Maths - orders of magnitude
Comparing sizes
The diagram shows the size of three organisms, different cells and other structures.
Sizes can be compared using a straightforward calculation.
For instance, the length of the leaf cell above is 10 times the diameter of a red blood cell.
The calculation would be:
\(\frac{70~\mu m}{7~\mu m } = 10\)
When comparing the size of a bacterium with the Human Immunodeficiency Virus (HIV), different units have to be converted to be consistent.
The length of the bacterium = 1 μm = 1000 nm
The diameter of the HIV = 100 nm
Therefore the length of the bacterium is \(\frac{1000}{100}\) = 10 times that of the bacterium.
Question
What is the width of a cheek cell compared with a Salmonella bacterium?
140 times the size.
Calculation:
\(\frac{70~\mu m}{0.5~\mu m} = 140\)
Order of magnitude
Differences in size are often described as differences in A number, to the base ten, often used to make comparisons. For each order of magnitude, a number is ten times the previous one.. That's the difference calculated in factors of 10.
Notation used to write numbers in multiples and sub-multiples of 10
| 1000 | = 10 × 10 × 10 | = 103 |
| 100 | = 10 × 10 | = 102 |
| 10 | = 10 | = 101 |
| 1 | = 1 | = 100 |
| 0.1 | = \(\frac{1}{10}\) | = 10-1 |
| 0.01 | = \(\frac{1}{100}\) | = 10-2 |
| 0.001 | = \(\frac{1}{1000}\) | = 10-3 |
| 1000 |
| = 10 × 10 × 10 |
| = 103 |
| 100 |
| = 10 × 10 |
| = 102 |
| 10 |
| = 10 |
| = 101 |
| 1 |
| = 1 |
| = 100 |
| 0.1 |
| = \(\frac{1}{10}\) |
| = 10-1 |
| 0.01 |
| = \(\frac{1}{100}\) |
| = 10-2 |
| 0.001 |
| = \(\frac{1}{1000}\) |
| = 10-3 |
If you increase a number by one order of magnitude, you are multiplying the number by 10.
If you decrease a number by one order of magnitude, you are dividing the number by 10, which is equivalent to 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
This means there is one order of magnitude between the height of a human being (2 m) and the height of an oak tree (20 m).
Notice that when dividing numbers in standard form, we subtract the powers.
When comparing orders of magnitude, actual distances can be approximated. It's the relative difference that is important.
Question
What is the difference in order of magnitude for a human hair and HIV?
The width of a human hair is 100 μm, while the width of the HIV is 100 nm.
The difference in order of magnitude is 3.
Calculation:
The width of the human hair is 100 μm = 10-4 m
The width of the HIV is 100 nm = 10-7 m
So, there are three orders of magnitude difference:
\(\frac{10^{-4}}{10^{-7}} = 3\)
Or, more simply:
4 - (-7) = -4 + 7 = 3
