Chain 1: Why relative errors add in multiplication (not absolute)
Absolute error means "how far off in actual units" → When you multiply two quantities, a 1% error in A combines with a 1% error in B → The result has roughly 1% + 1% = 2% error relative to the product → If you tried to add absolute errors (e.g., ±0.1 m × ±0.1 kg), you'd get wrong units → Therefore, for multiplication, relative (fractional) errors are the correct currency to add.
Chain 2: Why dimensional analysis cannot find constants
A constant like 1/2 or 2π has no units → In dimensional terms, it is [M^{0}$$L^{0}$$T^{0}] → When you match dimensions on both sides of an equation, you are equating powers of M, L, and T → A dimensionless constant contributes 0 to all three exponents → So no matter what value the constant has, dimensions balance → Therefore, dimensional analysis is blind to dimensionless constants.
Chain 3: Why leading zeros are not significant
Leading zeros in 0.00450 simply indicate the position of the decimal point → They carry no information about measurement precision → Rewrite as and the leading zeros vanish entirely → Only the 4, 5, 0 remain (3 sig figs) → Therefore, leading zeros are placeholders, not precision indicators.
Chain 4: Why averaging reduces random error but not systematic error
Random errors are equally likely positive or negative → When you average N readings, positive and negative fluctuations cancel progressively → Mean converges to true value as N increases → Systematic errors shift all readings in the same direction → Averaging biased readings gives a biased mean → The bias (systematic error) survives averaging → Therefore, only recalibration (not averaging) removes systematic error.
Chain 5: Why G in SI → CGS requires changes to all three base units
G has dimensional formula [M^{-1}$$L^{3}$$T^{-2}] → CGS changes mass (kg → g, factor 10^{3}), length (m → cm, factor 10^{2}), and time is unchanged → Applying conversion: (10^{3})^{-1} × (10^{2})^{3} × (1)^{-2} = 10^{-3} × 10^{6} × 1 = 10^{3} → So G_CGS = ^{1} × 10^{3} = → Changing only one base unit would give a wrong answer → Therefore, all relevant base-unit conversions must always be applied.