Linear factors: multiply and divide
Most unit conversions are a single factor because the scales share a zero. One inch is exactly 0.0254 metres, one pound is 0.45359237 kilograms, one mile is 1,609.344 metres. Converting is multiplication by the factor in one direction and division in the other, and the factor is exact because the units are defined that way.
Digital storage is the trap: a kilobyte is 1,000 bytes in SI and a kibibyte is 1,024 bytes in the binary convention. A “500 GB” drive is about 465 GiB as an operating system displays it, and both numbers are correct under their own definitions. The converter labels the convention rather than picking one silently.
Offset scales: temperature is not a factor
Celsius, Fahrenheit and Kelvin do not share a zero, so a single multiplier cannot convert them. The conversion is affine: multiply by the ratio of degree sizes and add the offset. 0 °C is 32 °F and 273.15 K; a change of 1 °C is a change of 1.8 °F, but the absolute values are not in that ratio. This is why a converter that treats temperature like length produces nonsense at any value except by coincidence.
Ratios: the scale is the shape
An aspect ratio is a comparison, not a unit. 1920 × 1080 simplifies to 16:9 by dividing both sides by 120; 1080 × 1920 is 9:16, the same shape rotated. Scaling one side to a target size keeps the ratio: for a 16:9 box at 800 pixels wide, the height is 800 × 9 ÷ 16 = 450. Fitting inside a box means taking the smaller of the two scale factors so neither side exceeds the target.
The common mistake is treating the ratio as a size. “16:9” does not say how many pixels; a 16:9 thumbnail and a 16:9 cinema screen share a shape and nothing else. Resize first to the ratio, then to the pixel dimensions the slot needs, and remember that a ratio is lost when an image is stretched to a different one.
Averages with weights: grades
A GPA is a weighted average: each course's grade points are multiplied by its credits, the products are summed, and the total is divided by the credits. Measured: grades of 4.0 and 3.0 in courses worth 3 and 4 credits give (12 + 12) ÷ 7 = 3.429. A simple average would give 3.5 and overstate the result, because the better grade was in the smaller course.
Weighted and unweighted GPAs use different scales and different credit totals, so a 3.7 on one is not a 3.7 on the other. A target calculation has to use the same scale as the inputs, and prior coursework has to be combined by credits rather than by averaging two GPAs.
Precision and rounding
Factors are exact, but the displayed result is rounded. The converter stores the factor at full double precision and rounds only the output, which is why 1 mile shows as 1.609344 km rather than 1.61. If a value is going into a calculation, use the full-precision result rather than the rounded display; rounding twice compounds the error. The same rule applies to grades: a GPA rounded to two decimals and then averaged again is not the same as averaging the unrounded values.
What the three share
| Scale type | Operation | Example |
|---|---|---|
| Linear unit | × factor | 1 mi = 1,609.344 m |
| Offset unit | × factor + offset | 0 °C = 32 °F |
| Ratio | divide both sides | 1920:1080 = 16:9 |
| Weighted average | Σ value × weight ÷ Σ weight | (4.0×3 + 3.0×4) ÷ 7 |
The shared discipline is to name the scale and the zero. A conversion is exact only when both are known; a ratio is meaningful only when the shape is preserved; an average is meaningful only when the weights are the right ones.