Small and Large Units of Measurement

Why these prefixes exist

Nobody wants to write that a computer chip switches in $0.000000002$ seconds, or that a hard drive holds $2\,000\,000\,000\,000$ bytes. The prefixes give the number a sensible size: $2$ nanoseconds, $2$ terabytes.

Every step in the chain is a factor of 1000.

The chain, smallest to largest

Prefix	Symbol	Multiplier	In powers of ten
nano	n	$\div 1\,000\,000\,000$	$10^{-9}$
micro	$\mu$	$\div 1\,000\,000$	$10^{-6}$
milli	m	$\div 1000$	$10^{-3}$
(base unit)	—	$1$	$10^{0}$
kilo	k	$\times 1000$	$10^{3}$
mega	M	$\times 1\,000\,000$	$10^{6}$
giga	G	$\times 1\,000\,000\,000$	$10^{9}$
tera	T	$\times 10^{12}$	$10^{12}$

Two details worth noting: micro uses the Greek letter $\mu$ (mu), and milli is a lower-case m while mega is a capital M. Getting those two confused changes the answer by a factor of a billion.

The only rule you need

Moving to a smaller unit makes the number bigger. Moving to a larger unit makes the number smaller.

That is it. If you are converting metres to millimetres, millimetres are smaller, so the number must grow: multiply. If you are converting bytes to gigabytes, gigabytes are bigger, so the number must shrink: divide.

Do the conversion, then look at your answer and ask whether it moved in the direction you expected. This catches every error.

Counting the steps

Each step in the table is $\times 1000$ or $\div 1000$ . For a jump of more than one step, apply it once per step:

kilo → giga is two steps up: $\div 1000$ twice, so $\div 1\,000\,000$ .
milli → nano is two steps down: $\times 1000$ twice, so $\times 1\,000\,000$ .
milli → kilo is one step up from milli to the base and one more to kilo — count through the table carefully, it is $\div 1\,000\,000$ .

Writing out the chain and ticking off the steps takes five seconds and is far safer than guessing.

Choosing a sensible unit

A good unit gives a number that is easy to say — roughly between 1 and 1000.

Quantity	Sensible	Not sensible
Thickness of paper	$0.1\ \text{mm}$	$0.0000001\ \text{km}$
A film file	$1.4\ \text{GB}$	$1\,400\,000\,000$ bytes
Distance to the Sun	$150$ Gm	$150\,000\,000\,000\ \text{m}$
A dose of medicine	$500\ \text{mg}$	$0.0005\ \text{kg}$

Both columns are correct; only one is useful.

Key rules and formulas

Going smaller

$milli (m) \times 1000 micro (μ) \times 1000 nano (n)$

Each step down makes the unit a thousand times smaller, so the number gets a thousand times bigger. Going back the other way, divide by 1000 each time.

Going larger

$kilo (k) \div 1000 mega (M) \div 1000 giga (G) \div 1000 tera (T)$

Each step up makes the unit a thousand times bigger, so the number gets a thousand times smaller.

The prefixes as powers of ten

nano

= 1 0^{- 9}

, micro

= 1 0^{- 6}

, milli

= 1 0^{- 3}

, kilo

= 1 0^{3}

, mega

= 1 0^{6}

, giga

= 1 0^{9}

, tera

= 1 0^{12}

The sense check

Ask yourself: is my answer bigger or smaller than what I started with? Converting to a smaller unit must give a bigger number. Getting this the wrong way round is the only real mistake available here.

Worked examples

Worked example 1

Nanoseconds to microseconds

Convert

34000

nanoseconds to microseconds.

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

Gigabytes to megabytes

Convert

4 GB

to megabytes.

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

Two steps at once

Convert

800000000 km

to gigametres.

Try it yourself first, then reveal the solution one step at a time.

Worked example 4

Milligrams to micrograms

Convert

70 mg

to micrograms.

Try it yourself first, then reveal the solution one step at a time.

Small and Large Units of Measurement

In this lesson you will learn to…

Why these prefixes exist

The chain, smallest to largest

The only rule you need

Counting the steps

Choosing a sensible unit

Key rules and formulas

Going smaller

Going larger

The prefixes as powers of ten

The sense check

Worked examples

Nanoseconds to microseconds

Gigabytes to megabytes

Two steps at once

Milligrams to micrograms

Practice: units of measurement