Which statement describes how a digit's value is determined in a positional numeral system?

Which statement describes how a digit's value is determined in a positional numeral system?

• A. All digits have Equal Weight
• B. Digits AreWeighted by Primes
• C. The Digit's Value Depends on Its Position and the Base
• D. The Value Is Always the Same Regardless of Position

Answer: (C)

In a positional numeral system, the place a digit sits determines how much it contributes to the overall value, and the base sets how those place values grow. Each digit represents itself times the base raised to the power of its position, so the value of a number is the sum of these contributions: digit_i × base^i. For example, in decimal, 345 equals 3×10^2 + 4×10^1 + 5×10^0, which is 300 + 40 + 5. In binary, the number 1011 equals 1×2^3 + 0×2^2 + 1×2^1 + 1×2^0, which is 8 + 0 + 2 + 1 = 11. This shows how the same digits can contribute different amounts depending on their position and the base. That’s why the statement describing the digit’s value as depending on its position and the base is the correct one. It captures the essential mechanism behind positional systems. The idea that all digits have equal weight ignores the changing place values; weighting by primes isn’t a general rule for positional systems, since the base (not primality) determines the weights; and saying the value is always the same regardless of position contradicts how moving a digit to another place changes its contribution.

In a positional numeral system, the place a digit sits determines how much it contributes to the overall value, and the base sets how those place values grow. Each digit represents itself times the base raised to the power of its position, so the value of a number is the sum of these contributions: digit_i × base^i. For example, in decimal, 345 equals 3×10^2 + 4×10^1 + 5×10^0, which is 300 + 40 + 5. In binary, the number 1011 equals 1×2^3 + 0×2^2 + 1×2^1 + 1×2^0, which is 8 + 0 + 2 + 1 = 11. This shows how the same digits can contribute different amounts depending on their position and the base.

That’s why the statement describing the digit’s value as depending on its position and the base is the correct one. It captures the essential mechanism behind positional systems. The idea that all digits have equal weight ignores the changing place values; weighting by primes isn’t a general rule for positional systems, since the base (not primality) determines the weights; and saying the value is always the same regardless of position contradicts how moving a digit to another place changes its contribution.