To convert the decimal number \(64.005125_{10}\) into binary, we need to handle the integer and fractional parts separately.

### Step 1: Convert the integer part (64) to binary
64 in decimal is an integer. To convert it into binary, we can repeatedly divide it by 2 and record the remainders:

\[
\begin{aligned}
64 \div 2 &= 32 \quad \text{remainder} = 0 \\
32 \div 2 &= 16 \quad \text{remainder} = 0 \\
16 \div 2 &= 8 \quad \text{remainder} = 0 \\
8 \div 2 &= 4 \quad \text{remainder} = 0 \\
4 \div 2 &= 2 \quad \text{remainder} = 0 \\
2 \div 2 &= 1 \quad \text{remainder} = 0 \\
1 \div 2 &= 0 \quad \text{remainder} = 1 \\
\end{aligned}
\]

Now, reading the remainders from bottom to top, we get:

\[
64_{10} = 1000000_2
\]

### Step 2: Convert the fractional part (0.005125) to binary
To convert the fractional part \(0.005125_{10}\) into binary, we repeatedly multiply it by 2 and record the integer parts:

\[
0.005125 \times 2 = 0.01025 \quad \text{integer part} = 0
\]
\[
0.01025 \times 2 = 0.0205 \quad \text{integer part} = 0
\]
\[
0.0205 \times 2 = 0.041 \quad \text{integer part} = 0
\]
\[
0.041 \times 2 = 0.082 \quad \text{integer part} = 0
\]
\[
0.082 \times 2 = 0.164 \quad \text{integer part} = 0
\]
\[
0.164 \times 2 = 0.328 \quad \text{integer part} = 0
\]
\[
0.328 \times 2 = 0.656 \quad \text{integer part} = 0
\]
\[
0.656 \times 2 = 1.312 \quad \text{integer part} = 1
\]
\[
0.312 \times 2 = 0.624 \quad \text{integer part} = 0
\]
\[
0.624 \times 2 = 1.248 \quad \text{integer part} = 1
\]
\[
0.248 \times 2 = 0.496 \quad \text{integer part} = 0
\]
\[
0.496 \times 2 = 0.992 \quad \text{integer part} = 0
\]
\[
0.992 \times 2 = 1.984 \quad \text{integer part} = 1
\]

Now, combining the integer parts, the binary equivalent of the fractional part is approximately:

\[
0.005125_{10} \approx 0.00000010101_2
\]

### Step 3: Combine the integer and fractional parts
Finally, combining the binary representations of the integer and fractional parts:

\[
64.005125_{10} \approx 1000000.00000010101_2
\]

### Final Answer:
\[
64.005125_{10} \approx 1000000.00000010101_2
\]

Would you like to go into further detail on any part of this conversion process?

---

**Relative questions:**
1. How do you convert other fractional decimals to binary?
2. What are the rules for multiplying fractions by 2 in binary conversions?
3. How accurate can a binary representation of a fraction be?
4. How does the process change for negative decimal numbers?
5. What are the common uses of binary numbers in computing?

**Tip:** To improve precision when converting fractions to binary, continue multiplying until you reach a repeating pattern or a desired precision limit.

This tutorial walks through converting the decimal number 64.005125 to binary, breaking down both the integer and fractional parts. Learn how to use repeated division and multiplication methods to achieve the binary equivalent of this decimal number. Suitable for high school students, Grades 9-12.

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