Hexadecimal numbers are widely used in computer science and digital electronics due to their compatibility with binary numbers. While converting whole hexadecimal numbers to decimal is a common practice, converting hexadecimal fractions is equally important but slightly more complex. This article will delve into the process of converting hexadecimal fractions to decimal, complete with examples for clarity.

A hexadecimal number system, or base-16 system, uses sixteen symbols to represent values: 0-9 for values zero to nine, and A-F (or a-f) for values ten to fifteen. Hexadecimal fractions extend this system beyond the decimal point, using negative powers of 16.

For example, the hexadecimal fraction 1.AB2 can be broken down as:

• 1: $16^0$
• A: $16^{-1}$
• B: $16^{-2}$
• 2: $16^{-3}$

### Conversion Process

To convert a hexadecimal fraction to a decimal fraction, follow these steps:

1. Identify the hexadecimal digits and their positions: Each digit’s position determines its power of 16.
2. Convert each hexadecimal digit to its decimal equivalent: Use the decimal values of the hexadecimal digits.
3. Multiply each digit by 16 raised to the power of its position: The position is negative and starts from -1 after the decimal point.
4. Sum the results: Add all the products obtained in the previous step to get the decimal equivalent.

### Example: Converting Hexadecimal Fraction to Decimal

Let’s convert the hexadecimal fraction 1.AB2 to decimal.

1. Identify the hexadecimal digits and their positions:

• 1 (position 0)
• A (position -1)
• B (position -2)
• 2 (position -3)
2. Convert each hexadecimal digit to its decimal equivalent:

• 1 = 1
• A = 10
• B = 11
• 2 = 2
3. Multiply each digit by 16 raised to the power of its position:

• $1 \times 16^0 = 1 \times 1 = 1$
• $10 \times 16^{-1} = 10 \times \frac{1}{16} = 10 \times 0.0625 = 0.625$
• $11 \times 16^{-2} = 11 \times \frac{1}{256} = 11 \times 0.00390625 = 0.04296875$
• $2 \times 16^{-3} = 2 \times \frac{1}{4096} = 2 \times 0.000244140625 = 0.00048828125$
4. Sum the results:

• $1 + 0.625 + 0.04296875 + 0.00048828125 = 1.66845703125$

So, the decimal equivalent of the hexadecimal fraction 1.AB2 is 1.66845703125.

### Another Example: Converting 0.F4 to Decimal

1. Identify the hexadecimal digits and their positions:

• F (position -1)
• 4 (position -2)
2. Convert each hexadecimal digit to its decimal equivalent:

• F = 15
• 4 = 4
3. Multiply each digit by 16 raised to the power of its position:

• $15 \times 16^{-1} = 15 \times \frac{1}{16} = 15 \times 0.0625 = 0.9375$
• $4 \times 16^{-2} = 4 \times \frac{1}{256} = 4 \times 0.00390625 = 0.015625$
4. Sum the results:

• $0.9375 + 0.015625 = 0.953125$

So, the decimal equivalent of the hexadecimal fraction 0.F4 is 0.953125.

### Practical Applications

#### Precision in Computation

In computing, precision is crucial. Hexadecimal fractions are often used in defining precise values for various computational processes, such as graphics rendering and scientific calculations.

#### Floating-Point Numbers

Hexadecimal fractions are instrumental in representing floating-point numbers in programming languages and systems. Understanding their conversion helps in debugging and optimizing code.

### Conclusion

Converting hexadecimal fractions to decimal is a straightforward but detailed process that involves understanding the positional value of each digit in the hexadecimal system. By breaking down the fraction into its components and applying the appropriate power of 16, one can accurately convert any hexadecimal fraction to its decimal equivalent. This skill is essential for tasks requiring precision in computing and digital systems.

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