When you first learned mathematics, it was all about numbers and counting. One, two, three, and so on. A big problem arose when you started to realize that while you often had direct access to things you could easily count (two eyes, two ears, ten fingers on a hand), there were situations where the amount of stuff was so large that your eyes glazed over just thinking about how you would go about counting such a huge number. Find someone with a full head of hair and ask how many hairs they have? Counting that many hairs seems like a chore no one is prepared to do.

But it’s still a meaningful question. How many hairs are on a human head?

In this appendix, we are going to approach counting in a slightly different fashion so that we can make estimates for these kinds of questions. We’re going to develop a counting language that will allow us to specify, with as little or as much work as we are willing to spend, anything we would encounter no matter how large or small.

We will start by counting rabbits.

A New Way To Count

We start by counting rabbits because of their (deserved) reputation for reproducing. Imagine giving a breeding pair of rabbits enough food to feed them and all the offspring they have while somehow preventing all rabbit deaths. Within about six weeks, your population of domestic rabbits will double. At such a rate, in two years time, you will have a number of rabbits that rivals the number of hairs on a human head. How can we keep track of them all?

Instead of counting individual rabbits, let’s count generations of rabbits. There will then be an easy way for us to switch back and forth between the generation we counted and the actual number of rabbits. See “Table 1” where the rabbits in each of nine generations are illustrated so that you can translate between the two counting systems. Crucially, this method of counting generations allows us to know the number of rabbits without having to count each individual one, something that’s not hard to do when the number of generations is small, but gets increasingly difficult as the number of generations gets larger and larger. This method may be called “logarithmic counting”, though it goes by many other names.

The rabbit counting game starts with a base of two rabbits and proceeds from there. But how do we communicate that this is what we are doing? We could write out this whole story, but this technique is common enough that mathematicians have developed a notation that uses the numerals with which you are already familiar, but uses the form of exponential notation. How many rabbits are in the final row of Table 1? The answer is succinctly written as: 2⁹, because we started with a “base” of 2 rabbits and we let them multiply for 9 generations. The rules for this notation are simple: write down whatever base you start with, and then the number of generations (or replications) is written in superscript to the right of that base.

Exponentiation

You probably learned that this superscript “generational” number is called an “exponent”. You might also recall that this indicates the number of times you multiply. This is true, and importantly, this is a new way to count or to make rapid calculations. There are any number of ways to translate the number into the old system you’re used to (the 1,2,3,4,5… one). You could multiply the base number, repeating this by the exponential number: . Or you could count up the number of rabbits in the ninth row of Table 1. Either of these approaches would end up with the number five hundred twelve. Even if you were able to count as fast as the world-record counter, counting one-by-one would take you more than four minutes.

Or you could simply be satisfied with the knowledge that the ninth generation of rabbits has 2⁹ rabbits. This really does uniquely specify the number of rabbits and does so in a way that is easy to understand and manipulate in mathematical equations.

Table 1: The number of rabbits in each generation.

Inverses, Zero, and Negative Numbers

Up until now, we’ve been counting starting at one and ever increasing. But the number zero is an even more fundamental starting place than the number one. This concept is not trivial! As it is written today, “zero” was invented for use in number-writing about fifteen hundred years ago by an enterprising scribe from India. The idea of zero is older, but it was not clear that we needed a separate symbol for “nothing” or “zero” until this time (there are examples of people simply using blank spaces to mean zero, but that system is obviously pretty confusing). Now we get to the first counterintuitive result in our new counting system. If we start at zero in our counting of generations of rabbits, this corresponds to 2⁰ rabbits. How does such an absurd concept even exist? I can understand a “first generation”, but what would a “zeroeth generation” be? To answer this question, we’re going to engage in a kind of abstract game that mathematicians regularly do: we will look for a pattern or a rule that we can follow that can then be extended to situations that go beyond our initial intent.

After each generation, the number of rabbits doubles which is to say that we multiply the number of rabbits we start with by (the base number, 2). So when moving from generation two to generation three, the number of rabbits increases from to , or . This doubling means that the numerical increase from one generation to another is dependent on where we start. If I ask the question, “How many more rabbits are added to the population after each generation?”, it depends on which generation you start with!

What is true is that no matter where you start, when we move from one generation to the next one in order, the number of rabbits doubles.

But what about going the other direction? If I go from generation three to generation two, what happens to the number of rabbits? The number decreases and I end up with rabbits when I started with . This is a confusing game to play because we have to imagine going backwards in time, something which simply does not happen in the real world. Nevertheless, I can imagine running a movie of rabbit population explosion in reverse. And anyway, there are scenarios where populations of rabbits decrease after a certain amount of time, so this idea of decreasing the number does have practical utility as well. This is the first instance of a very important concept that helps us calculate in mathematics: taking the inverse. When you first learned to count, at some point someone asked you to try counting backwards. Now we’re going to do the same thing with generations of rabbits. This is “inverse logarithmic counting”.

There is a simple rule with this inverse counting. The number of rabbits after each generation is halved, or divided by two. Just as with our rule of doubling when counting each generation, when counting backwards, it matters where I start if I want to determine where I end up.

After all this careful consideration, we can finally say what the “zeroeth” generation is. We take the rule I found for inverse logarithmic counting and apply it to the first generation. In the first generation there are two rabbits. So if we count backwards one generation, we should halve the number of rabbits we started with. Let’s add a new row to our table to illustrate.

Table 2: Add the first (0^th) row to Table 1 for a more complete picture.

The practical utility of this may be unclear. After all, it takes two rabbits to produce offspring, which means that a zeroeth generation of rabbits breaks our understanding of how the world works. But we will find many other situations where we don’t need to have a clear first-generation starting point like we do with our rabbits (you can, if the occasion calls for it, start counting at zero).  For now, we can treat this simply as a definition, something that has to be true if we think what’s fundamental about this whole game are the rules of transition between generations. In such a case, it absolutely must be the case that the zeroeth generation corresponds to one rabbit.

In fact, the general logic of this system requires zeroeth generations to always be associated with the number . It does not matter what base you start with.

At this point, we can keep counting backwards and arrive at a whole new set of numbers called “negative numbers”. If you are convinced zero exists, then there is nothing to keep you from continuing to count backwards. While these “negative numbers” seem entirely abstract (how can you have a number that is less than zero?), they are really just taking the concept of “inverse” applying them to the numbers we have encountered up to now.

In bookkeeping, where mathematics had its origins, negative numbers are known as “debits” and are traditionally written in red ink. They act in exactly the inverse way that credits, written black ink, do. So when you add a debit to an account, the amount of money in the account decreases. This is intimately related to the concept of “subtraction”, but we will find it convenient to associate the inverse with the number itself. So instead of saying that “subtraction is the inverse of addition”, we might say, “the inverse of adding a positive number is adding a negative number”.

What happens if we count backwards from the 0^th generation? We would have the negative first generation, the negative second generation, and so on. Again, this abstraction may seem absurd right now, but let’s apply the same rules as before and see what amounts we end up with for each generation.

To move from the 1^st generation to the 0^th, we took the base number and divided in half to get . Thus to move from the 0^th to the negative 1^st, we should divide in half to get . We now have entered the world of “fractions”, a concept that is intimately tied with division — the inverse of multiplication.

This sets up a beautiful symmetry of concepts that is worth taking a closer look at. We are asking you to switch between counting individual things as you learned when you were a child and counting generations. The counting you learned as a child is called “additive” while the logarithmic counting of generations is “multiplicative”. However, we will see that there is a complete correspondence between these two systems that allow us to translate back and forth with relative ease and, in the process, expand our mathematical horizons and give us access to calculation tools beyond those you learned in primary school.

We’ll start with inverses. What is the inverse of a number in the additive universe? It’s the negative number. So, in the additive universe, the opposite of two is minus two.

What about in the multiplicative universe? What is the inverse of a number that lives there? It’s the division by that number otherwise known as the “reciprocal”. So in the multiplicative universe, the opposite of two is one half.

The notation we’ve already developed allows us to write some beautiful shorthand that highlights the deep relationships between these two systems of counting. A way to state the previous paragraph using our shorthand is

2^-1 = ½

It’s worth unpacking this. In this situation, just as in the case with our rabbits, our base is the number two. By indicating a negative one in the exponent, we are really asking ourselves the question, “What is the opposite of multiplying a number by two?” the answer is multiplying a number by one half, or, equivalently, dividing by two.

We can continue with this system and count backwards to negative two, negative three, and so on. In the context of logarithmic counting, the actual amounts these exponents correspond to are smaller and smaller fractions, in the case of our base-two generations, each is one half of the previous one, so one fourth, one eighth, and so on. The beauty of understanding these inverse relationships is that if you know the translation between a positive exponent and its actual value, you immediately know the relationship between the negative exponent and its actual value — just take the value’s reciprocal (one divided by the value).

Writing and Talking About Large and Small Numbers

What is further astounding is that the rules for our logarithmic counting system in the multiplicative universe give us a new way to write down and talk about large and small numbers with an elegance and efficiency that is not present in the additive universe. “Two to the ninth” is, surprisingly, an easier concept to work with than “five hundred and twelve”. Lessons from human development give us insight into this: an average two-year-old child has the cognitive capability to understand the numbers “two” and “nine” while most children don’t learn the concept of “hundred” until they are at least five years old.

The decimal writing system you learned actually has a form of our logarithmic counting concept already embedded into it, although it is really set up to make bookkeeping easier instead of the task we are interested in which is describing extremely large and small numbers. Instead of base two like our rabbit example, our decimal system is based on the number ten. The generation is indicated by which “place” the numeral lies in the representation. So, to take the example of five hundred twelve

512

This number is really just telling us that there are five instances of the “second generation”, one instance of the “first generation” and two instances of the “zeroeth generation”. While we teach children that the number farthest to the right is in the “ones” place, the next number to the left is in the “tens” place, and so on, another way to describe this is to say that, for a base ten system, one is the zeroeth generation, ten is the first generation, and hundred is the second generation.

Let’s investigate the base ten system a little more carefully in Table 3.

Table 3: The logarithmic progression of numbers in base ten.

As you can see, really big and really small numbers written in decimal notation become problematic very quickly. Alternatively, our system with bases and exponents (we call it, “exponential notation”) is much more compact and gives us useful, relevant information at a glance. Isolated, it would be difficult to distinguish between one billion and one trillion written in decimal notation without sitting down and counting the zeroes:

1000000000000 or 1000000000?

But our eyes can see the difference between a nine and a twelve without much difficulty:

10¹² or 10⁹

One complaint about exponential notation might be that the exponent is arguably a more important numeral than the base as it tells you what generation you are in, but the exponent is almost always written typographically smaller than the base. This is just something you have to get used to. You may find that certain calculators use a different notation where an “E” or “EE” is used before giving the exponent (for example, 1E9 or 1EE12). Unfortunately, this form seems to have only really caught on with engineers and calculator displays.

Exploring the universe of base ten logarithmic counting further, there are some key linguistic patterns seen. One noticeable feature is the progression million, billion, trillion, and so on. Keen observers may recognize the sequence as being suspiciously similar to the Latin numeral prefixes (for example, unicycle, bicycle, tricycle, etc.). The ordered progression follows by multiplying the previous number in the sequence by one thousand, but, historically the British preferred a system where each successive number was multiplied by one million. Also, although our table ends at trillion, the system in principle continues on, with words like “quadrillion” and “quintillion” that are not in common use. In fact, most newspaper style guides request a clumsy repetition of “-illion” words to describe large numbers in a style made famous by Carl Sagan (e.g., “the universe contains ten billion trillion stars”).

Another interesting feature is the suffix “-th” which shows up on all the numbers less than one. This is an inheritance from the so-called “ordinal numbers” which are used to describe sequentially ordered things (first, second, third, etc.) Other than the fractions one half and one quarter, a form where the number “one” is followed by an ordinal number simply means the reciprocal of the number. Since most ordinal numbers use the suffix “-th” in English, this means that, for our purposes, adding a “-th” to the end of a number makes it a reciprocal. Thus, tenth, hundredth, thousandth, and so on. It provides a very convenient symmetry around our anchor at one.

Metric Prefixes

The final two columns of Table 3 are the so-called “metric prefixes” which we frequently use as our exploration continues. Suffice to say that these prefixes are completely synonymous with the number in question. Whenever you see the prefix “kilo” you can immediately replace that with “one thousand” and keep the same meaning. This similarly applies to the prefix symbol in the last column. For example, the unit “kilogram” is “one thousand grams” which is to say

kg = 10³ g

The Power of Powers

A final word on terminology. Because we motivated this whole discussion with a look at rabbits multiplying, we adopted the conceptual term “generation” to describe our logarithmic counting system. You may have noticed that I’ve already introduced a synonym for “generation” when talking about these steps in counting: “exponent”. Another synonym for this is “logarithm”. As long as we specify what base we are using (and we’re going to stay in base ten for the time being), then if I ask about the logarithm of a number, it’s the same thing as asking what the exponent is. Thus, I could say “the log of one thousand is three” or, if I wanted to be extra clear, “the log base ten of one thousand is three”. This is the full sense of what a so-called “logarithm” is and it’s why I call the counting system we developed, “logarithmic counting”.

In 1977, the Eames Office in Chicago (the ones who designed the Eames Chair) produced a short film called “Powers of Ten” which explores what it means to logarithmically count lengths, both forward and backward. The popularity of this film with physical science teachers has made it a somewhat common to call the generational numbers “powers of ten”, and indeed they are as another word for raising an exponent is “taking a power”. You can find this film discussed in Chapter 1 of this text.

But perhaps the most common way of describing these generations in base ten is to refer to “orders of magnitude”. This provides an excellent shorthand for comparing different scales. If I say “this measurement is an order of magnitude greater than the other”, that means this measurement is ten times the measurement of the other. If I say “this measurement is three orders of magnitude less than the other”, that means that this measurement is one thousand times smaller (or multiplied by a fraction of one thousandth).

Getting familiar with these linguistic tricks of describing numbers is useful. It allows you to start thinking in the language of the new counting system and, in so doing, will let you describe numbers that are awkward to do so in any other way. Rather than saying that there are ten billion trillion stars in the universe, you can just say that there are ten to the twenty-two stars. This is an evocative description. It means that I’ve gone through twenty-two generations of multiplication by ten to get to the number of stars in our universe. The number is unfathomably gigantic and almost defies any means of conceptualization, but it is something you can now talk about and even ponder because while 10²² is a silly-large number you may not have encountered until our discussion, the numbers ten and twenty-two are old friends.