Cut The Knot!

An interactive column using Java applets
by Alex Bogomolny

Between Arithmetic and Algebra

January 1998

What is the difference between the arithmetic 3+5 = 5+3 and the algebraic a+b = b+a? One is a specific fact, another is a pattern valid in a multitude of situations. While arithmetic may hint at some regularities, algebra, as a language, gives expression to acknowledgement of patterns as such. How did people express general ideas before the advent of algebra in the 15th-17th centuries? Diophantus of Alexandria (c. 250 A.D.) is credited with the invention of syncopated (shorthand) notations. Beforehand, it was geometry. For example, off the ubiquitous diagram on the right one may read (a+b)2 = a2+2ab+b2. And beforehand?

While many artifacts discovered during archeological digs indicate that man mastered some counting many thousand years ago, the Rhind papyrus and the competing Moscow papyrus are the first documents that provide a clue of the level of mathematical knowledge of our ancestors. The Rhind papyrus is a copy made about 1650 B.C. of a document from two centuries earlier. This is a collection of arithmetic problems and their solutions. But consider an example:

Problem 25: A quantity and its half added together become 16. What is the quantity?

Solution: Assume 2. Then the whole is 2, the half is 1, the total is 3. As many times as 3 must be multiplied to give 16, so many times 2 must be multiplied to give the required number...

Obviously, the author attempts to pattern a certain kind of problems which is exactly why the papyrus is regarded as a mathematical text. Our ancestors teach us that one should not wait until Algebra I or Geometry classes to be able to grasp and, probably, express abstract ideas that make math Mathematics. Deliberate search for patterns is an entertaining activity suitable for all ages.

Remember also that pattern recognition is an endemic human ability at which man outperforms the fastest among modern computers with all their awesome calculating power. So let's look for patterns in places that in our culture became symbols of dreadful rote associated with the first encounters of mathematics in school. Let's look at the addition and multiplication tables. As an important feature, Multiplication Table appears (among other pieces of Useful Information) on the inner cover of my notebook (The Mead Company, Dayton, Ohio 45463 U.S.A.) Addition and multiplication tables are also available on a separate page.

Start with the simplest - addition. One feature that stands out is that every diagonal that crosses the main diagonal consists of cells carrying one and the same number. For example, 1+7 = 2+6 = 3+5 = 4+4 =... Abusing notations somewhat, {I} + {IIIIIII} = {II} + {IIIIII} = {III} + {IIIII} = ... which is supposed to remind you of counting sticks; perhaps, matches - an inevitable evolutionary activity. Up the evolutionary ladder, we realize that counting the number of sticks in a big pile is considerably simplified by splitting the pile into two and handling each separately. Then, adding up the results. Lo and behold, the result does not depend on how the pile is split into two. As Lucretius has stated, naught from naught can be created. Let's follow another pattern (What worked once has a good chance to work once more) and apply the approach recursively. What transpires is the associative law of addition.

The table is also symmetric. This is an expression of commutativity of addition. Looking at the diagonals parallel to the main, the entries, when followed left to right, grow by 2 with each step. This is still a simple one. Is there anything deeper than that?

Consider 2x2 arrays of contiguous cells. Determinants of such arrays are always -1. 3x3 and higher order determinants are all 0. The sum of elements in the arrays uniquely determines all the elements. We can do one better. Solve the Rook Problem for one of the NxN arrays, i.e., select N elements - one in each row and one in each column of the array. The sum of thus selected N elements uniquely determines all its NxN entries. A similar fact holds for monthly calendar tables. Since in the calendar tables all entries are distinct, the value of an element defines its location in the table. As we already saw, in the addition tables, associativity of addition - an otherwise useful feature - causes a deficiency in this respect.

Now let's turn to the multiplication table. This one too is symmetric. A combination of associativity and commutativity is demonstrated by the multiplicative "calendar" game in which multiplication replaces addition. Also, 2x2 determinants are all 0 as are all higher order ones. If, in the definition of determinant, we replace multiplication with addition, then the result will always be 1. If we restrict ourselves to the 2x2 arrays whose diagonals lie on the main diagonal of the table, then the sum of four numbers in the array is always a complete square. (Quite reminiscent of the geometric proof above.) Now, the fact remains true for 3x3 and higher order "diagonal" arrays: the sum of elements in a "diagonal" NxN array is a complete square. As another generalization, consider the sum of elements in an off-diagonal 2x2 array. This is a composite number easily found from the row and column captions with the assistance of the distributive law. The same holds for higher order arrays. Lastly, in this vein, arrays must not comprise contiguous rows or columns: an array of cells formed at the intersection of arbitrary N rows and N columns serves to demonstrate the distibutive law.

Multiplication tables are rich in patterns. Put your finger on an entry on the main diagonal. Move the finger one step in either North-East or South-West direction. The entry you'll get is 1 less than the diagonal entry. As every calculating prodigy will tell, there is no faster way to compute 24*26 but to remember that 252 = 625 and, then, subtract 1: 24*26 = 624. Move your finger one step further and you'll have to subtract 4 (=22).

Entries in the last row between columns 2 and 9 are mirror images of each other. In the last row, the sum of entries equidistant from columns 2 and 9 always sum up to 99. Looks like another case for the distributive law. One row up, the sum is 88, then 77, and so on.

Look at the L-shaped combination of three cells with the "corner" one on the main diagonal. This diagonal entry is the average of the other two.

Some facts come up on observing addition and multiplication tables simultaneously. Say, both are 10x10. In the lower right corner of the tables, the entries 18 and 81 are mirror images of each other. Did I get carried away? Of course, 18 and 81 are mirror images of each other but where is the pattern? So far it was pure arithmetic. To see the pattern we have to handle several tables simultaneously. An applet provides addition and multiplication tables in several (2-36) number bases. Now we can claim a pattern: lower right entries in the multiplication and addition tables for the same base are mirror images of each other. In the addition tables, it is always a two digit number with 1 followed by the penultimate digit of the system. In the last row of addition tables, while the first digit increases by 1, the second digit always decreases by 1 - an intuitive introduction into the notion of casting out 9s in the decimal system and a meaningful generalization to other bases.

The idea may not be all that ridiculous to present various number systems before algebra (or programming) classes. Children familiar with clocks and watches will not be startled at counting in base 12 or 60. Double digit numbers in base 5 are easily represented on two hands. Moreover, such representation accommodates decimal numbers up to 30. Quite a gain compared to the customary 10! Imagine huge savings in paper waiste attained after the customary "" counting practice is replaced with the new system. (Ah, if I only lived 100 years ago.) With five finger hands we can actually count in base 6 and reach as far as 35 using only our natural facilities. In this we would follow the Japanese who modified the Chinese suan pan into the soroban by removing redundant counters. As to the patterns, all the properties of addition and multiplication we just discussed are shared by tables in all other bases.

In his small book, Mathsemantics, the author Edward MacNeal tells a childhood story of how, at the age of 8, he and his 10 year old brother started counting in the base 12 system. The twelve digits were called zero, one, two, three, four, five, six, seven, eight, nine, zip, zap. The count went on with ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen, zipteen, zapteen, twenty. There were zipty, zapty, and zapty-zap followed by one hundred. Mr. MacNeal did not become a mathematician but turned out a successful business consultant.

References

  1. H. Eves, Great Moments in Mathematics Before 1650, MAA, 1983
  2. The History of Mathematics, ed J.Fauvel and J.Gray, The Open University, 1987
  3. Lucretius, On the Nature of Things, C.E.Bennett, trans., Classics Club, 1946
  4. E. MacNeal, Mathsemantics, Making Numbers Talk Sense, Penguin Books, 1995

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