Step 1: Understanding the Question:
The topic of this problem is Number System Conversion, specifically converting a number from the Binary system (Base-2) to the Octal system (Base-8). Understanding the relationship between different bases is a core concept in computer science, as computers store data in binary, but humans often use octal or hexadecimal to represent that data more compactly.
Step 2: Key Formulas and approach:
The key relationship between binary and octal is $2^3 = 8$. This mathematical fact implies that exactly three binary digits (bits) are required to represent one octal digit.
The approach for conversion is as follows:
1. Take the binary number and group the bits into sets of three, starting from the right (the Least Significant Bit).
2. If the leftmost group has fewer than three bits, add leading zeros.
3. Convert each 3-bit group into its equivalent decimal value (ranging from 0 to 7), which represents the octal digit.
Step 3: Detailed Explanation:
We are given the binary value $100101_2$.
Start grouping from the right side: The first group (rightmost) is $101$. The second group (moving left) is $100$.
Now, we convert the first group $(101)_2$ to decimal: $(1 \times 2^2) + (0 \times 2^1) + (1 \times 2^0) = 4 + 0 + 1 = 5$.
Next, we convert the second group $(100)_2$ to decimal: $(1 \times 2^2) + (0 \times 2^1) + (0 \times 2^0) = 4 + 0 + 0 = 4$.
We place these digits in order to find the octal result. The leftmost group gave us 4, and the rightmost group gave us 5.
Combining them, we get $45_8$.
This method is much faster than converting the binary number to decimal first and then to octal, as it relies on simple bit-pattern recognition.
Step 4: Final Answer:
The binary number $100101_2$ is equivalent to 45 in the octal system.