6.2.2Radiation
Significant Figures and Scientific Notation
5 min read 5 parts Lesson 14 of 14
Section 1 of 5Scientific Notation
Scientific notation makes very large and very small numbers easier to read and to enter into a calculator.
Form
- A number in scientific notation is written as a × 10ⁿ, where a lies between 1 and 10 and n is a whole number.
Large numbers
- Move the decimal point left until one non-zero digit remains in front, and the number of places moved becomes a positive power.
Small numbers
- Move the decimal point right and the number of places moved becomes a negative power, so 0.00032 = 3.2 × 10⁻⁴.
Multiplying
- Multiply the numbers in front and add the powers of ten, then adjust to standard form if necessary.
Dividing
- Divide the numbers in front and subtract the powers of ten.
Section 2 of 5Significant Figures
Significant Figures
Non-zero digits
- All non-zero digits are significant, so 4 623 has four significant figures.
Zeros between
- Zeros between non-zero digits are significant, so 5 003 has four significant figures.
Leading zeros
- Zeros at the start of a number such as 0.0042 only show the position of the decimal point and are not significant.
Trailing zeros
- Zeros after the decimal point at the end count, so 2.50 has three significant figures.
Rounding
- Look at the digit after the last one you keep; if it is 5 or more, round the last kept digit up.
| Number | Significant figures | Rounded to 2 s.f. |
|---|---|---|
| 0.00706 | 3 | 0.0071 |
| 2.50 | 3 | 2.5 |
| 4 623 | 4 | 4 600 |
| 3.0 × 10⁸ | 2 | 3.0 × 10⁸ |
Section 3 of 5Giving Sensible Answers
1
Carry all the figures on the calculator through a multi-step calculation without rounding in the middle.
2
Round only the final answer, so that rounding errors do not build up.
3
Give the final answer to the same number of significant figures as the least precise piece of data in the question.
4
In practice at National 5, answers to two or three significant figures are usually acceptable.
5
Always include the unit with the final value, and write very large or small answers in scientific notation.
Section 4 of 5Worked Example
Question
- A radio wave has a frequency of 6.0 × 10⁵ Hz and travels at 3.0 × 10⁸ m/s.
- Calculate the wavelength and write it in scientific notation to two significant figures.
- Then write 0.000 072 in scientific notation.
- λ = v ÷ f = 3.0 × 10⁸ ÷ 6.0 × 10⁵.
- Divide the numbers3.0 ÷ 6.0 = 0.5, and subtract powers: 10⁸⁻⁵ = 10³.
- 0.5 × 10³ = 5.0 × 10² m, to two significant figures (500 m).
- 0.000 072 = 7.2 × 10⁻⁵ because the decimal point moves five places right.
Answer
- λ = 5.0 × 10² m;
- 0.000 072 = 7.2 × 10⁻⁵.
Section 5 of 5Exam Checklist & Tips
6.2.2 – Key Points for the Exam
- Scientific notation A way of writing numbers as a value between 1 and 10 multiplied by a power of ten.
- Significant figure A digit that carries meaning about the precision of a measurement.
- Precision How finely a measurement is made, shown by the number of significant figures.
- Rounding Reducing the number of digits in a value while keeping it close to the original.
- Power of ten The factor 10ⁿ in scientific notation.
Common Mistakes
- Counting leading zeros as significant figures.
- Rounding at every step of a calculation instead of only at the end.
- Writing 0.5 × 10³ and treating it as correct scientific notation.
- Giving a result to ten figures straight from the calculator.