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Close Enough

My college math dilemma

I was reading this post by John D. Cook and it got me thinking about math and measuring errors.

I distinctly remember the first time I was bothered by measurement error. It was in a college calculus class. The problem was something like “If a carton of milk has 1 PPM of bacteria, the bacteria  grows at a rate of 5 percent per day, and the milk is spoiled when it reaches 5 PPM of bacteria, how long before the milk is spoiled?”, but the details don’t matter.

I was used to problems like “If you have a wall 8 feet by 20 feet and paint covers the wall at a rate of 1 gallon per 100 square feet, how many gallons do you need?”

Can you see the difference that I saw?

Here is what I thought then. The bacteria problem is widely speculative. First of all, how do we know there are exactly 1 PPM bacteria? Secondly, we know bacteria reproduces at different speeds based upon temperature. What if the temperature varied even slightly over the course of the experiment? And certainly bacteria on average may reproduce at a rate of 5 percent per day, but that cannot be exact. They are living things. Some might be better than others at reproducing. Some might die quicker. How can we say “5 percent per day”?

However, the paint example is precise, right? You can measure a wall exactly. You can measure paint coverage rates exactly, right? They are just material properties. No variance depending upon growth rates or differences in cells.

So it really bothered me that I had to answer the milk problem. I mean, I could give an answer that was close, but this was math class. I needed an exact answer.

I went ahead and answered it, but it bothered me. As my time at college progressed and I took more and more engineering courses, I had to answer more and more questions like this. Why even bother with all this engineering and math stuff if we are only spitting out close answers, not right answers?

Well, let me tell you two things that I learned later, but wish I would have learned earlier.

You will never get the right answer

Yep. That’s right. Never. In Mathtopia you can get exact answers, but not here in the real world. You know why? GIGO. Garbage in, garbage out. From the paint example above, you will get the wrong answer if you say “1.6 gallons”. Why? Do you really know how much paint you have? Is it really exactly 1.6 gallons? Or is it maybe a little more or a little less? Who measured it? What about the wall size? Is there really a wall on earth that is exactly 8 feet by 20 feet? 96 inches by 240 inches?  2438.4 mm by 6096mm? Is it perfectly square? Are the 4 sides perfectly straight? Of course not.

When you come right down to it, this is just like the milk example. The only difference is that  I was accepting that in Mathtopia, we knew the exact size of the walls and the exact coverage of the paint. If we lived in Mathtopia, we could have exact answers. Unfortunately, the world of Mathtopia only exists in math books.

So where do we go from here? Why even bother if we can never get the right answer? That leads to point two.

It’s close enough

I don’t know about you, but I can only go to Lowe’s and buy paint in 3 sizes: 1 quart, 1 gallon, or 5 gallons. How much paint do we need for the problem above? 8 feet x 20 feet is 160 square feet, at 100 square feet per gallon means 1.6 gallons, right? Now go to Lowe’s and ask for 1.6 gallons of paint and see where that gets you. Maybe you can get exactly 1.6 gallons of paint in Mathtopia, but not at my Lowe’s.

In this case, you would get 2 gallons. Yes, you could technically buy 1 gallon and 3 quarts, 1.75 gallons, with .15 gallons to spare, but the 3 quarts will cost more than 1 gallon. I always buy a gallon, even if I only need a quart. It’s a well-known fact that when you own a house, it’s mandatory that you have lots and lots of leftover gallon paint cans. That way, when you need to touch up a spot, you can spend a few hours looking through them not quite sure which one is the proper match. And when you find the proper match, it will be mostly dried out and have rust from the lid floating in it. Trust me, you need two gallons. But I digress…

I haven’t addressed other questions that need to be answered, such as “Are you are painting with, a brush, roller, or sprayer?”,  “Is the wall textured?”,  “Are you covering a dark color with a light one?”,  “Is it just an experienced adult painting or are you letting your eight-year-old help?”, “Are you going to use one or two coats?”

How you answer every one of those questions will affect how much paint you need in the real world. So what you should probably do here is get two gallons and move on.


Oh, and if the milk is near the expiration date, doesn’t smell quite right, or you are just not sure, you should probably throw it out. No need to check the bacteria levels.


As a kid, I used to spend weeks with my cousins in Riverview Gardens, a suburb of St. Louis. Just a few miles away is a town called Ferguson.

In light of all the news about the upcoming grand jury decision in Ferguson, here is a good post from this summer from John Piper.