Comparing sizes

The diagram shows the size of three organisms, different cells and other structures.

The diagram shows the size of three organisms, different cells and other structures.

We can compare sizes using a straightforward calculation.

For instance, the length of the leaf cell above is ten times the diameter of a red blood cell.

The calculation would be:

\frac{70~\mu m}{7~\mu m } = 10

Question

What is the width of a cheek cell compared with a Salmonella bacterium?

100 times the size.

Calculation:

\frac{50~\mu m}{0.5~\mu m} = 100

Order of magnitude

Differences in size are often described as differences in order of magnitude. That's the difference calculated in factors of 10.

curriculum-key-fact
The 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
0.0l= \frac{1}{100}=10-2
0.001= \frac{1}{1000}=10-3

Any number to the zeroth power is one.

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

Hence there is one order of magnitude between the height of a human being (2m) 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

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