Converting Units across MeasurementsDate: 08/31/2010 at 19:40:33 From: Bally Subject: All About Conversions How do you convert units to a different measurement? For example, the metric system has kilograms and centimeters; and I know that "kilo-" means 1000 and "centi-" means 100. But when and why do you move the decimal spot? Date: 09/01/2010 at 10:01:16 From: Doctor Ian Subject: Re: All About Conversions Hi Bally, This is going to seem like a long explanation, but if you work through it, you'll find that you never have to worry about conversions again. I find the easiest way to think about conversions is to make the multiplication of units explicit. That is, instead of working with something like '3 cm,' I always work with '3 * (1 cm).' How does this help? Well, you can always multiply a quantity by 1, because it doesn't change the value of the quantity. We use this to find equivalent fractions, e.g., 3 5 15 - * - = -- 4 5 20 Going in the opposite direction, if we start with something like 15/20 and identify common factors ... 15 3 * 5 -- = ----- 20 4 * 5 ... we can cancel out pairs that appear in both the numerator and the denominator. That is, if we multiply by 5, and then divide by 5, those operations cancel, so we can get rid of them: 15 3 * 5 3 -- = ----- = - 20 4 * 5 4 Now, suppose we have something like 3 * (1 cm) I can multiply that by 1, but a special form of 1. That is, I can 'make 1' with a ratio of any two quantities that I know are equal to each other. For example, 1 * (1 in) 3 * (1 cm) * ------------- 2.54 * (1 cm) That is, I can do this because I know that 1 inch and 2.54 centimeters are defined to be the same distance. Now, since I'm multiplying by (1 cm) in one place, and dividing by (1 cm) in another place, those operations cancel out, leaving me with the equivalent distance in inches: 1 * (1 in) 3 * -------------- = (3/2.54) * (1 in) 2.54 That is, the unit (1 cm) is a common factor, just like 5 in my earlier fractions. And I can treat it the same way. Does this make sense? It's very, very general, and will work for any kinds of units whatever -- even units you've never heard of. If I tell you that something has a weight of 3 bizzles, and there are 12 bizzles in a dozzle, then you can do this: 1 * (1 dozzle) 3 * (1 bizzle) * --------------- = (3/12) * (1 dozzle) 12 * (1 bizzle) It also works for compound units, like speeds: 30 * (1 mile) 5280 * (1 foot) 1 * (1 hour) 1 * (1 min) ------------- * --------------- * ------------- * ------------ 1 * (1 hour) 1 * (1 mile) 60 * (1 min) 60 * (1 sec) Start with miles per hour Now I have feet per hour Now I have feet per minute And now feet per second After the units cancel, and I do the remaining operations, I've converted 30 miles per hour to an equivalent number of feet per second. (I happen to know that 60 miles per hour is the same as 88 feet per second, so this should work out to 44 feet per second. Try it, and see if that's what you get, just to check your ability to do this kind of calculation.) So far, so good? Now, what about metric units? Well, suppose I want to change centimeters into kilometers. I can change centimeters into meters, ... 1 * (1 m) 3 * (1 cm) * ------------ 100 * (1 cm) ... and then change meters into kilometers, 1 * (1 m) 1 * (1 km) 3 * (1 cm) * ------------ * ------------- 100 * (1 cm) 1000 * (1 m) ... and the result will be in kilometers. I could, in theory, keep stringing these together as much as I want: centimeters to inches, inches to feet, feet to yards, yards to furlongs, furlongs to miles, miles to light years, and so on. I'm just multiplying by 1 and canceling the units I don't want, until I get the ones I do want. Now, what makes metric units easy to deal with is that we're always working with powers of 10, as opposed to quantities like 2.54 (centimeters in an inch) or 12 (inches in a foot) or 4 (quarts in a gallon). So my digits don't change, and I just move the decimal point around. That eliminates a lot of work -- but it doesn't change the principles of what's happening. So what I can do now is cancel my units, 1 1 3 * --- * ---- * (1 km) 100 1000 and I can see that I'm dividing by 100,000. That tells me how far I have to move the decimal place, right? Is it making more sense now? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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