Physical Quantities and Units

Physical quantities, SI units, unit conversion and scientific notation.

Conceptual context

Physics cannot exist without measurement, and measurement cannot be communicated without a common language. For centuries, each region used its own units: the foot, the cubit, the pound. Results from different laboratories were incomparable. The central question of this topic is: how are physical quantities expressed, classified, and converted using the International System of Units? The answer builds the universal language of physics: a set of seven base quantities, a coherent unit system, notation rules, and standardized prefixes that allow working from the atomic scale to the scale of the universe.

🟢 Essential level

When you measure something — a distance, a mass, a time — you obtain a number together with a unit. That pair (number, unit) is a physical quantity. The International System defines seven independent base quantities: length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity. All others are obtained by combining these seven.

Unit conversion changes the number but not the physical quantity: one kilometer and one thousand meters are exactly the same distance. The factor connecting them, $f_{conv}$, is always positive and exact. When values are very large or very small, scientific notation expresses them as the product of a $m$ between 1 and 10 and a power of 10 with exponent $n$. SI prefixes such as kilo, mega, milli, or nano are equivalent shortcuts: $Q_{pref}$ with $f_{pref}$ gives the same $Q_{SI}$ as scientific notation.

Length, for example, is measured in meters; using miles or feet in the same scientific calculation introduces errors that are easily avoidable.

🔵 Formal level

Basic formulas

The three fundamental formulas of this leaf are:

Unit conversion:

Unit conversion using a conversion factor

Derived units inherit the factors of their base units raised to their exponent. In volume, the linear factor $f_{conv}$ appears cubed:

Conversion of volume units

That is why a small error in a length triples when going to volume, and why a known volume lets you recover the length unit with a cube root.

$f_{conv}$ is dimensionless and equal to $Q_{SI}$ / $Q_{orig}$. The line $Q_{SI}$ vs $Q_{orig}$ in the Coord graph has slope $f_{conv}$ and passes through the origin for every pure multiplicative conversion. When $f_{conv}$ > 1, the SI unit is smaller than the original unit and the number increases upon conversion. When $f_{conv}$ < 1, the SI unit is larger and the number decreases.

Scientific notation:

Scientific notation expression

where $m$ is in [1, 10) and $n$ is an integer. $n$ determines the order of magnitude and can be negative (quantities smaller than 1 in the base unit), zero (quantities of the order of the base unit), or positive (quantities larger than the base unit). The $m$ encodes precision: its significant figures are exactly those of the result.

Real student questions

Why multiply and not divide when converting from a small unit to a larger one? When converting from centimeters to meters (a larger unit), the number decreases: 100 cm is equal to 1 m. $f_{conv}$ is equal to 0.01, and you multiply: 100 × 0.01 is equal to 1. You divide by 100 only if you use the inverse of the factor. The confusion arises from mixing the factor and its inverse.

Can the mantissa be negative? Yes. Negative $m$ indicates that the physical quantity is negative (below-zero temperature, negative charge, position to the left of the origin). The sign belongs to $m$, not to $n$: -3.5 × 10^4 is correct; placing the negative sign separately from the mantissa is not standard notation.

Does zero have scientific notation? No. The logarithm of zero does not exist, so $n$ is not defined for $Q_{SI}$ is equal to 0. Zero is simply written as 0, without a power of 10.

Final synthesis

A physical quantity is the pair (number, unit); neither has meaning without the other. The SI unifies the language of physics through seven base quantities, exact conversion factors, standardized prefixes, and scientific notation. Expressing, classifying, and converting quantities is the first skill of every physicist.