Question

Suppose that, from measurements in a microscope, you determine that a certain bacterium covers an area of 1.50 μm2. Convert this to square meters.

Answer 1

Convert \(1.50 \mu m^2\) to \(m^2\)

I do unit conversion by fractions. Set up an identity relating the two units:

\(1 \mu m= 10^{-6} m\) Divide both sides by the unit you want to get rid of:

\(\dfrac{10^{-6}m}{1 \mu m} = 1\)

Now we can use that fraction to convert any micrometers we see into meters by multiplying. The value of the fraction is 1, so we don't affect the equation's truth at all, just the units. As we are converting square micrometers to square meters, we will have to multiply by the conversion fraction twice. The units will cancel properly if we set up the equation correctly, and there won't be any question about whether we divided when we should have multiplied, or vice versa.

\[1.50 \mu m^2 * \frac{10^{-6}m} {1 \mu m}*\frac{10^{-6}m} {1 \mu m} \]Observe that the units cancel the way we want:\[1.50 \cancel{\mu m^2} * \frac{10^{-6}m} {1 \cancel{\mu m}}*\frac{10^{-6}m} {1 \cancel{\mu m}} = 1.50 * 10^{-6} * 10^{-6} m^2\]

Can you finish the job?

Answer 2

Notice that if we had put the fraction upside down, or we didn't realize that we needed to multiply by it twice (due to the measurement being one of area rather than length), the units would not cancel out to give us a reasonable unit. We might end up with \(\mu m^4/m^2\) or \(\mu m\text{ }m\), and that would be the "canary in the coal mine" telling that the answer is incorrect. You don't get that protection if you just start multiplying and dividing by powers of 10 (for metric units) or strange numbers (for US/Imperial units). A little extra work in exchange for an answer which is more likely to be correct has always seemed like a good tradeoff to me!