Step 1: Understanding the Question:
This problem involves the conversion of a real number (containing both an integer part and a fractional part) from the Decimal system (Base-10) to the Octal system (Base-8). This is a two-step process because the integer and fractional components are handled using different mathematical operations. The topic is fundamental to digital logic and data representation.
Step 2: Key Formulas and approach:
The approach involves two distinct methods:
1. For the Integer Part: Use the "Successive Division" method. Divide the number by the target base (8) repeatedly and track the remainders.
2. For the Fractional Part: Use the "Successive Multiplication" method. Multiply the fraction by the target base (8) and track the resulting integer parts.
The integer remainders are read from bottom-to-top (Reverse), while the fractional integer results are read from top-to-bottom (Forward).
Step 3: Detailed Explanation:}
Step A: Integer Conversion (153):
$153 \div 8 = 19$ with a remainder of 1.
$19 \div 8 = 2$ with a remainder of 3.
$2 \div 8 = 0$ with a remainder of 2.
Reading the remainders upwards, we get $231_8$.
Step B: Fractional Conversion (0.513):
$0.513 \times 8 = 4.104$ (Integer is 4).
$0.104 \times 8 = 0.832$ (Integer is 0).
$0.832 \times 8 = 6.656$ (Integer is 6).
$0.656 \times 8 = 5.248$ (Integer is 5).
$0.248 \times 8 = 1.984$ (Integer is 1).
$0.984 \times 8 = 7.872$ (Integer is 7).
Combining the fractional results top-to-bottom, we get $.406517_8$.
Finally, merging the two parts: $231 + .406517 = 231.406517_8$.
Step 4: Final Answer:
The decimal number $(153.513)_{10}$ is equal to $231.406517$ in the octal system.