Chapter 4: Measurement, Units, and Mathematics for Technology
Measurement and the SI
Technology depends on measurement. The International System of Units (SI) has seven base units: metre (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol), and candela (cd). Since 2019, they are defined through fixed values of fundamental constants (Bureau International des Poids et Mesures [BIPM], 2019). Common derived units are the newton (N), joule (J), watt (W), pascal (Pa), hertz (Hz), coulomb (C), volt (V), and ohm (). Prefixes scale units: milli (), micro (), nano (), kilo (), mega (), giga ().
Unit conversions must be done carefully. Examples: ; ; . Always carry units through a calculation; they are a check on the answer.
Significant figures and scientific notation
The significant figures of a number are its meaningful digits. An answer cannot be more precise than the data it comes from. In multiplication and division, the result has as many significant figures as the least precise factor. Scientific notation writes numbers as with , which makes very large and small numbers manageable.
Measuring instruments
| Instrument | Typical least count | Use |
|---|---|---|
| Metre rule / tape | 1 mm | Lengths |
| Vernier caliper | 0.1 mm or 0.02 mm | Inner and outer diameters, depths |
| Micrometer screw gauge | 0.01 mm | Small thicknesses and diameters |
| Stopwatch | 0.01 s | Time |
| Electronic balance | 0.01 g or finer | Mass |
| Multimeter | Depends on range | Voltage, current, resistance |
| Thermometer | 0.5 °C or 1 °C | Temperature |
Good practice: check for zero error, avoid parallax (read the scale with the eye perpendicular to it), repeat readings, and record them with units. Random errors scatter readings and are reduced by repetition; systematic errors shift all readings in one direction and are removed by calibration. The percentage error is .
Algebra and proportion
Technologists constantly rearrange formulae. Example: from , and . Direct proportion means ; inverse proportion means . Percentages and ratios are used in mixtures, efficiency, and scale drawings. Simultaneous equations are solved by substitution or elimination. Indices and logarithms appear in scientific notation, decibels, and pH.
Trigonometry and geometry
In a right-angled triangle, , , , and (Pythagoras). Example: a 5 m ladder at to the ground reaches a height of m.
Mensuration:
| Shape | Area / surface | Volume |
|---|---|---|
| Circle | ; circumference | |
| Rectangle | Cuboid: | |
| Triangle | ||
| Cylinder | Curved area | |
| Cone | ||
| Sphere |
Example: a cylindrical tank of radius 0.5 m and height 2 m has volume L.
Graphs and data
A straight line is , with gradient and intercept . Plot data with a suitable scale, labelled axes with units, and a best-fit line. Rates (such as speed or the rate of a reaction) are gradients. Statistics: the mean is , the median is the middle value, the mode is the most frequent, and the standard deviation measures spread. Example: for 12, 15, 11, 18, 14, the mean is 14.
Common mistakes
Using and (or and ) without converting properly. and .
Reporting too many digits from a calculator.
Using degrees or radians incorrectly on the calculator.
Practice questions
Convert to . [15]
Find the volume of a sphere of radius 3 cm. [113 cm³]
A micrometer reads 2.35 mm with a zero error of mm. Find the true reading. [2.32 mm]