UK Secondary (7-11) Converting Units Very large and small
Lesson Some numbers are very large, like the distance between planets or how far light can travel in a second.  Some numbers are very small like the size of an atom, or the reaction time of a robot.  All of these measurements have units, and hence all also have units.

We already know how to convert units of length, area, capacity, mass and volume.  Now it's time to do these conversions using very large or very small numbers.

One of the easiest ways to do this, is using standard form, which we have already covered.

Recall

Standard Form requires us to use a particular number of significant figures and a multiplication by a power of 10.  This helps us write very large and very small numbers, using only a few digits and symbols.

#### Examples

$123958372=1.24\times10^8$123958372=1.24×108 in $3$3 significant figures

$0.0000029212=2.9\times10^{-6}$0.0000029212=2.9×106 in $2$2 significant figures

One of the easiest ways to connect all these ideas together is through an example, let's try a question.

#### Example

$0.0000621$0.0000621 light years is said to be the distance between Jupiter and the Earth.

a) write this in standard form

b) using the value $9.4605284\times10^{12}$9.4605284×1012 kilometres in one light year, how many kilometres is it between Jupiter and Earth, to $3$3 significant figures.

c) how many metres is this?

Think: to answer this question we need to remember how to write using standard form, how to multiply using standard form, and how many metres there are in a kilometre ($1000$1000).

Do:

a) $0.0000621$0.0000621 is $6.21\times10^{-5}$6.21×105

b) $6.21\times10^{-5}\times9.4605284\times10^{12}=5.87\times10^8$6.21×105×9.4605284×1012=5.87×108 km

c) $5.87\times10^8$5.87×108 km = $5.87\times10^8\times10^3$5.87×108×103 metres (convert to metres)

= $5.87\times10^{11}$5.87×1011 m

#### Worked Examples

##### QUESTION 1

A light year is defined as the distance that light can travel in one year. It is measured to be $9460730000000000$9460730000000000 metres.

1. Write this using standard form.

2. How many kilometres is this? Write this using standard form.

3. How many centimetres is this? Write this using standard form.

##### QUESTION 2

A micrometre (µm) is defined as being a millionth of a metre. This means that $1$1 µm is $0.000001$0.000001 m.

The size of a fog, mist or cloud water droplet is approximately $10$10 µm. How many would fit in a $9$9 cm sample?

The distance from London to Paris is $453$453 km.