In computing, the binary number system is based on which powers of a single base?

Introduction / Context:
This question tests knowledge of number systems used in computing. Binary is the foundational number system for digital electronics and computer architecture. Understanding which base underlies binary helps in converting between number systems, understanding bit representations, and working with low-level programming or digital logic design.

Given Data / Assumptions:
- The binary system uses only two digits: 0 and 1.
- Positional number systems represent numbers as sums of powers of a base multiplied by digits.
- We need to identify which base's powers are used in binary representation.

Concept / Approach:
In any positional number system, a numeral d_k d_{k-1} ... d_1 d_0 represents the value Σ d_i × base^i, where the base is fixed and the digits d_i are constrained by the base. For decimal, the base is 10; for binary, the digits allowed are 0 and 1. Since there are two distinct digit values, the base must be 2. Every binary digit (bit) represents a power of 2, starting from 2^0 at the rightmost bit.

Step-by-Step Solution:
Step 1: Recall that in decimal (our everyday system), we use digits 0 through 9, and each position represents a power of 10 (ones, tens, hundreds, etc.). Step 2: In binary, only digits 0 and 1 are allowed. Therefore the base must be 2, because we have exactly two digit symbols. Step 3: A binary number like 1011₂ represents 1 × 2^3 + 0 × 2^2 + 1 × 2^1 + 1 × 2^0. Step 4: All place values in binary are powers of 2: 1, 2, 4, 8, 16, and so on. Hence the system uses powers of 2. Step 5: Therefore the correct choice is that the binary system uses powers of 2.

Verification / Alternative check:
Consider 13 in decimal, which is 1101 in binary. We can verify: 1 × 2^3 = 8, 1 × 2^2 = 4, 0 × 2^1 = 0, 1 × 2^0 = 1. Summing these gives 8 + 4 + 0 + 1 = 13, showing explicitly that binary digits weigh powers of 2. This reinforces that the base is 2, not 0, 1, or 10.

Why Other Options Are Wrong:
Option 0: A base of zero would not allow meaningful positional representation and is not used in positional number systems.
Option 1: Using base 1 would collapse all positional distinctions and cannot represent general integers in a standard positional system.
Option 10: This corresponds to the decimal system, not the binary system.
Option None of these: Incorrect because powers of 2 are exactly the right choice and are listed as an option.

Common Pitfalls:
A frequent confusion is between the symbols used and the base. Some learners think that because the digits are 0 and 1, the base must be 1, which is incorrect. The base counts how many distinct digit symbols exist, not the maximum digit value itself. In binary there are 2 distinct digits, 0 and 1, so the base is 2 and the positional values are powers of 2.

Final Answer:
The binary number system uses powers of 2.