Determine the decimal value of the IEEE FPS pattern 0xc17a0000.

The given pattern is \( 0xc17a0000 \), which is a 32-bit value in hexadecimal. In the IEEE 754 standard for single-precision floating-point format, this is structured as follows:

- **1 bit** for the sign (S)
- **8 bits** for the exponent (E)
- **23 bits** for the fraction (F)

### Step 1: Convert the hex pattern to binary.

The hexadecimal value \( 0xc17a0000 \) in binary is:

\[
1100 \ 0001 \ 0111 \ 1010 \ 0000 \ 0000 \ 0000 \ 0000
\]

This divides into the sign, exponent, and mantissa as:

- **Sign (S)**: \( 1 \) (The first bit is 1, meaning the number is negative)
- **Exponent (E)**: \( 10000010_2 \) (The next 8 bits)
- **Mantissa (F)**: \( 11110100000000000000000_2 \) (The remaining 23 bits)

### Step 2: Interpret the components.

1. **Sign**: \( S = 1 \), so the number is negative.
2. **Exponent**: The exponent is \( 10000010_2 \), which is \( 130_{10} \).
   - IEEE 754 uses a bias of 127 for single-precision floating points, so the actual exponent is:
     \[
     \text{Exponent} = 130 - 127 = 3
     \]
3. **Mantissa**: The fraction (mantissa) is \( 1.111101_2 \) when normalized (we prepend a leading 1 for normalized numbers in IEEE 754 format).
   - In decimal, this is:
     \[
     1.111101_2 = 1 + 1 \times 2^{-1} + 1 \times 2^{-2} + 1 \times 2^{-3} + 1 \times 2^{-4} + 0 \times 2^{-5} + 1 \times 2^{-6}
     = 1 + 0.5 + 0.25 + 0.125 + 0.0625 + 0.015625 = 1.953125
     \]

### Step 3: Calculate the value.

Now that we have all components, the value of the number is:

\[
(-1)^S \times 1.M \times 2^E
\]

Substitute the values:

\[
(-1)^1 \times 1.953125 \times 2^3 = -1 \times 1.953125 \times 8 = -15.625
\]

### Final Answer:
The decimal value of the IEEE 754 single-precision floating-point pattern \( 0xc17a0000 \) is \( \boxed{-15.625} \).

Would you like more details on any of these steps? Here are some related questions:

1. How does the IEEE 754 standard represent denormalized numbers?
2. What is the difference between single-precision and double-precision formats?
3. Can you explain how the bias works in the exponent of IEEE 754?
4. How would the decimal value change if the exponent were increased by 1?
5. What happens if the mantissa has all zeros in IEEE 754?

**Tip:** Always remember that the exponent in IEEE 754 has a bias (127 for single precision) that must be subtracted when calculating the actual exponent.

Learn how to calculate the decimal value of the IEEE 754 single-precision floating-point pattern 0xc17a0000. This detailed explanation covers binary to decimal conversion, the role of the exponent and mantissa, and how to use the IEEE 754 formula. Suitable for students in Grades 11-12 and college.

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