To convert the hexadecimal number $6074548800000000$ into a real number represented by the IEEE 754 64-bit double-precision format, let's go through each step carefully.

Step 1: Write Out the Hexadecimal Number in Binary

IEEE 754 double-precision numbers are represented in 64 bits, where:

• The first bit is the sign bit (1 bit).
• The next 11 bits are the exponent.
• The remaining 52 bits are the fraction (or mantissa).

First, convert each hexadecimal digit into its 4-bit binary equivalent.

$\text{Hex: } 6074548800000000$

Breaking down each digit: $6 = 0110, \quad 0 = 0000, \quad 7 = 0111, \quad 4 = 0100, \quad 5 = 0101, \quad 4 = 0100, \quad 8 = 1000, \quad 8 = 1000, \quad 0 = 0000, \quad 0 = 0000, \quad 0 = 0000, \quad 0 = 0000, \quad 0 = 0000, \quad 0 = 0000, \quad 0 = 0000, \quad 0 = 0000$

So, the binary representation is: $0110\ 0000\ 0111\ 0100\ 0101\ 0100\ 1000\ 1000\ 0000\ 0000\ 0000\ 0000\ 0000\ 0000\ 0000\ 0000$

This gives us: $0110000001110100010101001000100000000000000000000000000000000000$

Step 2: Identify the Sign, Exponent, and Fraction

Based on the IEEE 754 structure:

1. Sign bit (first bit): $0$ (positive number).
2. Exponent (next 11 bits): $11000000111$.
3. Fraction (last 52 bits): $0100010101001000100000000000000000000000000000000000$.

Step 3: Convert the Exponent from Binary to Decimal

The binary exponent is $11000000111$. Converting this to decimal: $11000000111_2 = 1543$

In IEEE 754 double precision, the exponent has a bias of 1023. So, we find the actual exponent by subtracting the bias: $\text{Actual Exponent} = 1543 - 1023 = 520$

Step 4: Convert the Fraction to Decimal

The fraction part is $0100010101001000100000000000000000000000000000000000$. To interpret this fraction:

1. Start with an implicit leading $1$ (since the format uses normalized numbers).
2. Write the binary fraction as $1.0100010101001000100000000000000000000000000000000000$.

Now, convert this binary fraction to decimal by summing powers of $2^{-n}$ for each bit $n$ after the binary point: $1.0100010101001000100000..._2 \approx 1.2646484375$

Step 5: Put It All Together in the IEEE 754 Formula

The value of the number is given by: $\text{Value} = (-1)^{\text{sign}} \times 2^{\text{exponent}} \times \text{fraction}$ Substituting the values: $\text{Value} = (-1)^0 \times 2^{520} \times 1.2646484375$

This represents a very large positive number.

Would you like me to calculate this in decimal form, or is this breakdown sufficient for your needs?

---

Let me know if you'd like more details on any step or further help with this example.

5 Related Questions to Explore:

1. How does the IEEE 754 format handle negative exponents?
2. What are subnormal numbers in IEEE 754, and how are they represented?
3. Why is there a bias in the exponent, and how does it impact calculations?
4. What are the limitations of the IEEE 754 format in terms of precision?
5. How does rounding work in IEEE 754, particularly in double precision?

Tip:

When converting fractions in binary to decimal, remember that each binary place after the point represents powers of $2^{-n}$ (e.g., the first is $2^{-1}$, the second $2^{-2}$, and so on).