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Another concept that is important is that of units

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Unformatted text preview: label method is based on a very simple idea; the idea of unity. Yep, unity, the number one. One is the loneliest number that you’ll ever do. Two can be as bad as one, it’s the loneliest number since the number one. At least this is what “Three Dog Night” would have us believe, but in fact, one is the most powerful number that you’ll ever do. It has some wonderfully mystical and powerful properties, although these properties are so well known and so common that most have never even really considered them. If you multiply a number by one, the number is not changed. If you divide a number by one, the number is not changed. And the way that we write the number one can be varied, twisted and bizarre. Sometimes one is assumed. Take the number 12. The number 12 implies 12*1, 1*12, or even 12/1. It’s this last representation that we’ll dwell on. We like the representation 12/1, because it allows us to write division by 12 as a multiplication. For instance, 3/12=3*(1/12). So dividing by a number is identical to multiplying by the fraction 1 divided by that number. You’ll see how this idea becomes significant a little later, but first I would like to continue with another way in which we can represent the number one. As we know, twelve is a number. It is a number, in fact, equal to 12. But what is 12 divided by one dozen? Well, since 1 dozen=12, then Dakota State University page 219 of 232 Factor Label Method General Chemistry I and II Lab Manual 12 12 = =1 1dozen 12 Then, isn’t writing 12/dozen a form of writing 1? And since there are 2.54 cm in each inch, then isn’t 2.54 cm / 1 in another form of writing 1? Or 5,280 ft / mile, or 1 yd /3 ft all ways of writing the number one? And aren’t these constants all equal to dozen/12, 1 in/2.54 cm, 1 mile/5280 ft, or even 3 ft/ yd, respectively, since, after all, 1/1=1? We call these quantities “conversion factors.” A conversion factor is a fractional representation of the number 1. Since they are representations of 1, we are free to change the numerator and denominator at will, which you will eventually see is a very important characteristic in the factor label method. Another concept that is important is that of units being algebraic quantities. If we consider units to be algebraic quantities, then we can “cancel out” equivalent units in numerators and denominators. For instance, everybody knows that if we have 3 feet, and we want to find out how many inches that is, than we simply multiply 3 by 12. However, in the factor label, we multiply 3 feet by 12 inches/foot. Recognizing the equivalents of feet and foot (feet is just plural foots!), then we can write 3 foot * 12inch = 36inch 1 foot Notice how "foot" cancels, since it is in both the numerator and denominator, just as an algebraic quantity "x" would cancel if it were both in the numerator and denominator, as in x* y =y x The factor label method in action: So now we've got it. Conversion factors are equivalent regardless of what is in the numerator and what is in the denominator, and units cancel algeb...
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