Metric micro-scale definition:\r One micron (one micrometre) is numerically equal to which of the following?

Introduction:
Accurate unit conversion at micro-scales is essential in particle technology, filtration, microfluidics, and microscopy. The micrometre (micron) is a standard SI-derived unit for small lengths such as cell sizes and pore diameters.

Given Data / Assumptions:

• 1 micrometre = 1 μm by definition.
• We relate μm to meters, centimeters, and millimeters.
• Base conversions: 1 m = 100 cm = 1000 mm.

Concept / Approach:
By definition, 1 μm = 10^-6 m. Converting to centimeters: since 1 cm = 10^-2 m, then 10^-6 m = 10^-4 cm. Converting to millimeters: 1 mm = 10^-3 m, hence 1 μm = 10^-3 mm (not 10^-4 mm). Therefore, statements “10^-4 cm” and “10^-6 m” are both correct, while “10^-4 mm” is incorrect.

Step-by-Step Solution:
Start: 1 μm = 10^-6 m.To cm: 10^-6 m ÷ 10^-2 m/cm = 10^-4 cm.To mm: 10^-6 m ÷ 10^-3 m/mm = 10^-3 mm.Therefore, correct combined choice is both (b) & (c).

Verification / Alternative check:
Cross-check with a practical value: a 1000 μm object equals 1 mm, consistent with 1 μm = 10^-3 mm.

Why Other Options Are Wrong:

• 10^-4 mm: Off by a factor of 10.
• 10^-3 mm as a standalone option would be true, but it is not among the original correct pair requested here.

Common Pitfalls:
Mixing up exponents when converting between cm and mm; always convert via meters to avoid errors.

Final Answer:
both (b) & (c)