Adventures in Numbers, Measurement, and Math

Number Systems

We don't think about numbers very much, because we take the way that we write our numbers for granted. But there are many different ways of writing numbers, some very different, and we've settled on the way that we have for a very good reason. Even within the way that we write numbers (our "number system"), there are various different ways to formulate our numbers, such as the use of different digits or different bases. So, to cut through the trouble, this article makes an attempt to explain these terms in a simple way, to help the reader come to understand the concepts upon which he must form his opinion.


First, we'll talk about digits, because digits are a concept that span across all the others. Digits are simple the characters that we use when we're writing numbers, regardless of any other of the variables we'll later discuss. In other words, digits are just sequences of curved or straight lines, and nothing more.

Two simple illustrations will probably suffice to show what we mean by this. In the current dominant number system, we have ten digits, and they are as follows:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9
And that's the entirety of the issue. Digits don't tell you what they actually mean (think about it: the "9" in "94" doesn't really mean "nine," does it?), and they don't tell you how they should be put together to form other numbers. To prove this, let's look at another set of digits with which most Westerners are at least passingly familiar:
I, V, X, L, C, D, M
Very different symbols, of course, belonging to a very different number system; but they're digits all the same. Digits mean nothing by themselves; they're simply the building blocks that are used when constructing numbers in a given number system.

Number Systems

The next concept we'll investigate is the number system. The number system involves two things: (1) assigning numerical values to the chosen digits, and (2) making rules for how those digits are put together to form numbers. There are really an unlimited number of ways in which this could potentially be done; however, they essentially boil down to two main general methods. For convenience's sake we'll call them cardinal number notation and place notation. Place notation is the one we use most often; but cardinal number notation is also common.

Cardinal Number Notation

COne Hundred
DFive Hundred
MOne Thousand
Cardinal number notation is that used by Roman numerals, so most Westerners are, again, at least passingly familiar with it. Still, it behooves us to review that system in at least a little detail. Part one, assigning values to the digits, is easily explained. Essentially, this number system assigns symbols to what I've decided to call here, again for convenience's sake, "cardinal" numbers; that is, numbers deemed important enough to warrant it. These numbers are one, five, ten, fifty, one hundred, five hundred, and one thousand. The chart to the right shows the correspondence of the symbols to their values. There were variations on this system throughout history; one common one was the existence of a symbol "D" with a line overtop of it, which referred to "five thousand." However, this is the general pattern, and it is simple enough to serve well for our purposes.

Part two, making rules for how these digits are put together to form other numbers, is pretty easy, as well. Find the lowest cardinal number that is higher than the desired number; then take the next-lowest cardinal number and concatenate as many of them as possible without exceeding the desired number. If still less than the desired number, move to the next-lowest cardinal number and concatenate them in the same way. Continue until the sum of the digits (that is, all the digits added together) is equal to the desired number.

Most of us know how this works already; but an example, taken step by step, would probably be helpful.

Take the number "thirty-nine." The highest cardinal number we have in this number is ten, "X"; so, we put in tens until we can't put in another ten without exceeding our number. This gives us "XXX." Putting in another ten would give us forty, which is higher than we want, so we move down to the next cardinal number, five. We put in a five, "V," and find that we're at thirty-five, still below our number; however, we also find that putting in another "V" would again bring us to forty, higher than we want, so we leave it at one "V," leaving us with "XXXV." We then proceed to the next highest cardinal number, one, represented by "I." This is also the lower cardinal number, so we simply concatenate "I" until we reach the required number. This gives us "XXXVIIII," equal to thirty-nine.

(Later developments gave more complex notation, such as putting a lower cardinal number to the left of a higher one to indicate subtraction rather than addition. This allows constructions such as "IV" for "four," replacing the older "IIII." Our example here, though, does not require such complications, so I've rested with the older system.)

The first thing to notice about this system is that the value of a digit does not depend upon its place in the number. No matter where an "X" is in the number in Roman numerals, it always equals "ten." This is radically different from our other primary number system, place notation, which we will review shortly.

The next thing to notice is that this notation produces numbers which are nearly useless for calculation. We've all learned in school, using place notation, how to do a huge variety of mathematical operations by manipulating the digits of numbers in various ways. There is really no such system for Roman numerals, where the number system gives little aid in making calculations. Take the simple example of addition; let us add our prior number, "XXXVIIII," with "LXVII" (sixty-seven).

The fundamental rule to remember is that every digit in cardinal number notation is simply itself, and ever number in cardinal number notation is simply the sum of its digits. So the best way to do this is to simply run all the digits together, giving us this:

We can then sum these digits to get our resulting number. Summing them will be much easier, of course, if we order them in the way we're accustomed, from the highest cardinal number to the lowest:
Unwieldy at best. Let's continue the process by simplifying this:
But we can simplify still further:
And still further:
Giving us the answer that thirty-nine plus sixty-seven is one hundred and six.

In case that process wasn't amusing enough, let us now imagine multiplication. Not even multiplication by sixty-seven; let's do multiplication by a smaller number, say three. Multiplication is, of course, simply repeated addition; so that's precisely what we have to do here.

So thirty-nine times three is one hundred and seventeen. It's functional; but it's messy and far from convenient. Numbers in this number system are simply not useful for calculation; they are useful for recording the results of calculations, which are instead done on an abacus or some other device.

That's the reason that cardinal number notation is now limited to a very few, very ceremonial places (old-style clock faces; movie publication dates; occasionally inscriptions; and so on). Simply put, cardinal number notation fell by the wayside because civilization found a better way.

Place Notation

We all know place notation, as it's the system that we constantly use. At first glance, it seems more complicated, because the digits don't always mean what they look like they mean. However, in the end it's not only easier to read numbers, it's also much easier to use the numbers in the course of calculations.

Place notation is so called because the value of a digit is dependent upon its place in the number. We saw that in cardinal number notation every digit meant a given value no matter where it was, and one simply summed them up to read the number; in place notation, a digit doesn't necessarily mean just itself, but rather its own value multiplied by some other value. That other value depends upon the place.

Place Value
DigitValueExp. Val.Total Value
Once again, example will probably be the best teacher here. Let's take a random number, 4,638. Those who are used to place notation in base ten will easily read this as "four thousand, six hundred and thirty-eight." But where does that come from? The symbol "4" means "four," not "four thousand," doesn't it? And cannot similar questions be asked about the remaining digits in the number? As the chart to the right shows, however, the "4" does not always mean "four"; its meaning, and the meaning of all the digits, is dependent upon its place in the number. The mathematicians would say that the value of a digit in place notation is equal to the product of that digit and the base (in this case, ten) raised to the power of the number of digits that digit is to the left of the end of the number, if numeration is started at zero. But we can probably do it more simply than that.

Take the digit, and starting at zero, count how many digits to the left of the "decimal point" (or, if there isn't one, to the left of the last digit) it is. Counting in this way, we find that "8" is zero, "3" is one, "6" is two, and "4" is three. Remember that number: three. Now, take the value of the base (in this case, ten; we'll explain the concept of bases shortly) and raise that value to that power. Our number is three; therefore, we take the value of the base, ten, and raise it to the power of three. We write this as "103"; for those of us who don't remember, that means multiply ten by itself three times. 103 is 1,000; remember that number. Now, multiply the digit value, "4," by that value, "1,000." That gives us four thousand. Four thousand, then, is the value of the "4" in this particular number. Save that number for later.

Now repeat that process for each digit (remembering that any value raised to the power of one is itself, and any value raised to the power of zero is one), then add up the values. The sum of all those values is the number we're expressing, in this case "four thousand, six hundred and thirty-eight."

Some numbers, of course, have a "decimal point" and numbers following it, but this is the same concept and presents no added complication. Just count your exponent number backwards; so the first digit to the right of the "decimal point" will mean itself, multiplied by 10-1, or 0.1 (also voiced "one tenth). The next digit over will instead by multiplied by 10-2, or 0.01; and so on. When through with these, add them into the total like the others, and there you have it.

It all sounds heinously complicated, doesn't it? But you do it all the time, every time you read a number that's more than one digit long, and every time you read a number with a "decimal point." This system sounds difficult, but in fact its shockingly easy; it appears to really match the way we think about numbers, and consequently is quite simple to grasp. Furthermore, numbers written in this way are quite easy to use for all sorts of calculations; we've all gone through such calculations in school, so we need not multiply examples here. All in all, place notation is clearly the superior system of the two we've looked at here; and, though there are others, these are really the only two that are presently worth reckoning with.

The reader has doubtlessly noticed, however, that the number ten has seemed curiously predominant in this discussion. Each place's value depended upon some power of ten. Is this something about the nature of the number ten? That question brings us to the subject of bases.

The Base

The value of each successive place is not really a power of ten; it's a power of the base, and that's the best definition of the base: that number which determines the value of each position in the number. If this is confusing, let's look at another number, this time in the base of eight (frequently called "octal"). Consider the table below and right; the "value" columns give the value in decimal.

Place Value
DigitValueExp. Val.Total Value
As we can see, this is identical in form to the table we constructed above in decimal (base ten), except that rather than valuing each place as a power of ten, we value each place as a power of eight. Computers frequently use octal, or base eight, because it's an easy multiple of two (hexadecimal, or base sixteen, is also common for the same reason). Computers at their most basic level work in binary, or base two, because this is fundamentally how they analyze reality (a bit is either one or off; this is equivalent to having two digits, "0" and "1"). Other bases are often used for special purposes in this way; however, in daily life decimal, or base ten, is nearly universal.
Place Value
DigitValueExp. Val.Total Value
The base, obviously, is crucial to determining what a number actually means. The same number, 4627, means something considerably different in octal (base eight) than in decimal (base ten), and something even more different in dozenal (base twelve). The chart at right shows that number's value in base twelve, the value itself being displayed in decimal. The same digits, the same order, but an extremely different value. The value of the base is what makes the difference. When reading numbers, then, it is vitally important to know what base one is reading, or one will certainly read them completely wrongly. Such things are not infrequent in computer programming, where bases other than ten are common. Reading "10" as "ten," when it's actually a binary number equal to "two," can cause serious problems.

Digits in Bases

As we've seen, the same digits in the same order can have vastly different meanings depending on the bases. But not every base uses the same digits, nor the same number of digits. (The actual shape of the digits is quite arbitrary; we could write them down as different-colored troll dolls if we all agreed on it.) Octal, for example, requires only eight digits; decimal requires ten; dozenal requires twelve; hexadecimal requires sixteent; and binary requires only two.

How does this work? Our method of analyzing a number in place notation has already revealed this to us, but reading numbers in base ten is so engrained that some more examples may be helpful. Consider octal, as an example; let's simply count for a certain period in octal and see how it goes. We'll start with zero:

0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21...
Wait a minute! Ten doesn't come after seven, it comes after nine! And that's true, of course, in decimal as in every other base. But this is octal; "10" does not mean "ten," it means "eight." Remember place notation; that "1" there isn't a "one," it's a "one multiplied by the base raised to the power of one," or "eight." And then, when we get to "17," what we have is not "seventeen," it's "one eight and seven ones"; and since it's octal, we then have to go to two eights and zero ones, giving us "20."

This is precisely what we're doing in decimal, of course:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13...
We get to nine, and we find that adding one will make the number equal our base; so we add one to the place to the left, the tens place (since this is decimal), and then make the ones place zero, giving us "10," which in this case (and this case only) means "ten."

This is all well and good for bases equal to ten or less. But what about bases higher than ten? We don't have enough digits! And that's true. Consider us using the best base, twelve; how do we count in this base?

0, 1, 2, 3, 4, 5, 6, 7, 8, 9...
What do we use for "ten?" We can't use "10," because that, of course, means one twelve and zero ones, something very different from "ten." In any base, we need a number of digits equal in value to that base. So in dozenal, we need twelve digits; what digits will we use for the two missing ones, ten and eleven? Rather than spend time now on a full discussion of this issue, let us simply posit two digits. For purposes of this article, we will use "X," the familiar Roman numeral ten, for ten, and "E," the first letter of "eleven," for eleven. Our counting, then, proceeds thus:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, X, E, 10, 11...18, 19, 1X, 1E, 20, 21...
We can see a brief summary of how these new numbers should be pronounced elsewhere; for now, I will simply illustrate such counting as follows, using "qua" as shorthand for "twelve":
Zero, one, two, three...nine, ten, elv, onequa, onequa one, onequa two, onequa three...onequa nine, onequa ten, onequa elv, twoqua, twoqua one...

Congratulations! Now you've got a pretty thorough understanding of number systems, and particularly of place notation and the concept of the base. Keep exploring for more adventures in math!