To convert the decimal value 21.375 into its IEEE 754 single-precision floating-point (32-bit) representation, we follow these steps:

Step 1: Convert 21.375 to binary form

First, split the decimal number into its integer and fractional parts.

• The integer part is 21.
• The fractional part is 0.375.

1.1 Convert the integer part (21) to binary:

Divide 21 by 2 repeatedly:

$21 \div 2 = 10 \text{ remainder } 1$ $10 \div 2 = 5 \text{ remainder } 0$ $5 \div 2 = 2 \text{ remainder } 1$ $2 \div 2 = 1 \text{ remainder } 0$ $1 \div 2 = 0 \text{ remainder } 1$

So, the binary representation of 21 is $10101_2$.

1.2 Convert the fractional part (0.375) to binary:

Multiply the fractional part by 2 repeatedly:

$0.375 \times 2 = 0.75 \quad \text{(integer part = 0)}$ $0.75 \times 2 = 1.5 \quad \text{(integer part = 1)}$ $0.5 \times 2 = 1.0 \quad \text{(integer part = 1)}$

So, the binary representation of 0.375 is $0.011_2$.

1.3 Combine the binary representations:

The binary form of 21.375 is:

$21.375_{10} = 10101.011_2$

Step 2: Normalize the binary number

To normalize the number, move the binary point so that there is one non-zero digit to the left of the point.

$10101.011_2 = 1.0101011 \times 2^4$

Now, the number is in the form $1.xxxx \times 2^E$, where $E$ is the exponent.

Step 3: Determine the exponent and the bias

The exponent in this case is $4$, but in IEEE 754 format, the exponent is stored with a bias of 127. So:

$\text{biased exponent} = 4 + 127 = 131$

In binary, 131 is:

$131_{10} = 10000011_2$

Step 4: Determine the mantissa

The mantissa (fraction) is the binary digits to the right of the binary point in the normalized form. From the normalized number $1.0101011_2$, the mantissa is:

$01010110000000000000000$

(The mantissa is 23 bits long, so we pad with zeros.)

Step 5: Put it all together in IEEE 754 format

The IEEE 754 single-precision floating-point format is:

• 1 bit for the sign (0 for positive numbers)
• 8 bits for the exponent
• 23 bits for the mantissa

So, for 21.375:

• Sign bit = 0
• Exponent = $10000011_2$
• Mantissa = $01010110000000000000000_2$

Thus, the full 32-bit IEEE 754 representation is:

$0 \, 10000011 \, 01010110000000000000000$

Step 6: Convert to hexadecimal

Now, group the bits into 4-bit sections:

$0100 \ 0001 \ 1010 \ 1011 \ 0000 \ 0000 \ 0000 \ 0000$

Convert each group to hexadecimal:

$0100 = 4, \ 0001 = 1, \ 1010 = A, \ 1011 = B, \ 0000 = 0, \ 0000 = 0$

Thus, the hexadecimal representation is:

$\boxed{41AB0000}$

This is the IEEE 754 hexadecimal representation of 21.375.

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

1. How does IEEE 754 handle negative numbers?
2. What is the difference between single-precision and double-precision floating-point formats?
3. How can rounding errors occur in floating-point arithmetic?
4. What is the bias in the IEEE 754 floating-point representation, and how is it chosen?
5. How do you convert a hexadecimal floating-point representation back to decimal?

Tip: When converting decimal numbers to binary, it helps to split the number into integer and fractional parts and handle them separately.