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:

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.