# 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 | 10^{3} | 6,000 |

7 | 100 | 10^{2} | 700 |

8 | 10 | 10^{1} | 80 |

3 | 1 | 10^{0} | 3 |

6,783 |

*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 | 12^{3} | 10,368 |

7 | 144 | 12^{2} | 1008 |

8 | 12 | 12^{1} | 96 |

3 | 1 | 12^{0} | 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.