Standard form & significant figures
Physical Quantities & Units - OCR A-Level Physics
Key Definition
Standard form
A way of writing any number as $A \times 10^{n}$, where $A$ is a decimal between $1$ and $10$ (not including $10$) and $n$ is an integer. Standard form turns very large or very small physical quantities into a compact, comparable shape.
A way of writing any number as $A \times 10^{n}$, where $A$ is a decimal between $1$ and $10$ (not including $10$) and $n$ is an integer. Standard form turns very large or very small physical quantities into a compact, comparable shape.
$$x = A \times 10^{n}, \quad 1 \le A < 10, \quad n \in \mathbb{Z}$$
Procedure: into and out of standard form
Number bigger than one: count the digits after the first digit. That count is $n$. Place the decimal point after the first digit; drop any trailing zeros. Example: $25\,700\,000 = 2.57 \times 10^{7}$.
Number smaller than one: count the zeros up to and including the first non-zero digit. That count, made negative, is $n$. Example: $0.000\,062 = 6.2 \times 10^{-5}$.
Number smaller than one: count the zeros up to and including the first non-zero digit. That count, made negative, is $n$. Example: $0.000\,062 = 6.2 \times 10^{-5}$.
Significant figures
Significant figures (s.f.) are the meaningful digits in a measurement, counting from the first non-zero digit. They indicate precision: $4.50 \text{ m}$ (3 s.f.) is more precise than $4.5 \text{ m}$ (2 s.f.).
- Use standard form for very large or very small quantities: $c = 3.00 \times 10^{8} \text{ m s}^{-1}$, $e = 1.60 \times 10^{-19} \text{ C}$.
- The number of significant figures in your answer should match the least precise piece of data. If data is given to 3 s.f., quote your answer to 3 s.f.
- Intermediate steps should keep one extra figure to avoid rounding errorError introduced when a number is shortened to fewer digits. Compounds over multi-step calculations if rounding happens too early.. Only round at the final step.
- Leading zeros are not significant: $0.0042$ has 2 s.f.
- Trailing zeros after a decimal point are significant: $4.50$ has 3 s.f.
- In standard form the ambiguity disappears: $4.50 \times 10^{2}$ is clearly 3 s.f.
Common Mistake
MEDIUM
Wrong: Giving a final answer to 6 significant figures when the data was given to 2 s.f.; or rounding intermediate values to 2 s.f. and then losing accuracy by the final step.
Right: An answer cannot be more precise than the least precise piece of input data. Quote the final answer to the same s.f. as the data, but keep one extra digit in every intermediate step.
Right: An answer cannot be more precise than the least precise piece of input data. Quote the final answer to the same s.f. as the data, but keep one extra digit in every intermediate step.
Examiner Tips and Tricks
- If the question stem says "give your answer to an appropriate number of significant figures", match the s.f. of the input data.
- OCR usually penalises by one mark if your answer is more than one s.f. away from the mark scheme value.
- Set your calculator to display values to four or five significant figures during working; round only when writing the final line.