SI Base & Derived Units

The International System of Units (SI) defines seven base quantities. Every other physical quantity in physics is built from these. Understanding base units, derived units, and prefixes is the first step to handling any quantitative problem.

Concept

A base unit is one of seven independently defined units; a derived unit is built by combining base units.

Seven SI Base Units

Common Derived Units

SI Prefixes

Worked Example

Q: Express $1$ joule entirely in SI base units.

Solution: $1\,\text{J} = 1\,\text{N} \cdot \text{m} = (1\,\text{kg} \cdot \text{m}/\text{s}^2) \cdot \text{m} = 1\,\text{kg} \cdot \text{m}^2 / \text{s}^2.$

So $[\text{J}] = \text{kg}\,\text{m}^2\,\text{s}^{-2}$.

Q (bonus): Convert $3.6 \times 10^9$ pJ to MJ.

$3.6 \times 10^9 \text{ pJ} = 3.6 \times 10^9 \times 10^{-12}\,\text{J} = 3.6 \times 10^{-3}\,\text{J} = 3.6 \times 10^{-9}\,\text{MJ}$.

Common Confusions

• The kilogram is the base unit of mass, not the gram. So $\text{kg}$ has no implicit "kilo" prefix in formulas.
• Capitalization matters: $\text{m}$ (metre) vs $\text{M}$ (mega-), $\text{s}$ (second) vs $\text{S}$ (siemens).
• A litre ($\text{L}$) is not an SI base unit; it equals $10^{-3}\,\text{m}^3$.
• "Weight" is a force ($\text{N}$); "mass" is a kg quantity. They are different.

Key Takeaways

• Memorize the seven SI base units and the common derived ones with special names.
• Any physical equation must be dimensionally consistent in base units.
• Prefixes scale magnitudes by powers of 10; learn them up to $10^{\pm 15}$.
• Be careful with unit conversions in numerical problems — they are an easy source of mistakes.