### Dimensions and conversion of units

#### Definition

A *dimension* is a measurement of a particular kind, for example height or weight. Scientists use the SI Units to record measurements.

Knowing the dimensions of an object is important when performing calculations. You cannot add or subtract two amounts if they are not of the same dimension.

- For example, “£1 + 1kg” is undefined and has no meaning.
- However, 100kg - 50kg = 50kg.

You can however multiply or divide amounts that are not the same dimension.

An example is velocity: $V = \dfrac{\text {Mass}}{\text{Time}}$

#### Manipulating units

Sometimes you are given two measurements of the same kind, for example distance, but the units aren't the same (e.g. miles and kilometers). To perform calculations you will need to convert them into the same units.

For example to convert miles to kilometers. There are approximately 1.61 km in one mile so to convert 50 miles into km you multiply 50 by 1.61.

- i.e. $1.61\times50 \text{miles} \approx 80$km.

##### Worked example

Suppose you want to place an order for rabbit food for June (in kilograms), on average a rabbit eats 30 grams of concentrated food a day, you have 300 rabbits, how much concentrated rabbit food should you order?

##### Solution

Per day, your rabbits eat $30\text{g}\times300=9000$g of food.

here are 30 days in June so you need $9000\text{g}\times30=270000$g of food.

However the order has to be in kgs. There are 1000g in 1kg, so we need to divide by 1000.

Converting to kilograms: $\frac{270000}{1000}\text{g}=270 \text{g}$.

**You need to order $270$ kg of rabbit food for June**.

##### Worked example

If a particular compound has molecular weight (RMM) 62 then one mole weighs 62g. How many moles are there in 93μg?

##### Solution

To perform this calculation you need to convert 62g into μg or 93μg into grams.

There are $10^{-6}$g in one μg.

So $93$μg is $93 \times 10^{-6}$g.

$\frac{93\times10^{-6}}{62}$g = $1.5 \times 10^{-6}$g.

**There are $1.5\times10^{-6}$g or $1.5$ μmol in this compound.**

For more complex examples of this type see Molar calculations.

#### SI unit prefixes

k |
kilo |
1000 |

d |
deca |
One-tenth, $10^{-1}$ |

c |
centi |
One-hundredth, $10^{-2}$ |

m |
milli |
One-thousandth, $10^{-3}$ |

μ |
micro |
One-millionth, $10^{-6}$ |

##### Examples

- 1 kg = 1000 g
- 1 mg = 0.001 g
- 1 cm = 0.01 m
- 1 m = 1,000,000 μm

#### Test yourself

Try our Numbas test on manipulating units.

Try our Numbas test on Units of measurement.

#### See Also

For further information on units see: SI units table.