08 Lesson 7: Multiples and Submultiples of Units; Scientific Notation
- Physical quantities can be gigantic or tiny.
- The distance between stars is about 100 000 000 000 000 000 m, while the spacing of atoms in a solid is about 0.000 000 001 m.
- Such numbers are hard to read, so we use powers of ten (scientific notation): a × 10ⁿ with 1 ≤ a < 10.
- The two distances become 1 × 10¹⁷ m and 1 × 10⁻⁹ m.
The powers of ten have agreed names (prefixes):
1Prefix
10⁻⁹
- nano
10⁻⁶
- micro
10⁻³
- milli
10⁻²
- centi
10³
- kilo
10⁶
- mega
10⁹
- giga
2Symbol
10⁻⁹
- n
10⁻⁶
- µ
10⁻³
- m
10⁻²
- c
10³
- k
10⁶
- M
10⁹
- G
Method of conversion. Replace the prefix by its power of ten and simplify.
Worked example 1. Express 7 mA in µA. 1 mA = 10⁻³ A and 1 µA = 10⁻⁶ A. Dividing: 1 mA/1 µA = 10³, so 1 mA = 10³ µA. 7 mA = 7 × 10³ µA = 7000 µA.
Worked example 2. A wire is 0.35 mm in diameter. In metres: 0.35 × 10⁻³ m = 3.5 × 10⁻⁴ m.
✍️ Worked examples
Worked example 3. A hard disc stores 2.5 GB (gigabytes). In bytes: 2.5 × 10⁹ B.
Worked example 4. Convert 108 km/h to m/s. 1 km/h = 1000 m ÷ 3600 s = 5/18 m/s. So 108 × 5/18 = 30 m/s.
Writing numbers in scientific notation (practice).
- Mass of a car ≈ 1 200 kg = 1.2 × 10³ kg.
- Thickness of a leaf: 0.000 12 m = 1.2 × 10⁻⁴ m.
- Mass of the Earth ≈ 6 000 000 000 000 000 000 000 000 kg = 6 × 10²⁴ kg.