Place Notation for the Non-Mathematician

Place notation is something that we all do every day, and something that we take so completely for granted that we often don't realize we're doing it. It depends upon our ability to consider digits (what we use to write numbers, such as "1," "2," and so forth) not only as themselves, but as multiples of a given power of the base.

But this is an explanation for non-mathematicians, and that last sentence may have already scared those folks away. So lest we frighten off anyone else, let's look at this question in the easiest way possible, in the base that most people are most familiar with, ten. Consider the number 6,783.

	Place Value
Digit	Value	Exp. Val.	Total Value
6	1,000	103	6,000
7	100	102	700
8	10	101	80
3	1	100	3
			6,783

In the table to the right is a simple illustration of how place notation works. The digits in the number don't really mean themselves; they mean themselves multiplied by the value of its place in the number. To find our what value a particular place in the number has, go to the "decimal point" (or the end of the number, if there isn't one), and start counting leftward starting at zero. So in this case, "3" is zero, "8" is one, "7" is two, and "6" is three. Once you've got the place number, take ten and raise it to the power of that place number. So for "6" in our example number, take ten and raise it to the power of three. (This means multiply ten by itself three times.) Ten times ten times ten equals 1,000. Now, multiply the value of the digit by this number. For the "6" in our example, that means "multiply 6 by 1,000," which makes "6,000." That, then, is the value of that digit; remember this value for later. Repeat this process for each individual digit; when done, add them all together, and the result is the total value of the number, in this case "six thousand, seven hundred and eighty-three."

Digits after the decimal point are handled similarly, but the place number is counting backwards. In other words, the first digit to the right of the decimal point is counted backwards from zero, so its value is ten raised to the power of negative one, or 0.1 (also called one tenth). The second digit's value is ten raised to the power of negative two, or 0.01 (also called one hundredth). And so on. Once these place values have been multiplied by the digits in that place (e.g., for "0.03," multiply the 3's place value, 0.01, by three to make "three hundredths"), simply add that to the total in exactly the same way.

This all seems impossibly complicated when we go through it step-by-step in this way; however, it's really astonishly simple. Indeed, we do it every time we read a number with more than one digit, or with anything after the "decimal point." Place notation must be the way our brains think about numbers, because we are able to do it so effortlessly.

One will notice that we used the number "ten" an awful lot in this process; what is it about ten that makes it special? Nothing, really, except that it is the base that we're accustomed to using. The base in place notation is the number by multiples of which place values increase. The following shows the same number, 6,783, analyzed as if it were written with a base of twelve, rather than of ten. All columns other than the digit column give values in decimal, that being the system most of us are used to.

	Place Value
Digit	Value	Exp. Val.	Total Value
6	1,728	123	10,368
7	144	122	1008
8	12	121	96
3	1	120	3
			11,475

The value of each place, in other words, isn't really the number of that place multiplied by a power of ten, but the number of that place multiplied by a power of the base, whatever that base might be.

Understanding this means we have to change our habits of looking at numbers. Consider "10," for instance. We tend to look at this and think "ten," but that's not really correct; what if it's not written in base ten? If it's written in base eight, it's "eight"; if it's written in base twelve, it's "twelve." We can use literally any integer (whole number) as a base, as long as we have a number of digits equal to that base. Here, for instance, is a sequence of counting in base twelve, using "X" as the digit for "ten" and "E" as the digit for "eleven":

0	zero
1	on
2	two
3	three
4	four
5	five
6	six
7	seven
8	eight
9	nine
X	ten
E	eleven
10	twelve; a dozen
11	dozen one
12	dozen two
13	dozen three
...
19	dozen nine
1X	dozen ten
1E	dozen eleven
20	two dozen
21	two dozen one
...

Now that we know that place notation allows the use of any base, many of us find ourselves asking: is ten really the best base? Perhaps some other base is more effective, or easier to learn or use? To learn more about this question, we must compare various bases and see what their properties are.