Understanding the Exponent, and Other Ways Math Illiteracy Screws You Over
The Modern Survival Guide #70
This is the Modern Survival Guide, a guidebook I’m writing for things I think people need to know about living in the modern world. The views expressed here are mine, and mine alone. And I think that math is extremely important, in fact critical, to our lives, so in this article I want to talk a little bit about how math affects us.
GET BACK HERE, WE’RE NOT DONE YET. ALL I DID WAS SAY “MATH.”
Seriously though, there is a famous statement attributed to a conversation Stephen Hawking was having with his editor, prior to publishing his masterpiece A Brief History of Time, in which he was told that “every equation I included would halve the sales.” Why? Because we don’t like to deal with math. Math is hard, you guys. Math is for nerds. Math makes brain hurty.
So it’s kind of a damn shame that many of us are functionally math illiterate in an age when our survival is dependent on tolerance statistics in machinery, dosage limits in medicine, whatever insane black magic goes into computers, and investment portfolios with varying interest and return ratios. Not to mention things like nuclear physics, climatology estimates, and political science game theory.
Look, it’s like this: math is all around us. We can hide from it if we want to, but modern life demands some degree of competence in mathematics, at least if you want to live, live well, and avoid the traps and pitfalls of charlatans. It does us no good to pretend that we shouldn’t understand some basic mathematical concepts. So let’s see if we can fix that, shall we?
Here are some mathematical things that we all should know:
- Time management, and how that runs our lives
- Measurement conversions, and what they mean for making things
- Percentages, and how they affect our worldview
- Probability equations, and what they do to our standards for risk
- Exponents, and what they do to our investments (and debts)
I tried to rank these in order to complexity. This isn’t an exhaustive list, and I’ll try to get through these without undue reliance on equations. But there will be equations. Gird your loins, or whatever you normally do when faced with maths, but please do read on.
Time Management, and How it Runs Our Lives
Let’s start with a look at time management, and basic arithmetic. A lot of people think of time management as a function of willpower, but it’s actually more of a function of math. Just willing yourself to be on time doesn’t work. Calculating how long it takes to do things and then using that information to plan your day, on the other hand, does.
Let’s take a practical example: how long does it take you to get to work in the morning?
Well, for some people that calculation might look like this: wake up (5 min) + brush teeth (5 min) + take shower (10 min) + dress (10 min) + eat breakfast (30 min) + get to the car (5 min) + drive to work (45 min) + walk to work space (5 min) = total time of 115 minutes, or just under two hours. So if you’re due at work at 9, better set the alarm for 7.¹
The willpower part of time management is all about making a list of things you need to do, figuring out how much time they take, figuring out a plan to do all those things, and then sticking to the plan. But you can’t do that if you don’t have a plan, and you can’t have a plan if you don’t do the math to figure out how long everything is going to take.
I am an extremely punctual person, in the sense that I arrive everywhere pretty much when I want to get there (if I’m late I usually meant to be late, because a wizard arrives when he means to). I do this by keeping a running tally of how long it ought to take to do whatever I’m about to do, and then working backward to figure out when I need to start doing things. I couldn’t do that without basic arithmetic.
Measurement Conversions, and What They Mean for Making Things
Just about everything in life is measurable, and some things in life are intentionally designed to be measurable. Building materials and cooking ingredients, for example, are typically marketed, packaged, and used in standardized quantities.
The problem is, the standards aren’t yet standardized. As a result, we often need to use multiplication or division to figure things out when we’re making a cake or building a house.
In the US we have to use both Imperial and Metric measurements. Imperial measures are in terms of inches, feet, miles, ounces, pints, quarts, gallons, etc. Metric measures are in terms of millimeters, centimeters, meters, kilometers, grams, kilograms, liters, etc. In general, you can tell the different between them by noting that metric units of measure scale by increments of 10 (100 millimeters is a centimeter, 1,000 grams is a kilogram, etc.) and imperial units… don’t.
Seriously, why is a mile 5,280 feet? Why are there 16 ounces in a pound? What sense does that make?²
Now for the problem: since we use both measurement systems in the US, we often have to convert between them. This requires math because, since one measurement system is based on tens and the other one isn’t, there’s often no clean and easy way to convert between them.
This has some consequences if you’re not careful. If all you’ve got is imperial measuring cups, a cake recipe that calls for things in metric units is going to be interesting. How many grams of flour is in a cup? How many cups does the recipe call for? Let’s say the recipe calls for 1,000 grams of flour, and (I looked it up) a cup is 340 grams. That means our recipe calls for:
1,000 g / 340 g = 2.94 cups, and good luck getting that one right.
This is fun when you have to convert international recipes. It’s also fun when you’re trying to land spaceships on Mars. In case we all forgot, back in 1999 NASA lost a Mars lander because somebody in the engineering shop forgot to convert from imperial to metric measurements in their sensor software.
So yeah, watch your conversions. They’re easy to screw up, and it sometimes takes careful math to get them right. Measurement conversions usually require some knowledge of multiplication and division, or the ability to operate Google; ideally, both.
I’ve included a couple of equations at this point, by the way, so by Stephen Hawking’s estimate that means 25% of you are still reading this. So if you’re still reading this, congratulations! You beat out 75% of the population!
Percentages (And How They Affect Our Worldview)
What does it mean to consume 200% of your daily value of vitamin C in a glass of orange juice? Is that a good thing or a bad thing? What does it mean that 40% of the electorate thinks the President is doing a good job? Is that a good statistic or a bad statistic? What does it mean that 90% of the tap water tested in Flint, Michigan is below an “action level” for lead content? Is that good water quality or bad water quality?
Look, percentage measurements are all around us, they’re not going away, and they’re one of the most common ways we judge, well, everything. It behooves us to understand what they mean. So let’s break this down.
A “percentage” measurement means that you are measuring something in parts per hundred, with 0% being no parts per hundred and 100% being one hundred parts per hundred, or to put it another way, the whole thing. Percentage measures may also be expressed in decimals — 0.01 is the same as 1%.³
Some math teachers like to make the argument that you can’t have more than 100% of something, and that is correct but also wrong. You can have more than 100% of something, it’s just that after 100% you have more than one thing.
So — in the case of our orange juice, if you are consuming 200% of your daily value vitamin C from one serving, that means you’re getting twice as much vitamin C as you need per day from that serving. 100% would be your full daily value; 200% is twice the full daily value. 200% is therefore overkill, and you should find out whether consuming that much vitamin C is a good idea before you have another glass.⁴
In general, Americans are taught that more is better. This is why juice makers advertise that you’re getting 200% of your daily value of vitamin C (or whatever nutrient is the flavor of the week) in their products. And this highlights a key point about percentage measurements:
Percentages are a useless form of measurement unless you know what the hell you’re talking about.
Because if you don’t know the details of the idea, product, survey, or statistic in question, all you’re really finding out from a percentage measure is what part out of one hundred the thing you’re measuring is when compared to something else. This can very quickly become what my old stats professor liked to call “information-free content.”
So — when you see a percentage measurement, unless you know what the hell you’re talking about with regard to the thing being measured, do not make a judgement based on the percentage! Instead, figure out what the hell you’re talking about. Then and only then, make a judgement.
Probability and Risk Ratios
Most people are not good at probability assessment — they’re not good at looking at a situation, figuring out what is likely or unlikely to happen, and then acting based on that information. But in an increasingly complicated and uncertain world, probability becomes a very important thing to grasp, because life is risky and living long is all about decreasing the risks.
Let’s say, for example, that you want to make a choice between driving a car and driving a motorcycle. Which one is safer to drive? Well, if we look at the statistics, we can see that motorcycles are involved in fatal crashes at a higher rate than passenger cars.
What does this mean? Does that mean that motorcycles are grossly unsafe? Well, yes — for a given value of “unsafe.” According to 2006 data, there were roughly 13 fatal crashes per 100,000 cars, compared to roughly 72 fatal crashes per 100,000 motorcycles.
The way you calculate probability is to take the number of events that you want to track and divide that into the total number of events that happen. So if we say that there were 13 crashes per 100,000 cars, that translates to 13/100,000, or 0.00013, or a 0.013% chance of dying in a car crash for you, each year, if you drive a car. That compares to 72/100,000, or .00072, or a 0.072% chance of dying in a motorcycle wreck.
All of which is to say that the odds of getting in either a fatal car wreck or a fatal motorcycle accident aren’t that great — but it’s worse for motorcyclists.
Now for the hard part: how to do you make a decision based on this information? Well, if you want to make a decision based on risk reduction, you go with the car, hands down. And this is because this math isn’t really concerned with how likely it is that you get in a wreck, but rather with how likely it is that you die in a wreck.
Note that I at no time indicated how many motorcycle accidents actually happened in these calculations. I didn’t need to do this, in order to make a risk judgement. This is because probability calculations are usually normalized — they are designed to use common standards of comparison, in this case 100,000 accidents. Because I’m making a judgement on which vehicle is safer, not how many accidents happen, I don’t need to know the actual number of accidents.
Contrast this with the so-called “Cheney Doctrine” of the early 2000s, which stated that if there was even a 1% chance of an attack on America we were obliged to act preemptively to stop it, because eventually that 1% would come up. This was an original justification for the “War on Terror,” and it was dumb. But why do I say that? Didn’t we just say that we were prepared to make a risk assessment on a vehicle based on a 0.072% chance of a fatal accident (i.e. something that will happen to roughly 7 people out of every 10,000)?
Yes we did, but again — look at the assumptions. Our probability statement regarding cars vs. motorcycles was concerned with which one was safer if an accident occurred. For the purposes of that decision, we assumed the accident had already occurred when making judgement. Cheney’s doctrine makes the mistaken assumption that any event that has a tiny probability of occurrence will occur.⁵
This was a terrible risk management strategy, because the benefits of mitigating these tiny risks came with costs that far outpaced those of the risk itself. Let’s say that the CIA decides that Iran has a 1% chance of deploying a nuclear warhead against the US. Cheney’s response, if he had followed his doctrine, would have been to invade Iran — an act which, if they did have nukes, would virtually guarantee they would be used, and would in any case cost trillions of dollars and thousands of lives. For 1%.
These are key factors to understanding probabilities of any kind — before you make a decision, make sure you understand what is being measured. If you’re dealing with risks, make sure that your judgement is based on the risk, not the sum total of possible events. And make sure that your response is proportional to the possibility of risk, and not based an assumption that the risk will occur.
Exponents: What They Do to Investments and Debts
Last but not least, it has been said that compound interest is the most powerful force in the universe. However, unless you know how compound interest works, that statement is just elitist gibberish. So let’s talk about exponential progression.
No, seriously, we’re doing this, it’s the last one, just power through. But first here’s a cat video. I needed that. I think you did too.
An exponent is a numeric mark that shows the power to which a given number or expression is to be raised. In English, that means that this — ^2 — is an exponent, and if you append it to a number or formula that means, in math-speak, that you are multiplying that number or formula by itself once. Like this:
2 ^2 means 2 multiplied by 2, or 2 x 2, which equals 4.
2 ^3 means 2 x 2 x 2 = 8. And so on, and so forth. Cool? Cool.
You’ll also see exponents attached as superscripts, like this: 2²
Fine, so what does this have to do with interest rates and debt, you are no doubt asking while desperately searching for the willpower to continue reading. Well, this is how interest rates are calculated. Interest rates are calculated on a timed basis, usually annually, and the formula looks like this:
We can see here that the exponent is used to multiply the equation times itself, over and over, for however many years you need to calculate. What this means is that the longer you let an investment (or debt) accumulate interest, the faster it grows. And the rate of interest becomes super important, because the higher the interest rate, the faster it grows.
And this is how they get you. A high interest rate means a progressively higher number if you don’t pay off the principal, and that turns a debt into a lifelong burden. This is why 6% interest is much worse than 3%, and why you should never, ever get behind payments on a credit card with 25% interest.
Keep an eye on the exponents. As we saw in this example, they tend to mark things that get progressively bigger or smaller with time, and they tend to represent feedback loops — something getting bigger in an exponential equation will get bigger very, very quickly.
Everything is Math, Deal With It
Wrapping this up, let’s just accept it: almost everything that we do, day to day, has math stamped all over it. Even if you don’t think about it, or perhaps especially because most of us don’t recognize it, it’s important to realize that math is everywhere and everything.
We can’t afford to ignore it. We can’t afford to trivialize it. We can’t afford to hand it off to some dude with thick glasses and a graphing calculator, because eventually that guy will figure out asset-backed derivative stocks, and then we get the 2007 recession. We need to understand enough math to keep track of the world.
This isn’t easy; I’m not asking you to do easy things in this series. I personally do not like math. But I understand its importance, and I think that we all need to understand the ways that math affects our world if we want to stand much of a chance of thriving in the modern age.
Math is everywhere. Math is everything. And math illiteracy will screw you over. So buy some books, listen to podcasts, and open up the YouTube, because we don’t have an excuse not to understand it in the Information Age.
¹For ladies, add time for makeup, if you put on makeup.
²There’s actually a super long, complex answer to the mile question.
³Because 0.01 represents 1/100th of 1 by decimal measure — the 1 is in the “hundredths” place. Therefore 0.1 represents 10% (because the 1 is in the “tenths” place) and 1 represents 100%.
⁴This is a serious problem with vitamins, by the way — your body can only absorb so much per day, and consuming more than that amount doesn’t help you and can actually cause problems. You don’t want to consume too much vitamin A, for example — it can be toxic. This is also why vitamin pills aren’t particularly effective or medically advisable most of the time — they contain a higher concentration than your body can absorb in pill form, so you just eliminate most of the dose.
⁵The idea that any probability or statistical estimate will eventually come true is only true in very basic iterative games, which is to say events that repeat over and over again in the exact same way, with the exact same probability. Yes, if you keep flipping a coin for long enough, eventually you’ll get 50% heads and 50% tails. But life isn’t usually a coin flip.
