0.9 Conversions by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.

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## 0.9 Conversions

Conversions are one of the three essential mathematical skills, summarized on pp.532-533, that you need for success in this course.

I suggest you avoid memorizing lots of conversion factors between SI units and U.S. units, but two that do come in handy are:

1 inch = 2.54 cm

An object with a weight on Earth of 2.2 pounds-force has a mass of 1 kg.

The first one is the present definition of the inch, so it's exact. The second one is not exact, but is good enough for most purposes. (U.S. units of force and mass are confusing, so it's a good thing they're not used in science. In U.S. units, the unit of force is the pound-force, and the best unit to use for mass is the slug, which is about 14.6 kg.)

More important than memorizing conversion factors is understanding the right method for doing conversions. Even within the SI, you may need to convert, say, from grams to kilograms. Different people have different ways of thinking about conversions, but the method I'll describe here is systematic and easy to understand. The idea is that if 1 kg and 1000 g represent the same mass, then we can consider a fraction like

`(10^3 g)/(1 kg)`

to be a way of expressing the number one. This may bother you. For instance, if you type 1000/1 into your calculator, you will get 1000, not one. Again, different people have different ways of thinking about it, but the justification is that it helps us to do conversions, and it works! Now if we want to convert 0.7 kg to units of grams, we can multiply kg by the number one:

0.7 kg×`(10^3 g)/(1 kg)`

If you're willing to treat symbols such as “kg” as if they were variables as used in algebra (which they're really not), you can then cancel the kg on top with the kg on the bottom, resulting in

0.7 kg×`(10^3 g)/(1 kg)`=700 g.

To convert grams to kilograms, you would simply flip the fraction upside down.

One advantage of this method is that it can easily be applied to a series of conversions. For instance, to convert one year to units of seconds,

1 year×`(365 days)/(1 year)`×`(24 hours)/(1 day)`×`(60 min)/(1 hour)`×`(60 s)/(1 min)`=3.15×10^{7} s.

### Should that exponent be positive, or negative?

A common mistake is to write the conversion fraction incorrectly. For instance the fraction

`(10^3 kg)/(1g)`(incorrect)

does not equal one, because `10^3`kg is the mass of a car, and 1 g is the mass of a raisin. One correct way of setting up the conversion factor would be

`(10^(−3) kg)/(1 g)`(correct).

You can usually detect such a mistake if you take the time to check your answer and see if it is reasonable.

If common sense doesn't rule out either a positive or a negative exponent, here's another way to make sure you get it right. There are big prefixes and small prefixes:

big prefixes: | k M |

small prefixes: | m μ n |

(It's not hard to keep straight which are which, since “mega” and “micro” are evocative, and it's easy to remember that a kilometer is bigger than a meter and a millimeter is smaller.) In the example above, we want the top of the fraction to be the same as the bottom. Since `k` is a big prefix, we need to *compensate* by putting a small number like `10^(−3)` in front of it, not a big number like `10^3`.

◊ Solved problem: a simple conversion — problem 6

◊ Solved problem: the geometric mean — problem 8

##### Discussion Question

◊ Each of the following conversions contains an error. In each case, explain what the error is.

(a) 1000 kg×1 kg1000 g=1 g

(b) 50 m×1 cm100 m=0.5 cm

(c) “Nano” is `10^(−9)`, so there are `10^(−9)`nm in a meter.

(d) “Micro” is `10^(−6)`, so 1 kg is `10^6` μg.

0.9 Conversions by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.