Metric prefixes, converting units, and derived units
What this lesson covers
- The number of significant figures in a measurement is the number of digits that contribute to the degree of accuracy of the measurement, given as the number of digits known with certainty plus one uncertain digit.
- When calculating, it is best to use at least one more significant figure than is required in the final solution, and round at the end of the whole calculation.
- Conversions between different-sized metric units may be needed to give the appropriate units for a measurement or calculation.
- The metric prefixes kilo–, mega–, giga–, and tera– signify a unit that is one thousand, one million, one billion, and one trillion times larger than the base unit.
- The metric prefixes centi–, milli–, micro–, and nano– signify a unit one hundredth, one thousandth, one millionth, and one billionth the size of the base unit.
- Derived units (e.g. cm², km/h) reflect a relationship (a product or quotient) between two different measurements.
One system, not forty conversions
- A kilometre is not a different kind of length from a metre.
- It is a thousand of them. That is the only difference.
- So every metric conversion is the same move: a power of ten.
The big prefixes
- Kilo is a thousand. Mega is a million.
- Giga is a billion, tera is a trillion.
- As powers: three, six, nine, twelve. Steps of three.
| Prefix | As a power | In words |
|---|---|---|
| kilo | × 10³ | a thousand |
| mega | × 10⁶ | a million |
| giga | × 10⁹ | a billion |
| tera | × 10¹² | a trillion |
And the small ones
- Milli is a thousandth. Micro is a millionth.
- Nano is a billionth, which is where computer chips live.
- Same steps of three, going the other way.
| Prefix | As a power | In words |
|---|---|---|
| centi | × 10⁻² | a hundredth |
| milli | × 10⁻³ | a thousandth |
| micro | × 10⁻⁶ | a millionth |
| nano | × 10⁻⁹ | a billionth |
Which way does the number go?
- Ask one question: is the new unit smaller or bigger?
- Smaller unit, you need more of them, so the number goes up.
- Bigger unit, fewer of them, number goes down.
| Going | Because | Effect |
|---|---|---|
| Going to a smaller unit | you need more of them | number goes up |
| Going to a bigger unit | you need fewer | number goes down |
| 2.5 km to metres | metres are smaller | 2 500 |
| 450 mm to metres | metres are bigger | 0.45 |
Three conversions
- Two point five kilometres is two thousand five hundred metres.
- Four hundred and fifty millimetres is nought point four five metres.
- Kilometres to centimetres is two steps, not one.
| Conversion | Doing | Answer |
|---|---|---|
| 2.5 km → m | × 10³ | 2 500 m |
| 450 mm → m | ÷ 10³ | 0.45 m |
| 3.4 km → cm | × 10³ then × 10² | 340 000 cm |
| Count the steps | not the zeros |
The prefix that breaks the pattern
- Kilo, milli, micro and nano all step by a thousand.
- Centi does not. It is a hundredth.
- So a metre is a hundred centimetres but a thousand millimetres.
| Prefix | Step |
|---|---|
| kilo, milli, micro | all steps of 1 000 |
| centi | a hundredth, not a thousandth |
| So 1 m is 100 cm | but 1 000 mm |
| The habit of moving 3 places | breaks on centimetres |
Prefixes are powers of ten wearing names
- Three point two gigabytes is three point two times ten to the ninth.
- The prefix and the power are the same information.
- One is a word for talking, one is a number for calculating.
| Written | Because | Same as |
|---|---|---|
| 3.2 GB | giga is 10⁹ | 3.2 × 10⁹ bytes |
| 2 500 000 000 bytes | count back | 2.5 GB |
| Prefixes and powers | are the same idea | |
| One is a word | the other is a number |
Choosing a sensible prefix
- Nought point nought nought nought nought nought four metres.
- Nobody can read that. Four micrometres, and everyone can.
- Pick the prefix that puts the number between one and a thousand.
| Written badly | Problem | Better |
|---|---|---|
| 0.000 004 m | hard to read | 4 micrometres |
| 5 000 000 m | hard to read | 5 000 km |
| Pick the prefix | that gives 1 to 1 000 | |
| Same length | easier to say |
Units made out of other units
- A square centimetre is a centimetre times a centimetre.
- Kilometres per hour is kilometres divided by hours.
- The unit is telling you what calculation made it.
| Unit | Means | Kind |
|---|---|---|
| cm² | cm × cm | a product |
| km/h | km ÷ h | a quotient |
| m/s | metres ÷ seconds | a quotient |
| g/cm³ | grams ÷ cm³ | density |
Converting a rate
- Seventy-two kilometres per hour. Convert the top and bottom.
- Seventy-two thousand metres, in three thousand six hundred seconds.
- That is twenty metres per second.
| Part | Converting | Gives |
|---|---|---|
| 72 km/h | the top: 72 km | 72 000 m |
| the bottom: 1 h | 3 600 s | |
| So | 72 000 ÷ 3 600 | 20 m/s |
| Shortcut | km/h ÷ 3.6 | gives m/s |
Where the divide by 3.6 comes from
- A thousand metres over three thousand six hundred seconds.
- That ratio is the same every time, and it works out at 3.6.
- So the shortcut is not a separate rule. It is that division.
What a significant figure is on a ruler
- It is the digits you are certain of, plus one you are guessing.
- A tape marked in centimetres: you read the centimetres.
- Then you estimate the millimetre. That last digit is the uncertain one.
| Definition | In practice |
|---|---|
| Digits you are sure of | plus one you are not |
| A tape marked in cm | you read the cm, guess the mm |
| So that last digit | is always the uncertain one |
| Carry one spare figure | and round at the end |
Carry one spare figure
- If you need three figures in the answer, work with four.
- Then round once, at the very end of the whole calculation.
- Rounding early is where precision quietly disappears.
What to take away
- Seven lines, and the fourth one catches most people.
| Metric is one system | with a power of ten in front |
| kilo, mega, giga, tera | 10³, 10⁶, 10⁹, 10¹² |
| centi, milli, micro, nano | 10⁻², 10⁻³, 10⁻⁶, 10⁻⁹ |
| Centi is the odd one | a hundredth, not a thousandth |
| To a smaller unit | the number gets bigger |
| A derived unit | is a product or a quotient |
| km/h to m/s | divide by 3.6 |
Your turn: four questions
- Pause here and work all four.
- Question 1 is two steps, and the second is a hundred.
- Question 3 can be done either way round.
| Question | |
|---|---|
| 1 | Convert 3.4 km to centimetres. |
| 2 | Write 0.000 004 m with a prefix. |
| 3 | Convert 90 km/h to m/s. |
| 4 | Write 2 500 000 000 bytes with a prefix. |
Show answer
1. 340 000 cm. Kilometres to metres is times a thousand, giving 3 400, and metres to centimetres is times a hundred. Two steps, and the second one is a hundred rather than a thousand. 2. 4 micrometres. Micro is ten to the minus six, and 0.000 004 is 4 times ten to the minus six. 3. 25 m/s. Either convert both parts, 90 000 metres over 3 600 seconds, or use the shortcut and divide by 3.6. 4. 2.5 gigabytes. Giga is ten to the ninth, and 2 500 000 000 is 2.5 times ten to the ninth.
The answers
- Three hundred and forty thousand centimetres.
- Four micrometres.
- Twenty-five metres per second.
- Two point five gigabytes.
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