundamental skill in digital electronics and computer science. The question asks for the hexadecimal equivalent of the binary number (1110)$_2$. Binary numbers use base 2, meaning they only use digits 0 and 1. Hexadecimal numbers use base 16, using digits 0-9 and letters A-F to represent values 10-15.

Step-by-Step Binary to Hexadecimal Conversion for 1110

To convert a binary number to hexadecimal, we group the binary digits into sets of four, starting from the rightmost digit. Each group of four binary digits corresponds to exactly one hexadecimal digit.

For the binary number (1110)$_2$, we have exactly four digits. We can convert this single group directly.

Let's convert the binary group 1110 to its decimal equivalent first. In a binary number, each digit position represents a power of 2, starting from $2^0$ on the right.

• The rightmost digit (0) is in the $2^0$ position.
• The next digit to the left (1) is in the $2^1$ position.
• The next digit (1) is in the $2^2$ position.
• The leftmost digit (1) is in the $2^3$ position.

The decimal value is calculated as follows:

\( (1110)_2 = 1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 \)

Let's calculate the powers of 2:

• \( 2^0 = 1 \)
• \( 2^1 = 2 \)
• \( 2^2 = 4 \)
• \( 2^3 = 8 \)

Now, substitute these values back into the equation:

\( (1110)_2 = 1 \times 8 + 1 \times 4 + 1 \times 2 + 0 \times 1 \)

\( (1110)_2 = 8 + 4 + 2 + 0 \)

\( (1110)_2 = 14 \)

So, the decimal equivalent of the binary number 1110 is 14.

Mapping Decimal to Hexadecimal

Now, we need to convert the decimal value 14 to its hexadecimal equivalent. Hexadecimal uses digits 0-9 for values 0-9 and letters A-F for values 10-15. Here is the mapping for values 10 through 15:

From the table, we can see that the decimal value 14 corresponds to the hexadecimal digit E.

Therefore, the hexadecimal equivalent of the binary number (1110)$_2$ is E.

Summary of Conversion

Binary (1110)$_2$ is equivalent to Decimal 14, which is equivalent to Hexadecimal E.

Revision Table: Number System Equivalents

Additional Information on Number Systems

Number systems are ways of representing numbers using digits or symbols. The most common number systems used in computing are binary (base 2), decimal (base 10), and hexadecimal (base 16).

• Decimal (Base 10): This is the system we use daily. It uses digits 0-9. Each position represents a power of 10.
• Binary (Base 2): Used extensively in digital circuits and computers because it represents two states: on (1) or off (0). Each position represents a power of 2.
• Hexadecimal (Base 16): Often used in computing as a shorthand for binary. Because 16 is \(2^4\), four binary digits can be easily represented by one hexadecimal digit. This makes reading and writing long binary strings much simpler. It uses digits 0-9 and letters A-F. Each position represents a power of 16.

Understanding conversions between these systems is crucial for working with computer memory addresses, color codes (like in web design), and low-level programming.