SI units and changing units

Anything we can measure is a physical quantity: length, mass, time, speed, force. Love, anger or how cute a puppy is cannot be measured, so they are not physical quantities.

Every measurement has two parts: a number and a unit. "The pen costs ₹10" can also be written "the pen costs 1000 paise". The number changes with the unit, but the quantity itself never changes. So n₁u₁ = n₂u₂: a bigger unit gives a smaller number, and a smaller unit a bigger one (1 m = 100 cm).

Seven quantities are chosen as base quantities. Their units are the base units, and every other unit is built from them (a derived unit: speed in m s⁻¹, density in kg m⁻³).

Older systems fixed only three units: CGS (centimetre, gram, second), FPS (foot, pound, second) and MKS (metre, kilogram, second). The SI (Système International) keeps the MKS three and fixes all seven.

Two angles have units without dimensions:

• Plane angle, in radians: dθ = ds/r, arc divided by radius. An arc as long as the radius makes 1 radian at the centre.
• Solid angle, in steradians: dΩ = dA/r². Think of the cone of an ice-cream cone, or a slice cut out of a watermelon: a patch of area r² on a sphere makes 1 steradian at the centre.

Some practical units: 1 fermi = 10⁻¹⁵ m, 1 angstrom = 10⁻¹⁰ m. A light year is the distance light travels in one year: 3 × 10⁸ m s⁻¹ × (365 × 24 × 60 × 60) s ≈ 9.46 × 10¹⁵ m. The astronomical unit (the average Earth–Sun distance) is about 1.496 × 10¹¹ m.

To change units, replace each unit by its value in the new units and multiply out. For example, g = 9.8 m s⁻² in km h⁻²: 1 m = 10⁻³ km and 1 s = 1/3600 h, so 9.8 × 10⁻³ × 3600² ≈ 1.27 × 10⁵ km h⁻².