Numbers dominate our lives. Over Thanksgiving Day weekend an estimated 43.6 million people traveled several billion miles to consume approximately 46 million turkeys across the nation . If we did nothing but count the number of turkeys raised in the United States this year alone (254,000,000 in total) on the very hands we use to trace them, it would take nearly a decade. Yet, despite all this, very few of us actually understand where our numbers come from, how they are found or what they mean.
It’s quite likely that not a single reader stopped to question (let alone verify) whether the numbers stated above were true. Luckily for all of us, I don’t intend to pull one over on anyone and these statistics come straight from the American Automobile Association and the U.S. Census Bureau. However, this only leads us to ask the next and more pertinent question: How did they arrive at their numbers? As with just about any number, there are three methods: One can count, one can calculate or one can approximate.
Counting is the first approach we’re taught. It consists of finding the number of elements for some finite set of objects by increasing a counter by a set unit for each element. This is a more intuitive concept than the above definition would lead you to believe. If you have some number of turkeys plopped down in front of you, if you wish to count them, it’s necessary to mentally represent each turkey with a single digit and then go up the number scale by one for each new turkey. Whether the number is represented in binary in a computer or counted by a robot in a turkey factory, counting establishes a one-to-one correspondence between the elements of one set (number of turkeys) and the elements of another set (fingers on a hand).
Inherent to counting is the idea of “ordinality,” which states that of two given values, one can either be greater than, less than or equal to the other. Thus, a certain “order” is established with respect to their values. This can be honed further to the idea of “cardinality,” which more explicitly reveals the value or quantity of something. This is how we can go from saying there are more turkeys over there than here to saying there are five turkeys over there and only four over here.
But to know the one-turkey difference requires a whole new technique. And that technique is calculation.
It’s usually at this point that most people end their mathematical career. They found a tool for arriving at most numbers they feel will ever concern them (after all, Johnny is probably not going to give 7, 243 apples to Jane). However, calculation — the ability to transform inputs to outputs via mathematical operation — equips us with a tool to understand every number that could ever exist. When numbers like the amount of people who traveled (43.6 million) and the number of turkeys consumed (46 million) are incomprehensible to the human brain, calculation allows us to manipulate them into terms we can fathom — for each person who traveled, 1.055 turkeys were consumed.
But to know that one turkey is a more informative answer requires a whole new technique. And that technique is approximation.
Approximating — representing an inexact (though useful) value in place of an exact one — is when we balance the ordinality and cardinality of calculated numbers with the time and resources necessary to find and understand them. Typically one approximates when information is difficult to procure (How many grains of sand are there in the world?) or when further specificity does not radically alter the answer (the number of atoms in your body). The exactness required of an approximation is a function of the context in which the value is placed. Sometimes it’s better to be close enough than exact.
From the one turkey whose outline resides on our hands to the millions at the center of this past holiday, numbers run our world through and through. For as informative as they can be, they can also mislead, obfuscate and deceive. With three tools presented here — counting, calculating and approximating — we can more thoroughly question and answer the world around us. Only then — only by understanding how they are brought about — can we truly grasp their meaning.
Barry Belmont is an Engineering graduate student.