An interactive column using Java applets
Cut The Knot!
by Alex Bogomolny
Place value is an important concept that is often misunderstood and sometimes misplaced. The Penguin Dictionary of Mathematics 1989 has no corresponding entry. The Harper Collins Dictionary of Mathematics (1991) has this:
place value, the particular power of the base of a counting system that is represented by a particular position in a PLACE-VALUE NOTATION; for example, units, tens, hundreds, etc. in the decimal system, so that in 5374 the place-value of the 3 is 100.
The definition correctly places the concept of place value as an attribute of positional systems, not solely the decimal one. However, the particular example obviously misses the point.
Here's an entry from an older (1959) The Dictionary of Mathematics from D. Van Nostrand Company:
PLACE, decimal place. See DECIMAL. place value. The value given to a digit by virtue of the place it occupies in the number relative to the units place. In 423.7, 3 denotes merely 3 units, 2 denotes 20 units...
The example is correct, but the direct association of place value with the decimal system, while correct, undermines the value of the definition.
Importance of place value has been emphasized in the NCTM Standards:
In grades K-4, the mathematics curriculum should include whole number concepts and skills so that students can -
- understand our numeration system by relating counting, grouping, and place-value concepts;
This is of course important to understand our numeration system. It is also true that place value arises from one of several possible representations of grouping in the counting process. (Note, though, that it is by far more efficient to group by 5s than by 10s. The reason is in our ability to subitize quantities of small groups of objects.) But why, in general, does mathematics education shy away from placing our decimal numeration into the framework of positional systems? I can think of three reasons to reconsider the common trend that mostly shuns all bases but 10.
First, it may be (and likely is) easier to construct the right concept from a multitude of examples than from just 1 or 2. In this context, it is an indisputable fact that the decimal numeration is only one example of a large family of positional systems. Many concepts are introduced through multiple examples; why not place value? Second, it is important for children to grasp the idea that various entities, as a rule, admit several different representations. For example, it is another of the modern mathematics education tenets that functions have 4 different representations: graphical, tabular, algebraic and (more recently) even verbal. But what about numbers? Children who failed to develop the concept of there being distinction between an entity and its representation are bound to have difficulties with fractions - both common and decimal. And many do. Many children [Ginsburg] identify numbers with numerals, a misconception that might be avoided by representing numbers in various systems.
The third reason is that study of positional systems in general provides a rich source of powerful tools, patterns, and mathematical reasoning, for children to learn and to explore. Even so basic devices, such as the addition and multiplication tables, conceal a great many patterns valid across all positional systems. Below I give another example.
Self-documenting sentences of the sort offered by the applet, have been invented by Raphael Robinson [Hofstadter, p 27 and p 389 and on; see also Gale, p 10]. The puzzle is to fill in the blanks in "This sentence contains _ 0's, _ 1's, _2's, _ 3's, _ 4's, _ 5's, _ 6's, _ 7's, _ 8's, and _ 9's" so that the sentence become true. A curious approach to solving the puzzle is by iterations. Fill blanks arbitrarily, count the number of 0's, 1's and so on, and feed thus obtained numbers back into the sentence in place of the initial selection. Again count the number of various digits and substitute the result into the sentence. The iterations are documented in the drop-down box at the bottom of the applet. (The "Clear" button clears the contents of the box.) The truth to be said, such iterations do not always lead to solutions. Sometimes they settle into cycles. Moreover, there are solutions of the repelling sort that can't be obtained by the iterations. So not all fun can be automated. (The "Iterate" button performs a single iteration. The "Auto" button runs until "Stop"ed or until a solution is found. Numbers that fill the blanks can be changed up or down manually by clicking a little off their center line.)
The original puzzle appeared in the decimal system. The apparent difficulty of handling 10 digits simultaneously led me to thinking of those sentences in smaller bases. In base 2, there are only 2 digits to count; in base 3 there are only 3 of them, and so on. In base 3, there are three solutions:
- 1 0, 11 1's, and 2 2's.
- 2 0's, 2 1's, and 10 2's.
- 10 0's, 10 1's, and 2 2's.
A short investigation showed that the first solution is remarkable in that it forms a pattern that solves the puzzle in other bases as well. More accurately, we have the following
Let b > 2 be the base at hand. Then the numbers 1, 11, 2, ..., where the ellipsis are filled by a sequence of 1's as needed, solves the puzzle.
The proof can be run by direct verification (e.g., the quadruple 1, 11, 2, 1 solves the puzzle in base 4.) or may be arrived at after a little pondering. The clue is of course in that the numeral 11 stands for different numbers in different number systems. Put it another way: the place value of the left 1 depends on base b. Here's another pearl:
With base b greater than 6, the sequence 1, (b - 3), 3, 2, ..., 2, 1, 1, where ellipsis are replaced as needed with 0 or more 1's solves the puzzle.
- D. Gale, Tracking the Automatic Ant, Springer, 1998
- H. P. Ginsburg, Children's Arithmetic, II edition, pro-ed, 1989
- D. R. Hofstadter, Metamagical Themas, Basic Books, Inc., 1985