Math Problem Statement
Convert Decimal 64.005125 to binary
Solution
To convert the decimal number 64.005125 into binary, we handle the integer and fractional parts separately:
Step 1: Convert the integer part (64) to binary
The integer part is 64. We'll repeatedly divide by 2 and note the remainders.
32 \div 2 = 16 \text{ remainder } 0 \\ 16 \div 2 = 8 \text{ remainder } 0 \\ 8 \div 2 = 4 \text{ remainder } 0 \\ 4 \div 2 = 2 \text{ remainder } 0 \\ 2 \div 2 = 1 \text{ remainder } 0 \\ 1 \div 2 = 0 \text{ remainder } 1$$ Reading the remainders from bottom to top, the binary representation of **64** is: $$64_{10} = 1000000_2$$ ### Step 2: Convert the fractional part (0.005125) to binary For the fractional part, we multiply by 2 and keep track of the integer part. We stop when the fraction becomes zero or repeat after a reasonable number of steps. 1. $$0.005125 \times 2 = 0.01025$$ (integer part = 0) 2. $$0.01025 \times 2 = 0.0205$$ (integer part = 0) 3. $$0.0205 \times 2 = 0.041$$ (integer part = 0) 4. $$0.041 \times 2 = 0.082$$ (integer part = 0) 5. $$0.082 \times 2 = 0.164$$ (integer part = 0) 6. $$0.164 \times 2 = 0.328$$ (integer part = 0) 7. $$0.328 \times 2 = 0.656$$ (integer part = 0) 8. $$0.656 \times 2 = 1.312$$ (integer part = 1, fractional part = 0.312) 9. $$0.312 \times 2 = 0.624$$ (integer part = 0) 10. $$0.624 \times 2 = 1.248$$ (integer part = 1, fractional part = 0.248) 11. $$0.248 \times 2 = 0.496$$ (integer part = 0) 12. $$0.496 \times 2 = 0.992$$ (integer part = 0) 13. $$0.992 \times 2 = 1.984$$ (integer part = 1, fractional part = 0.984) 14. $$0.984 \times 2 = 1.968$$ (integer part = 1, fractional part = 0.968) The binary for the fractional part goes: $$0.0000001011...$$ ### Step 3: Combine the integer and fractional parts Now, combine the binary parts: $$64.005125_{10} \approx 1000000.0000001011_2$$ This binary representation is an approximation, as the fractional part continues infinitely. Would you like to see more details or have any further questions? --- Here are five related questions you might want to explore: 1. How do you convert any decimal fraction to binary? 2. What are the advantages of using binary in computers? 3. How do you convert a binary fraction back to decimal? 4. How do you perform binary addition with fractional numbers? 5. How do floating-point numbers work in binary? **Tip**: When converting fractions, be mindful that some decimal fractions cannot be exactly represented in binary.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Binary Conversion
Number Systems
Fractions
Formulas
Integer to binary: Repeated division by 2
Fraction to binary: Repeated multiplication by 2 and noting the integer part
Theorems
Positional number system
Base-2 (binary) system
Suitable Grade Level
Grades 9-12