🔭 Big numbers: positive power
Move the decimal point left until one digit is in front of it. Count the moves. 56 000 = 5.6 × 104 (four moves).
Maths Skills » Maths Skills: Numbers and Data
What you'll learn this session
Study time: 30 minutes
Edexcel spec: Mathematical skills (Appendix 4), first part
- Write and use numbers in decimal and standard form
- Use ratios, fractions, percentages, powers and roots, and estimate without a calculator
- Use sin and sin−1 in degrees, and give answers to sensible significant figures
- Find a mean, read charts and tables, use simple probability and give an order of magnitude
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Unlock This CourseWhy maths skills matter in physics
None of this is new physics. These are skills for the physics you already know, each with one short example.
Standard form
Physics deals with huge and tiny numbers. Writing 300 000 000 or 0.000 000 5 in full is slow and easy to get wrong. Standard form fixes this.
Key terms:
- Standard form: a number written as A × 10n, where A is at least 1 and less than 10, and n is a whole number (positive or negative).
🔬 Small numbers: negative power
Move the decimal point right until one non-zero digit is in front of it. Count the moves. 0.00032 = 3.2 × 10−4 (four moves).
To go back to a normal number, do the reverse: 2.4 × 105 = 240 000 and 7.5 × 10−3 = 0.0075.
The number part must be between 1 and 10. So 0.5 × 10−6 is not in standard form. It becomes 5 × 10−7.
On your calculator: use the EXP or ×10x key. To enter 3.0 × 108, press 3.0, then EXP, then 8. Do not type × 10 and then the power as a separate number: typing 3.0 × 10 8 gives 240, not 3.0 × 108.
Worked example: standard form
A rocket sled travels 3.0 × 105 m in 6.0 × 102 s. Work out its average speed.
Speed = distance ÷ time = (3.0 × 105) ÷ (6.0 × 102)
3.0 ÷ 6.0 = 0.5 and 105 ÷ 102 = 103, so the answer is 0.5 × 103 m/s.
That is not standard form, so move the point: speed = 5.0 × 102 m/s.
Decimals, ratios, fractions and percentages
Decimals use place value: 0.045 A is smaller than 0.45 A.
⚖️ Ratio
A ratio compares sizes. A current of 5 A splits between two branches in the ratio 3 : 2. Add the parts (3 + 2 = 5), so one part is 1 A. The branches carry 3 A and 2 A.
🍰 Fraction
Find a fraction of an amount by dividing by the bottom and multiplying by the top. 3/4 of 80 J: 80 ÷ 4 = 20, then 20 × 3 = 60 J.
Percentages
Percentage = (part ÷ whole) × 100. A machine takes in 120 J and transfers 30 J usefully. Useful energy as a percentage of the total = (30 ÷ 120) × 100 = 25%.
Powers and roots
A power says how many times to multiply a number by itself: 42 = 16 and 103 = 1000. A square root undoes a square: √81 = 9. Example: a square of side 0.40 m has area 0.402 = 0.16 m2. If v2 = 36 then v = √36 = 6 m/s.
Estimating without a calculator
To estimate, round each number to one significant figure, then calculate. It checks your calculator answer is sensible.
Worked example: estimating
A van travels 19.6 m in 4.9 s. Estimate its speed.
Round: 19.6 is about 20 and 4.9 is about 5.
Speed = 20 ÷ 5 = 4 m/s. The exact answer is 4.0 m/s, so the estimate is close.
Using sin and sin−1 on a calculator
You met sine and inverse sine in Refractive Index. Here the skill is using the keys correctly. Check the calculator shows DEG (degrees), not RAD. In radian mode sin 30 gives −0.988 instead of 0.5.
- To find sin x, press sin, type the angle, close the bracket. sin 40° = 0.643 (3 s.f.).
- To find an angle from a sine value, press shift then sin (this gives sin−1). sin−1(0.5) = 30°.
Worked example: sin and sin−1
A 2.0 m ladder reaches 1.6 m up a wall. Find the angle x between the ladder and the ground, using sin x = height ÷ length.
sin x = 1.6 ÷ 2.0 = 0.80
x = sin−1(0.80) = 53.13°, so x = 53° (2 s.f.).
Significant figures
Significant figures (s.f.) are the digits that carry meaning. Count from the first non-zero digit.
Key terms:
- Significant figures: the digits in a number, counted from the first non-zero digit, that show how precise it is.
✍️ Counting them
0.004 582 has 4 s.f. (4, 5, 8, 2). The zeros before the 4 only hold the place. 12 345 to 3 s.f. is 12 300. A number like 2.0 has 2 s.f.
✅ Rounding
Look at the next digit. 5 or more: round up. Less than 5: leave it. 0.004 582 to 2 s.f. is 0.0046.
Worked example: significant figures
A cyclist travels 25 m in 7.0 s. Speed = 25 ÷ 7.0 = 3.5714... m/s.
The data has 2 s.f., so give the answer to 2 s.f.: 3.6 m/s. Never copy every digit from the calculator.
Rule: give your answer to no more significant figures than the least precise number in the question.
The mean
You used the mean in Practical: Investigating Motion: add up the values, then divide by how many there are.
Worked example: mean
Five timings of a swing: 0.52 s, 0.48 s, 0.55 s, 0.50 s, 0.45 s.
Total = 2.50 s. Mean = 2.50 ÷ 5 = 0.50 s.
Bar charts, frequency tables and histograms
Key terms:
- Bar chart: a chart with separate bars, used when the data falls into categories, such as types of lamp. The height of each bar shows the value.
- Frequency table: a table showing how many times each value, or each group of values, occurs. Here, frequency means a count, not waves per second.
- Histogram: a chart for continuous data (a measurement such as time) grouped into equal ranges. The bars touch, with no gaps.
To draw a bar chart: put the categories along the bottom, the measured quantity up the side with a label and unit, start the scale at zero, and leave equal gaps between bars.
Here is a frequency table of 20 reaction times:
| Time (s) | 0.10 to 0.20 | 0.20 to 0.30 | 0.30 to 0.40 | 0.40 to 0.50 |
|---|---|---|---|---|
| Frequency | 3 | 9 | 6 | 2 |
To draw the histogram, make four touching bars of equal width, with heights 3, 9, 6 and 2. The tallest bar shows the most common group: 0.20 to 0.30 s. Check by adding: 3 + 9 + 6 + 2 = 20.
Simple probability
Key terms:
- Probability: how likely an event is, found as the number of ways it can happen ÷ the total number of equally likely outcomes. It is a number from 0 (impossible) to 1 (certain).
Worked example: probability
A box holds 3 alpha sources, 2 beta sources and 5 gamma sources. You pick one without looking. Probability it is gamma = 5 ÷ 10 = 1/2 (or 0.5, or 50%).
Radioactive decay is random. You cannot say when one nucleus will decay, only how likely it is. Each nucleus has a probability of 1/2 of decaying within one half-life.
Order of magnitude
Key terms:
- Order of magnitude: a number rounded to the nearest power of ten. It tells you roughly how big something is.
Write the number in standard form. If the number part is under about 3, the order of magnitude is that power of ten. If it is over about 3, go up one power.
Worked example: order of magnitude
Light travels at 3 × 108 m/s. Sound in air travels at about 340 m/s. How many times faster is light, to the nearest order of magnitude?
(3 × 108) ÷ 340 = 8.8 × 105.
The number part, 8.8, is over 3, so go up a power: the order of magnitude is 106. Light is about a million times faster.
Common mistakes
Common mistakes
- Leaving an answer like 0.5 × 10−6. The number part must be between 1 and 10.
- Having the calculator in RAD mode for sin questions.
- Writing every digit from the calculator instead of rounding to sensible significant figures.
- Leaving gaps between the bars of a histogram, or starting a bar chart scale above zero.
Exam-style question
(a) Write 0.000 072 m in standard form. [1 mark]
(b) A student times a trolley rolling down a ramp four times: 1.42 s, 1.38 s, 1.46 s and 1.34 s. Calculate the mean time. [2 marks]
(c) A cyclist travels at 4.8 m/s for 19.6 s. Estimate the distance travelled without a calculator. [2 marks]
(d) A ray of light enters glass. Sin of the angle of refraction is 0.60. Calculate the angle of refraction. Give your answer to 2 significant figures. [2 marks]
Model answer
(a) 7.2 × 10−5 m (1)
(b) Total = 1.42 + 1.38 + 1.46 + 1.34 = 5.60 s (1). Mean = 5.60 ÷ 4 = 1.40 s (1)
(c) Round to 5 m/s and 20 s (1). Distance = 5 × 20 = 100 m (1)
(d) Angle = sin−1(0.60) (1) = 37° (1)
Exam tip
In part (d), check your calculator shows DEG before you press sin−1. Wrong mode is the most common way to lose these marks.