Measurement accuracy and scientific notation

No measurement is perfect

When measuring a length with a ruler marked in mm, it is not possible to give a result with infinite precision. The precision of a measurement depends on the instrument used. A ruler marked in mm allows readings to the nearest mm, but not to a smaller value.

Giving a result with too many decimal places (for example, 12.34567 cm using a ruler marked in mm) is an error: it suggests a level of precision that the instrument cannot achieve.

Scientific notation

In physics, some values are very large (the distance between the Earth and the Sun) or very small (the size of a bacterium). To write them simply, we use scientific notation:

a × 10^n, where 1 ≤ a < 10

Examples

Usual value	Scientific notation
150,000 m	1.5 × 10⁵ m
0.0034 kg	3.4 × 10⁻³ kg
8,000,000 s	8 × 10⁶ s

To find the exponent, count the number of places by which the decimal point is moved: to the left → positive exponent, to the right → negative exponent.

Common pitfalls

• Forgetting the condition 1 ≤ a < 10: writing 15 × 10⁴ instead of 1.5 × 10⁵ is not the correct form.
• Forgetting the unit at the end of the result: a value without a unit is meaningless in physics.
• Confusing the instrument’s precision with the number of decimal places displayed by the calculator: the calculator may show 10 digits, but the instrument often only warrants 2 or 3.