Maths - number, size and scale
Comparing sizes
We can compare size using a straightforward calculation.
For instance, the length of a leaf cell is ten times the diameter of a red blood cell.
That is:
\(\frac{\text{70 μm}}{\text{7 μm}}=10\)
In scientific calculations it is essential to remember that you must work in the same units. In the calculation above both measurements are in micrometres. However in the calculation below it is much easier if both values are shown in the same units. So 1 micrometre is converted into 1000 nanometres.
When comparing the size of a bacterium with the Human Immunodeficiency Virus (HIV):
The length of the bacterium = 1 μm = 1000 nm.
The diameter of the HIV = 100 nm.
The length of the bacterium is \(\frac{1000}{100}\) = 10 times that of the virus.
Question
What is the width of a cheek cell compared with a Salmonella bacterium?
Calculation:
\(\frac{\text{70 μm}}{\text{0.5 μm}}=140\)
Order of magnitude
When two numbers are similar, we say they have the same order of magnitude.
Differences in size can be described as differences in order of magnitude. The difference is often calculated in factors of 10.
If you increase a number by one order of magnitude, you are multiplying the number by 10.
For example, we would say that the numbers 200 and 300 are of the same order of magnitude whereas the numbers 200 and 2000 are of different orders of magnitude.
200 and 300 are both in the magnitude of 102 whereas 2000 is in the magnitude of 103.
If you decrease a number by one order of magnitude, you are dividing the number by 10, or multiplying by 0.1.
For instance, there is a one order of magnitude difference between a person 2 m tall, and an oak tree, 20 m tall.
The person's height = 2 m = 2 × 100
The oak tree's height = 20 m = 2 × 101
The oak tree is approximately 10 times bigger than the person. We can also say this as there is an order of magnitude between the height of a human being (2 m) and the height of an oak tree (20 m).
When comparing orders of magnitude, actual distances can be approximated. It's the relative difference that is important.
How to convert between different scientific units
| Unit | How many in a metre? | In standard form |
| Centimetre | 100 | 1/100 or 1 × 10−2 |
| Millimetre | 1000 | 1/1000 or 1 × 10−3 |
| Micrometre | 1 000 000 | 1/1 000 000 or 1 × 10−6 |
| Nanometre | 1 000 000 000 | 1/1 000 000 000 or 1 × 10−9 |
| Unit | Centimetre |
|---|---|
| How many in a metre? | 100 |
| In standard form | 1/100 or 1 × 10−2 |
| Unit | Millimetre |
|---|---|
| How many in a metre? | 1000 |
| In standard form | 1/1000 or 1 × 10−3 |
| Unit | Micrometre |
|---|---|
| How many in a metre? | 1 000 000 |
| In standard form | 1/1 000 000 or 1 × 10−6 |
| Unit | Nanometre |
|---|---|
| How many in a metre? | 1 000 000 000 |
| In standard form | 1/1 000 000 000 or 1 × 10−9 |
Centimetres are odd units in they don't fit the pattern of reducing in size by 1000 each time. There are one thousand micrometres in one millimetre, and one thousand nanometres in one micrometre.
