Materials Science · Units, loads, stress and strain
Units, scales and orders of magnitude
Compare a thin coating with a sheet, convert lengths, areas and volumes without changing their physical meaning, and use powers of ten to catch scale errors.
Useful background: Materials, questions and levels of evidence. Each worked case explains the quantities it uses.
Back to the chapterAfter reading, you can
- Explain why the number and unit must be read together, and interpret positive and negative powers of ten.
- Convert lengths, areas and volumes using factors equal to one.
- Use a dimensionless ratio to compare scales and count factors of ten.
- Distinguish a unit conversion from a change of object size or an improvement in measurement precision.
A thin coating and a thick sheet
A coating 2 micrometres thick is much thinner than a 2 millimetre sheet. The repeated number does not make them equal: a micrometre is one millionth of a metre, while a millimetre is one thousandth. Convert both into the same unit before comparing.
A measurement value joins a number to a unit. Writing the sheet thickness as 2 mm, 2000 µm or 0.002 m changes the number used to describe it; the sheet stays the same thickness. A smaller unit requires more units to cover the same length.
Powers of ten shorten these statements. The expression 10³ means 10 × 10 × 10 = 1000. A negative exponent means a reciprocal: 10⁻³ = 1/1000 = 0.001. Thus 2 × 10⁻³ m means 2 multiplied by one thousandth of a metre. It is a small positive length, not a negative length.
| Unit | Size in metres | Another way to read it |
|---|---|---|
| millimetre, mm | 10⁻³ m | one thousandth of a metre |
| micrometre, µm | 10⁻⁶ m | one millionth of a metre |
| nanometre, nm | 10⁻⁹ m | one billionth of a metre |
Convert by multiplying by one
Since 1 mm = 10⁻³ m, the ratio (10⁻³ m)/(1 mm) equals one. Multiplying a length by this ratio changes its written unit without changing the physical quantity. For example, 2 mm × (10⁻³ m/mm) = 0.002 m: the mm symbols cancel.
Choose the ratio the other way up when you need the other conversion. To turn 0.002 m into millimetres, multiply by (1000 mm/m). Keep the units in the calculation until the cancellation is visible. This is more reliable than remembering whether to move a decimal point left or right.
Only quantities of the same kind can be directly converted this way. A length in millimetres cannot be converted into an area in square millimetres without geometric information. Units are part of the reasoning, not labels added after the arithmetic.
An order of magnitude is a multiplication
For two lengths, divide the larger by the smaller after expressing them in the same unit. The unit cancels, leaving a ratio. A ratio of 10 is one factor-of-ten step, 100 is two steps and 1000 is three steps. We call these differences of one, two and three orders of magnitude.
For the sheet and coating, (2 × 10⁻³ m)/(2 × 10⁻⁶ m) = 1000. Their common leading number cancels; the powers leave 10³. The sheet is three orders of magnitude thicker. This says nothing about which material is stronger or better: it is a comparison of thickness alone.
When a ratio is not an exact power of ten, report the ratio and make any approximation explicit. A ratio of 600 lies between 100 and 1000, so the difference is between two and three orders of magnitude. It is not exactly three. The logarithmic scale below gives equal spacing to equal multiplicative steps.
Scale factors act on every dimension
A length conversion affects area twice and volume three times. One square millimetre is 10⁻⁶ square metres; one cubic millimetre is 10⁻⁹ cubic metres. An order of magnitude is a factor of ten, not an added ten units. Keep powers attached to the unit as well as the number.
To see why, imagine a square with sides 1 mm. Each side is 10⁻³ m, so its area is (10⁻³ m) × (10⁻³ m) = 10⁻⁶ m². A cube contains a third length factor: (10⁻³ m)³ = 10⁻⁹ m³. The prefix belongs to the unit before that whole unit is squared or cubed.
A real change of size is a different operation. If every length of a geometrically similar solid doubles, its area grows by 2² = 4 and its volume by 2³ = 8. Merely converting the same solid from mm to m changes neither area nor volume. Keep physical resizing separate from rewriting a measurement.
| Quantity | Conversion factor | Result for one unit |
|---|---|---|
| Length | 10⁻³ | 1 mm = 0.001 m |
| Area | (10⁻³)² = 10⁻⁶ | 1 mm² = 0.000001 m² |
| Volume | (10⁻³)³ = 10⁻⁹ | 1 mm³ = 0.000000001 m³ |
Check the kind, size and precision of the answer
Before accepting a result, ask three questions. Is the final unit the right kind, such as m² for area? Does a change to a smaller unit give the expected larger number? Does converting back recover the starting value? These checks often expose a missing squared or cubed conversion factor.
Conversion factors between these SI prefixes are exact, but an actual thickness measurement is not made more accurate by conversion. If 2 mm is a rounded reading, writing 0.002000000 m does not establish extra measured detail. Keep measurement precision and unit size as separate ideas; later measurement lessons develop uncertainty explicitly.
Worked examples
Count the scale difference before adding thicknesses
A sheet is 2 mm thick and receives a 2 µm coating on one face. Treat the given dimensions as idealised values. How many times thicker is the sheet than the coating, and what is their combined thickness?
- Convert the coating: 2 µm = 2 × 10⁻⁶ m = 0.002 mm. Keep the sheet as 2 mm so both thicknesses now use the same unit.
- Divide like quantities: sheet/coating = 2/0.002 = 1000. Since 1000 = 10 × 10 × 10, the difference is three orders of magnitude.
- Add only after conversion: 2 mm + 0.002 mm = 2.002 mm. The coating adds one thousandth of the sheet thickness, so the result should be only slightly greater than 2 mm.
- The sum follows from the stipulated dimensions; a real measurement report would also consider their rounding and uncertainty. Adding the original numbers as 2 + 2 would instead confuse two different unit sizes.
The sheet is 1000 times thicker, or three orders of magnitude thicker. The idealised total is 2.002 mm.
Convert a cross-section two ways
Convert a rectangular cross-section 2 mm wide and 3 mm thick into square metres.
- Calculate in the starting units: area = 2 × 3 = 6 mm².
- Convert the complete area unit: 6 mm² × (10⁻³ m/mm)² = 6 × 10⁻⁶ m². Two mm factors cancel.
- Check by converting the sides first: (0.002 m) × (0.003 m) = 0.000006 m², the same result.
- Using only one factor of 10⁻³ would give 0.006 m², a thousand times too large. It would fail to convert both dimensions.
The area is 6 × 10⁻⁶ m². Converting each side or the complete area gives the same physical area.
A cube checks a cubed conversion
A cube has edges of 2 mm. Find its volume in mm³ and m³. A worksheet instead reports 8 × 10⁻³ m³; how can you detect the error?
- Calculate V = 2 × 2 × 2 = 8 mm³.
- Each edge is 0.002 m, so V = (0.002 m)³ = 8 × 10⁻⁹ m³. Equivalently, multiply 8 mm³ by (10⁻³ m/mm)³.
- The worksheet used a length conversion once where a volume requires it three times. Its reported volume is one million times the correct value.
- As a reverse check, 8 × 10⁻³ m³ is the volume of a cube with 0.2 m edges, or 200 mm edges. That is not a 2 mm cube.
8 mm³ = 8 × 10⁻⁹ m³. The incorrect result describes a cube 100 times larger in every length.
Try the reasoning
A feature is 350 nm wide. Express its width in µm and compare it with a 1.4 µm feature. Does the ratio imply a whole-number difference in orders of magnitude?
Show the answer and reasoning
350 nm = 0.350 µm; the 1.4 µm feature is four times wider. The difference is less than one order of magnitude.
There are 1000 nm in 1 µm, so divide 350 by 1000. The ratio is 1.4/0.350 = 4, which is smaller than 10. A size ratio need not be an exact power of ten.
A cube doubles every edge. Another description of the original cube simply converts every edge from mm to m. Compare what happens to volume in these two operations.
Show the answer and reasoning
Physical doubling increases volume eightfold. Unit conversion leaves the original physical volume unchanged.
The first operation multiplies three physical lengths by 2, giving 2³ = 8. The second changes the number and unit together; its cubed conversion factor preserves the same volume.
A coating patch is 4 mm long and 500 µm wide. A learner writes its area as 2000 mm². Find the area in mm² and m², and explain the mistake.
Show the answer and reasoning
500 µm = 0.5 mm; the area is 2 mm² = 2 × 10⁻⁶ m².
The product 4 × 500 has units mm·µm, not mm². Convert the width first, or convert that compound unit correctly. The learner kept the number from mixed units and silently changed its meaning.
One layer is 20 µm thick and another is 2 mm thick. Find their ratio and order-of-magnitude difference. Why does subtracting the written numbers fail?
Show the answer and reasoning
The thicker layer is 100 times thicker, a difference of two orders of magnitude.
2 mm = 2000 µm, and 2000/20 = 100 = 10². Subtracting 2 and 20 mixes units; even subtracting matched-unit values gives an absolute difference, not the multiplicative ratio that defines the scale comparison.
What to carry forward
- A number and its unit describe one quantity; changing units does not resize the object.
- Negative powers of ten describe reciprocals, and each order-of-magnitude step is a factor of ten.
- Convert a whole area or volume unit with a squared or cubed factor.
- Use matching units for a ratio, and check both dimensional consistency and plausible size.
References
Teaching resources and research supporting this lesson’s explanations and examples.