Units are your Friends

Copyright © Karl Dahlke, 2023

In science, and physics especially, they teach you to track and write down all your units, and you can have points taken off your test it you don't include the units. If it's pounds say pounds, if it's kilograms say kg. Just a number won't do. A lot of folks find this an unnecessary bother, but … in 1999 we had a 400 million dollar spacecraft crash into Mars, because two teams were working together and neither one of them wrote down the units. One thought it was newtons of thrust, and one thought it was pounds, and that's all it took. So the units are important after all.

My high school chemistry teacher, better than most of the profs I had in college, said, “Units can help you. They can be your friend. You can know the formula is right. It's not just a number at the end of a calculation, you can know you're doing it right.” I've carried that wisdom ever since.

Let's look at a real world example. Say you have 7 gallons, and your car gets 25 miles per gallon, how far can you go? The word per means divide, it means a fraction, it means a slash if you have to write it in line. Miles per gallon, m divided by g, m above g if you want to write it on paper as a fraction, m/g in this inline ascii notation. The performance of your car is 25, but not just 25, 25m/g, 25 miles per gallon. If you knew the miles you drove, and you divided it by gallons consumed, you'd always get 25. Except for traffic jams of course. 😀

Now let's go back to our question: you have 7 gallons, how far can you go? Remember the units, they are your friends. Start with 7 gallons, or 7g. You might think multiplying 7 by 25 would be right, but how do you know? Carry the units along, in a process called dimensional analysis. They will tell you.

7g times 25m/g = 175mg/g

this would be clearer if I could write it out as fractions on a piece of paper. When you multiply fractions you multiply the tops and bottoms. The top of 7g is just 7g. The top of 25m/g is 25m. The bottom of 25m/g is g. Multiply the tops and get 7g times 25m = 175mg. The bottom is the g from the 25m/g. Ok, so 175mg/g, or 175mg over g.

The cool thing about fractions is, the same thing on the top and bottom you can cross out. It's like this. 5 divided by 5 is 1. 12 divided by 12 is 1. 198 divided by 198 is 1. Anything divided by itself is 1. So g divided by g is 1. The g on top and the g on the bottom go away. That leaves 175m, or 175 miles. The unit on your answer is miles, which is what you want, so your formula is right. That's how units can help you. It gives you that extra confidence. It's not just a bother and waste of time, it's a help.

Next pretend you want to know how many gallons it takes to go 300 miles. This is the problem in reverse. Now you want an answer with g on top, you want to know gallons. You have miles and want gallons.

Ok here's something else about fractions. If you want to divide by two thirds, 2/3, you flip it over and multiply by three halfs, by 3/2. It's the rule for dividing fractions. To divide by x/y, you multiply by y/x. Example, your recipe calls for 4 cups of flour, and dangit you can't find your one cup measure. If you had it, just use 4 of those, and you'd be done, but you can't find it. Only the 2/3 cup measure, that's all you have. How many of those do you need? How many 2/3 cups are in 4 cups? Let's divide, and don't forget the units, c for cups. 4 cups divided by 2/3 cups. 4c divided by 2c/3. Dividing by a fraction is multiplying by its reciprocal. Flip the second one over and multiply. So 4c times 3/2c. Multiply tops and get 12c. Multiply bottoms, well only the second has a bottom, that is 2c. Thus 12c over 2c. The c's go away. Cups over cups. You are left with the number 6, and no units. There should be no units cause you are just counting how many cups are in cups. So that's right. The answer is 6, 6 of those 2/3 cups gives you 4 cups.

Now let's return to the mileage problem, and carry the units. Divide your distance, 300 miles, by 25 mpg. Divide 300m by 25m/g. That means we flip it over. Multiply 300m by the fraction g over 25m. Remember we multiply tops and bottoms. The top becomes 300mg. The bottom is 25m. The m on top and the m on the bottom cancel. The top is 300g, and the bottom is 25. So it's time for the math part, 300 divided by 25 is 12. The g is still there, and you need 12 gallons. You need 12 gallons to go 300 miles. The units help you know you're doing it right. Carry the m's and the g's along and the unit should come out as you want, in this case g for gallons.

Now, I like to know an easy number for miles per dollar. It's my own measure I suppose, but it's the one that counts. How far can you go for this much money? So it would be nice to have m/d, miles per dollar. Let's say you knew m/d, maybe it's 6, and you have 200d that is $200, and multiply to see how far you can go. Carry the units along.

200d times 6m/d

The top is 200m times 6d = 1200md. The bottom is just the d. d cancels d, the answer is 1200m, m is miles, m is what you want, the formula is right, the answer 1200 miles.

But how did I get the handy dandy ratio of miles per dollar in the first place? Again, units are your friends. You know your car's mpg. I've been calling it 25. Gas prices change by the day, but say 4 dollars per gallon. You sort of want a worst case number here, so you don't get stranded. We write that as 4d/g. 4 dollars per gallon. We look at what we have, we have a number with units m/g and a number with units d/g. What can we do with those? Divide the first by the second. 25m/g divided by 4d/g. Dividing by a fraction means multiply by its reciprocal. Flip it over. 25m/g times g/4d Multiply tops and bottoms. The top is 25mg. The bottom is 4gd. There's a g on top and on bottom. Cancel them out. The top is 25m, and the bottom is 4d. The units are miles per dollar. That's what I want! So finish the math. 25 divided by 4 is 6.25. At high gas prices, and mediocre highway driving, you get a little over 6 miles for each dollar spent. As I say, this is rather a worst case. You can calculate a more accurate m/d ratio based on your car's mileage and current gas prices.

Units are helpful when converting between English and metric systems, or between any two comparable systems. The conversion between miles and kilometers is approximately 1.6 kilometers per mile. This is something you would look up, it's not necessarily something you would know or could figure out. Two separate systems for measuring distance evolved, and there is a way to convert from one to the other. Your friend is in a 10k run, i.e. 10 kilometers. You want to know how many miles he is running. Do you multiply by 1.6, or divide by 1.6? Not sure? The units will help you. The distance is 10k, 10 kilometers. If you multiply by 1.6k/m you'll get 16kk/m, which isn't at all what you want. Divide by 1.6k/m and get 6.25km/k, or 6.25m, which is miles, which is what you want. He is running 6 and a quarter miles.

There are even conversions within the metric system itself. The joule and the calorie are both units of heat, or energy. Remember that heat is just a form of energy, and at the end of the day, almost all energy winds up as heat. When you watch television, the screen emits light, but that light enters your eyes and becomes chemical energy in your retinas and your brain. A tiny fraction of this energy is stored in memory, if you remember what you watched, but most of it becomes heat. The sound enters your ears, and most of that becomes heat as well. Two years later, when you have forgotten the show completely, those memories degrade into heat. All is heat.

Physicists developed the joule, a unit of energy based on meters and seconds and mass, while chemists developed the calorie, a unit of heat based on mass and temperature. Since energy is heat, these units are interoperable. The calorie is more intuitive; it is the amount of heat needed to raise one gram of water one degree celsius. A gram of water is about the size of a sugar cube, so picture that much water, and raise its temperature by 1 degree C, or almost 2 degrees F. That is the heat of a calorie. A joule is the energy of a newton of force acting through a distance of one meter. As I say, it's less intuitive. There is a conversion, which you would look up. 0.239 joules in every calorie, 0.239 joules per calorie, 0.239j/c. We'll use this conversion in the next chapter.