Math Problem Statement

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

Solution

The given pattern is 0xc17a00000xc17a0000, 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 0xc17a00000xc17a0000 in binary is:

1100 0001 0111 1010 0000 0000 0000 00001100 \ 0001 \ 0111 \ 1010 \ 0000 \ 0000 \ 0000 \ 0000

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

  • Sign (S): 11 (The first bit is 1, meaning the number is negative)
  • Exponent (E): 10000010210000010_2 (The next 8 bits)
  • Mantissa (F): 11110100000000000000000211110100000000000000000_2 (The remaining 23 bits)

Step 2: Interpret the components.

  1. Sign: S=1S = 1, so the number is negative.
  2. Exponent: The exponent is 10000010210000010_2, which is 13010130_{10}.
    • IEEE 754 uses a bias of 127 for single-precision floating points, so the actual exponent is: Exponent=130127=3\text{Exponent} = 130 - 127 = 3
  3. Mantissa: The fraction (mantissa) is 1.11110121.111101_2 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:

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

Substitute the values:

(1)1×1.953125×23=1×1.953125×8=15.625(-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 0xc17a00000xc17a0000 is 15.625\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.

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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