Let's break down the 32-bit binary number shown and analyze it according to the IEEE 754 single-precision floating-point format:

Step 1: Identify the different parts of the binary number

The 32-bit IEEE 754 number consists of:

• 1 bit for the sign (S)
• 8 bits for the exponent (E)
• 23 bits for the mantissa (fraction) (M)

The binary number provided is:

0 10000011 10111000000000000000000

• Sign bit (S): The first bit is 0, indicating a positive number.
• Exponent bits (E): The next 8 bits are 10000011.
• Mantissa bits (M): The remaining 23 bits are 10111000000000000000000.

Step 2: Decode each part

1. Sign (Blank 1)

Since the sign bit (S) is 0, the number is positive. So:

• Blank 1: P

2. Exponent (Blank 2)

The 8-bit exponent is 10000011, which is in binary. We convert this to decimal:

• 10000011 in binary is 131 in decimal.
• This is the biased exponent stored in the IEEE format.
• Blank 2: 131

3. Unbiased Exponent (Blank 3)

The unbiased exponent is found by subtracting the bias from the stored exponent. For single-precision floating-point numbers, the bias is 127.

• Unbiased Exponent = 131 - 127 = 4
• Blank 3: 4

4. Mantissa (Fraction) (Blank 4)

The mantissa in the IEEE format has an implicit leading 1. So, the mantissa is:

• 1.10111000000000000000000 in binary.

Now convert this binary fraction to decimal:

• 1.10111 (we can ignore the trailing zeros).

Breaking it down:

• 1 in decimal is 1.
• .1 is 0.5.
• .001 is 0.125.
• .00011 is 0.09375.

Thus, the mantissa is:

• 1 + 0.5 + 0.125 + 0.09375 = 1.71875
• Blank 4: 1.71875

5. Calculate the Final Value (Blank 5)

The final number is calculated using the formula: $\text{Value} = (-1)^S \times 2^{\text{Unbiased Exponent}} \times \text{Mantissa}$

Substitute the values:

• Sign = 0 (positive)
• Unbiased Exponent = 4
• Mantissa = 1.71875

So: $\text{Value} = 1 \times 2^4 \times 1.71875$ $\text{Value} = 16 \times 1.71875 = 27.5$

Thus, the stored decimal real number is:

• Blank 5: 27.5

Summary of the Blanks

1. Blank 1: P
2. Blank 2: 131
3. Blank 3: 4
4. Blank 4: 1.71875
5. Blank 5: 27.5

Would you like further details on any step? Feel free to ask any questions!

Related Questions:

1. What is the range of numbers that can be represented by IEEE 754 single-precision?
2. How does the biasing mechanism work in IEEE floating-point numbers?
3. What are the differences between single-precision and double-precision floating-point formats?
4. How do subnormal numbers (denormals) work in IEEE floating-point?
5. What are common causes of floating-point precision errors in computing?

Tip: Always check the exponent bias in different IEEE floating-point formats, as it changes depending on the precision (e.g., 127 for single-precision, 1023 for double-precision).