Number systems: Why do binary numbers generally require more digit places to represent the same quantity compared with decimal numbers?

Introduction / Context:
Different positional numeral systems use different bases. The number of digits needed to express a fixed value depends strongly on the base (radix). Understanding this helps when estimating memory sizes and data widths in computing.

Given Data / Assumptions:

• Binary uses base 2; decimal uses base 10.
• We compare how many digits (places) are needed for the same magnitude.

Concept / Approach:

In a positional system, the count of unique symbols per digit equals the base. Lower bases have fewer symbols per place, so more places are required to cover the same numeric range. Since binary has only two symbols (0 and 1), it needs more positions than decimal for equal values.

Step-by-Step Solution:

Verification / Alternative check:

Use logarithms: digits_needed ≈ floor(log_base(value)) + 1. For base 2, log2(255) ≈ 7.99 → 8 digits; for base 10, log10(255) ≈ 2.4 → 3 digits.

Why Other Options Are Wrong:

• 'Always big numbers': representation length is about base, not size bias.
• Leading zeros: can be added in any base and do not explain typical length.
• Spacing requirement: physical spacing has nothing to do with digit count.
• None: incorrect because 'Binary base is small' explains it.

Common Pitfalls:

• Confusing magnitude with representation method.
• Overlooking that octal/hex reduce length by using larger bases.

Final Answer:

Binary base is small.