Interesting Math Column - V

Another column in the NY Times series by Cornell mathematician Steven Strogatz. This week it's The Joy of X and about algebra.  One thing that popped out for me is his mention of thinking about how units will cancel out when discussing a conversion of feet into yards problem.

Converting units involves dimensional analysis (it's easier than it sounds).  One of the complaints among science faculty at the community college where I teach is that students coming out of high school have no idea how to do dimensional analysis and needlessly struggle through what should be very simple algebraic problems.  A common refrain heard around here is "What exactly are they teaching these kids in high school science classes?"

Even in my relatively math-free introductory geology classes, students often struggle to do the simplest problems.  For example, working with maps, we might have to convert feet to meters.  Maybe we have an elevation of 750 ft and we need to know how many meters that equals.  Their lab manual may list a conversion factor 2.54 cm = 1 in.  What now?  Well, it's assumed that people know there are 12 inches in 1 foot and that there are 100 centimeters in 1 meter.  Many students will then take 750 ft and proceed to semi-randomly multiply and divide numbers until they get some sort of answer. 

Sometimes, they even write down, as an answer, a numerically ridiculous answer like 0.246062992 m (they did all division and reported all the decimals on their calculator screen) or 2,286,000 m (they did all multiplication).  I've seen it over and over again.  That's what the calculator said so it must be true.

I always stress that these problems become trivially easy if you simply multiply through by 1 and pay attention to units you're crossing off.  What do I mean by multiplying by 1?  Well, if there are 2.54 centimeters in 1 inch (2.54 cm = 1 in), then we can set up the fraction (2.54 cm / 1 in) or (1 in / 2.54 cm) and both fractions equal 1.  Why do they equal 1?  Because the top and bottom terms are equal to each other.  We can set up similar fractions for 12 in = 1 ft - (12 in / 1 ft) or (1 ft / 12 in) - and 100 cm = 1 m - (100 cm / 1 m) or (1 m / 100 cm).  Now we just set up a series of fractions so we can cross off units going from feet to meters.

750 ft x (12 in / 1 ft) x (2.54 cm / 1 in) x (1 m / 100 cm) = Answer in meters

Why does this work?  Because units cross off as you go left to right.  First feet (ft):

750 ft x (12 in / 1 ft) x (2.54 cm / 1 in) x (1 m / 100 cm) = Answer in meters

Then inches (in):

750 x (12 in / 1) x (2.54 cm / 1 in) x (1 m / 100 cm) = Answer in meters

Then centimeters (cm):

750 x (12 / 1) x (2.54 cm / 1) x (1 m / 100 cm) = Answer in meters

And, finally, the only units left are meters (m).

750 x (12 / 1) x (2.54 / 1) x (1 m / 100) = Answer in meters

Now just multiply all the top terms together, all the bottom terms together, and divide top by bottom:

(750 x 12 x 2.54 x 1 m) / (1 x 1 X 100) = 22,860 m / 100 = 228.5 m

So 750 feet is equal to 228.6 meters.

It's a little more laborious to write out this way, but you'll never again wonder "Do I multiply by 100 or divide by 100?"  It's obvious if you set the units up so that cross out top and bottom.  Here's a website that has more simple examples.