Question

Which of the following is decimal number representation of \((10000)_2\)?

• A. 2
• B. 4
• C. 8
• D. 16

Hint:

Binary weights are powers of 2: $2^0=1, 2^1=2, 2^2=4, 2^3=8, 2^4=16$. A '1' followed by $n$ zeros in binary equals $2^n$. Here $n=4 \implies 2^4 = 16$.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Positional number systems express numbers as a sum of products of each digit and the base (radix) raised to the power corresponding to its position index.
Key Formula or Approach:
\[ (b_n b_{n-1} \dots b_1 b_0)_2 = \sum_{i=0}^{n} b_i \cdot 2^i \]

Step 2: Detailed Explanation:
Given the binary number \((10000)_2\):
Expanding in powers of base 2 from right to left (indices 0 to 4):
\[ (10000)_2 = 1 \times 2^4 + 0 \times 2^3 + 0 \times 2^2 + 0 \times 2^1 + 0 \times 2^0 \]
\[ = 1 \times 16 + 0 + 0 + 0 + 0 = 16 \]

Step 3: Final Answer:
Thus, the decimal equivalent of \((10000)_2\) is 16, corresponding to option (D).