Prefixes
Key idea: Use O-Level Physics prefixes from nano to tera, convert units with powers of ten, and handle squared and cubed units correctly.
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The core idea
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Learning objectives
- Represent a physical quantity with a numerical magnitude and unit
- Recall the six prescribed SI base quantities and their units
- Use the prescribed SI prefixes from nano to tera
- Compare orders of magnitude from a typical atom to the Earth
- Select and justify measuring instruments by range and precision
- Distinguish scalar and vector quantities and give examples
- Add two vectors graphically to determine a resultant
1. Definitions
A prefix placed before a unit represents a power-of-ten multiplier. For example, 1 km = 10³ m.
2. Key Ideas
| Factor | Prefix | Symbol |
|---|---|---|
| 10⁻⁹ | nano | n |
| 10⁻⁶ | micro | μ |
| 10⁻³ | milli | m |
| 10⁻² | centi | c |
| 10⁻¹ | deci | d |
| 10³ | kilo | k |
| 10⁶ | mega | M |
| 10⁹ | giga | G |
| 10¹² | tera | T |
Prefix symbols are case-sensitive: m means milli, while M means mega.
3. Detailed Explanations
- Replace the prefix with its power of ten.
- Carry out the multiplication.
- Write the requested unit and sensible significant figures.
Examples:
25 cm = 25 × 10⁻² m = 0.25 m
250 ms = 250 × 10⁻³ s = 0.250 s
Squared and cubed units
Apply the power to the conversion factor as well as the unit:
1 cm² = (10⁻² m)² = 10⁻⁴ m²
1 cm³ = (10⁻² m)³ = 10⁻⁶ m³
Standard form
Standard form is a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. Move the decimal point until the first factor lies in this range, then count the places moved.
Comparing orders of magnitude
An order of magnitude gives the approximate power-of-ten scale of a quantity. It is useful for comparing sizes even when exact values vary. If a question asks you to compare exponent scales, write each value in standard form and compare the exponents. If it instead defines “nearest power of ten”, follow that stated definition.
| Example | Approximate size | Order-of-magnitude scale |
|---|---|---|
| typical atom | about 10⁻¹⁰ m | atomic scale |
| person | about 10⁰ m | metre scale |
| Earth diameter | about 10⁷ m | planetary scale |
These are scale comparisons, not exact measurements. A change of one order of magnitude means a factor of about ten.
4. Common Mistakes
- Moving the decimal point from memory without writing the prefix as a power of ten.
- Applying a length conversion only once to an area or volume unit instead of squaring or cubing the conversion factor.
- Keeping a prefixed input and an unprefixed input in the same substitution.
5. Exam Tips
- Replace every prefix by its power of ten before substituting into an equation.
- Put squared or cubed conversion factors in brackets before applying the exponent.
- Convert the final answer to the requested unit and check that the direction of the size change is sensible.
6. Worked Examples
Modelled example 1
Small length
Problem
Study the worked solution
Replace the micro prefix
Method
Write μ as 10⁻⁶.Reason
A prefix is a multiplier attached to the unit.Working
13 μm = 13 × 10⁻⁶ mNormalise the coefficient
Method
Move the decimal one place left and increase the exponent by one.Reason
Standard form requires a first factor from 1 up to, but not including, 10.Working
13 × 10⁻⁶ = 1.3 × 10⁻⁵ m
Guided practice 2
Volume conversion
Problem
Apply the conversion to all three dimensions
Hints
Hint 1: bracket the length conversion
Hint 2: cube the whole factor
View solution step by step
Convert the cubic unit
Method
Cube the centimetre conversion factor.Reason
A volume contains three length dimensions.Working
40.0 cm³ = 40.0 × (10⁻² m)³Simplify in standard form
Method
Combine the coefficient with 10⁻⁶.Reason
The requested unit is cubic metres.Working
40.0 × 10⁻⁶ = 4.00 × 10⁻⁵ m³
Common misconception 3
Milli is not mega
Learner response
Read the prefix symbol exactly
View solution step by step
Identify the prefix
Method
Read uppercase M as mega.Reason
Prefix symbols are case-sensitive; lowercase m means milli.Working
M = 10⁶, m = 10⁻³Apply the correct multiplier
Method
Multiply 3.0 by 10⁶.Reason
Removing the mega prefix expresses the same power in watts.Working
3.0 MW = 3.0 × 10⁶ W
Examiner practice 4
Compare orders of magnitude
Examination question
Compare exponents, not coefficients
View solution step by step
Identify both exponents
1 markMethod
Read -3 for A and 4 for B.Reason
Both coefficients already satisfy the standard-form range.Working
n_A = -3, n_B = 4Find the separation
1 markMethod
Subtract the smaller exponent from the larger.Reason
Each exponent step represents one factor-of-ten scale change.Working
4-(-3) = 7 orders of magnitude
Self-mark with the mark scheme
Compare your response with each mark point. Select a point only when your response contains that evidence.
Self-mark the exponents and their separation.
Challenge 5
Density conversion
Compound-unit transfer
Convert numerator and denominator separately
Hints
Hint 1: convert mass
Hint 2: convert cubic centimetres
View solution step by step
Replace both unit factors
Method
Convert grams in the numerator and cubic centimetres in the denominator.Reason
Density is a compound unit, so both parts must become consistent.Working
7.8 g cm⁻³ = 7.8 × (10⁻³ kg)/(10⁻⁶ m³)Combine the powers
Method
Divide the two powers of ten.Reason
10⁻³/10⁻⁶ = 10³.Working
7.8 × 10³ kg m⁻³
Further mistakes to diagnose
- Confusing m (milli) with M (mega).
- Reversing the power when changing unit.
- Squaring or cubing the unit but not its conversion factor.
- Writing standard form with a first factor outside 1 ≤ a < 10.
- Applying a “nearest power of ten” rule when the question asks for the exponent scale, or comparing exponents without first writing both values in standard form.
7. Mind Stretchers
Mind stretcher 1: Convert an areaExtension
A square has side 3.2 cm. Calculate its area in m².
Show answer
Convert the length before squaring:
3.2 cm = 3.2 × 10⁻² m
A = (3.2 × 10⁻²)² = 1.024 × 10⁻³ m²
To two significant figures, A = 1.0 × 10⁻³ m².
Mind stretcher 2: Compare exponent scalesExtension
The radius of Earth is about 6.4 × 10⁶ m and an atom’s diameter is about 1.0 × 10⁻¹⁰ m. Using the exponents in standard form, how many orders of magnitude separate their scales?
Show answer
The exponents are 6 and -10:
6-(-10) = 16
Their exponent scales are separated by 16 orders of magnitude. This does not mean their exact size ratio is precisely 10¹⁶; the coefficients also affect the ratio.
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Course and syllabus information
- Course
- SEC G3 Physics
- Edition
- SEC G3 Physics 2027