MCQ
Q.
Conversion of binary number $(101110)_2$ to octal is
Correct Answer: C
The correct answer is The correct option is C, which represents the octal number 56. The given binary number $(101110)_2$ is converted to octal by grouping its digits into sets of three, starting from the right, and then converting each group into its equivalent octal digit. The binary group '101' converts to octal '5', and the binary group '110' converts to octal '6'. Combining these gives 56 in octal..
🔑 Key Points
- Octal numbers use a base of 8, meaning they use digits from 0 to 7.
- Binary numbers use a base of 2, meaning they use digits 0 and 1.
- To convert a binary number to an octal number, group the binary digits into sets of three, starting from the rightmost digit.
- If the leftmost group has fewer than three digits, pad with leading zeros to complete the group.
- Each group of three binary digits corresponds to exactly one octal digit.
🧮 Step-by-Step Solution
To convert the binary number \((101110)_2\) to octal, follow these steps: 1. **Group the binary digits into sets of three, starting from the right.** Binary number: \(101110\) Grouped: \(101 \ 110\) 2. **Convert each group of three binary digits to its equivalent octal digit.** * For the first group (from the left): \(101_2\) \(1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 1 \times 4 + 0 \times 2 + 1 \times 1 = 4 + 0 + 1 = 5_{10}\) So, \(101_2 = 5_8\) * For the second group: \(110_2\) \(1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 1 \times 4 + 1 \times 2 + 0 \times 1 = 4 + 2 + 0 = 6_{10}\) So, \(110_2 = 6_8\) 3. **Combine the octal digits to form the final octal number.** Combining \(5\) and \(6\) gives \(56_8\). Therefore, the binary number \((101110)_2\) is equal to \((56)_8\). The correct option is C.
📄 Additional Information
- Number systems are fundamental in computer science and digital electronics. Binary is the native language of computers, while octal and hexadecimal are often used as more human-readable representations of binary data, as they are compact and easy to convert from binary. Octal is particularly convenient because each octal digit perfectly maps to three binary digits (since \( 2^3 = 8 \)). Similarly, hexadecimal maps to four binary digits (since \( 2^4 = 16 \)).