UGC NET Questions (Paper – 1)

For a positive integer N, the number of decimal digits reflects the position of the most significant non-zero digit. The magnitude of N lies between powers of 10, and if 10^(k−1) ≤ N ≤ 10^k − 1, then N has k digits. This inequality rearranges to k = ⌊log₁₀N⌋ + 1. Hence the standard formula used in computer science and mathematics for the digit count is ⌊log₁₀N⌋ + 1.

Option A:
Option A uses the floor function to ensure that fractional logarithm values yield the correct integer digit count. For any N in the interval [10^(k−1),10^k−1], log₁₀N lies in [k−1,k), so ⌊log₁₀N⌋ = k−1 and adding 1 gives k. This matches the definition of decimal digit length.

Option B:
Option B, ⌈log₁₀N⌉, would overestimate the digit count whenever N is exactly a power of 10, such as 100. For N = 100, log₁₀N = 2, and ⌈2⌉ = 2, which ignores that 100 has 3 digits. Hence, the ceiling function alone is not appropriate for this formula.

Option C:
Option C, log₁₀(N−1), lacks both integer rounding and the +1 term, so it does not produce an integer digit count reliably. It also fails for small values like N = 1, where log₁₀(0) is undefined. Thus, this expression is mathematically unsuitable for the purpose.

Option D:
Option D, log₁₀N + 1, can give non-integer, fractional values, which do not correspond directly to a digit count. It also overestimates digits for many values, since there is no rounding mechanism to correct for the logarithm's fractional part.