Math Problem Statement
Determine the decimal value of the IEEE FPS pattern 0xc17a0000.
Solution
The given pattern is , 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 in binary is:
This divides into the sign, exponent, and mantissa as:
- Sign (S): (The first bit is 1, meaning the number is negative)
- Exponent (E): (The next 8 bits)
- Mantissa (F): (The remaining 23 bits)
Step 2: Interpret the components.
- Sign: , so the number is negative.
- Exponent: The exponent is , which is .
- IEEE 754 uses a bias of 127 for single-precision floating points, so the actual exponent is:
- Mantissa: The fraction (mantissa) is when normalized (we prepend a leading 1 for normalized numbers in IEEE 754 format).
- In decimal, this is: = 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:
Substitute the values:
Final Answer:
The decimal value of the IEEE 754 single-precision floating-point pattern is .
Would you like more details on any of these steps? Here are some related questions:
- How does the IEEE 754 standard represent denormalized numbers?
- What is the difference between single-precision and double-precision formats?
- Can you explain how the bias works in the exponent of IEEE 754?
- How would the decimal value change if the exponent were increased by 1?
- 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.
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Math Problem Analysis
Mathematical Concepts
IEEE 754 Floating-Point Standard
Binary to Decimal Conversion
Exponent and Mantissa Interpretation
Formulas
(-1)^S * 1.M * 2^E
Exponent = E - Bias (127 for single precision)
Mantissa = 1 + F (normalized)
Theorems
IEEE 754 Single-Precision Floating-Point Representation
Suitable Grade Level
Grades 11-12, College Level
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