🧮 Physics
Dimensional Arithmetic Calculator
Do arithmetic with units instead of bare numbers. Enter two quantities, choose an operation, and the calculator tracks length, mass, time and six other base dimensions.
About the Dimensional Arithmetic Calculator
Every physical quantity has dimensions: length, mass, time, electric current, temperature, amount of substance, luminous intensity and, in this calculator, data. A speed has length divided by time; a force has mass times length divided by time squared. Dimensional arithmetic is the bookkeeping that keeps those relationships straight.
Addition and subtraction only make sense when both operands have identical dimensions. You can add 3 metres to 40 centimetres, but you cannot add 3 metres to 2 seconds. This calculator refuses the second operation instead of silently producing a meaningless number. Multiplication and division are always allowed: the dimension exponents are added for multiplication and subtracted for division.
The result is shown in a unit chosen automatically to keep the number readable, together with a table of every compatible unit in the registry. For example, 60 km/h multiplied by 2 hours gives 120 km, while 500 W multiplied by 3 hours gives 1.5 kWh. The dimension display uses the SI base symbols m, kg, s, A, K, mol, cd and B.
Temperature is the one deliberate limitation. Celsius and Fahrenheit are affine scales with offsets, not pure multiplicative units, so 20 °C × 2 is not 40 °C in any useful sense. This page accepts kelvin only. The same caution applies to any unit whose zero point is arbitrary.
How the calculation works
- Addition and subtraction: dimensions must match exactly; add or subtract the values in SI base units.
- Multiplication: base value = a × b; dimension exponents = exponents(a) + exponents(b).
- Division: base value = a ÷ b; dimension exponents = exponents(a) − exponents(b).
- Example: 5 m + 30 cm = 5 m + 0.3 m = 5.3 m.
- Example: 60 km/h × 2 h = (16.667 m/s) × (7,200 s) = 120,000 m = 120 km.
- Example: 2 m × 3 m = 6 m²; 6 m² ÷ 2 m = 3 m.
Reference values
| Operation | Dimension rule | Worked example |
|---|---|---|
| Addition | Same dimensions only | 5 m + 30 cm = 5.3 m |
| Subtraction | Same dimensions only | 90 min − 1 h = 30 min |
| Multiplication | Exponents add | 60 km/h × 2 h = 120 km |
| Division | Exponents subtract | 6 m² ÷ 2 m = 3 m |
| Power × time | Energy | 500 W × 3 h = 1.5 kWh |
| Force × distance | Energy | 10 N × 2 m = 20 J |
| Mass ÷ volume | Density | 2 kg ÷ 0.5 m³ = 4 kg/m³ |