By HAJIRA KHAN & SAMEEN AHMED KHAN
Arithmetic is the oldest and the most elementary branch of mathematics. Its primary purpose is to study numbers and perform the four basic operations namely: addition, subtraction, multiplication and division. These operations are encountered in everyday life. For example, one may collect fruits from a tree and put them together. Addition helps us to keep the count of the fruits in the heap. At the end of the day, one may give away the fruits from this heap, which will require the knowledge of subtraction to keep the account. Suppose we have many trees and many heaps. Then keeping an account of the total fruits would involve big numbers. One can still manage with addition.
But the third arithmetic operation namely the multiplication takes care of the large numbers. Multiplication of whole numbers can be thought of as repeated addition. For illustration, let there be six heaps with eight fruits in each of them. Then there are fruits. This can be seen either as counting ‘six heaps eight times’ or equivalently ‘eight sets six times’. Two approaches are possible because . Order does not matter in addition and multiplication.
These statements may appear to us as obvious but are of great mathematical value. This property of the order not mattering has a special name and we say that ‘addition is commutative’ and ‘multiplication is commutative’, respectively. To visualise the division, let us consider a heap of 20 fruits, which has to be distributed among four persons. Symbolically, this task is stated as . Each person gets 5 fruits. After the distribution, one can get back the size of the original heap as . Hence, the division is sort of opposite of multiplication. We note, that subtraction and division are not commutative. This can be checked by considering the example with the numbers say, 2 and 5; as and .
Since, ancient times there have been attempts to obtain mastery of the four basic arithmetic operations. Addition and subtraction are relatively easy. The multiplication is facilitated with the help of tables. Division is the most difficult and an expertise in multiplication is a must. The oldest known multiplication tables date back to four thousand years and were used by the Babylonians (in the modern day they include parts of Iraq, Kuwait, Syria, Turkey and Iran). The Babylonian tables used base 60 (same as our clocks use for time, 1 hour = 60 minutes and each minute = 60 seconds). Such tables have been also found in other ancient civilizations from later periods.
In this article, we shall have a closer look at the modern day base 10 ‘multiplication table’ (also called the ‘times table’) and some of its many properties. In our school books, we are introduced to a multiplication table.
The figure has a table. We note some basic properties. Multiplication with 1 does not change the number. Multiplication with 10 only appends a zero in the end. This is the advantage of using the base 10. The diagonal distributes the table into two identical triangles having the same set of numbers, with the diagonal acting as a mirror. The two triangles have the same set of numbers because the multiplication is commutative. The diagonal has the perfect squares: 1, 4, 9, 16 and so on. This reduces the burden of memorising the multiplication table to a little more than the half. One has to memorise either of the triangles and the diagonal only.
Now, we note some advanced properties of the multiplication table. Numbers within the diagonal do not repeat. But numbers within each of the triangles repeat. For example, within each triangle, the numbers 6, 8, etc. repeat. The numbers in the diagonal and the triangle also have an overlap. For example, 4, 9 etc. occur both in the diagonal and the triangles.
Because of these repetitions and overlaps, it becomes difficult to count the numbers in the multiplication table! The repetitions and the overlaps only increase with the size of the multiplication table, that too in an unpredictable manner. Using these properties of the table, we note the sets of numbers in the multiplication tables of different sizes. The sets designated by contain the numbers occurring in the multiplication table. For completeness, we start with a table of size 1. The first few sets are
Sets of larger size completely contain all the sets of lower size. The size or order of the sets denoted by grows as 1, 3, 6, 9, 14, 18, 25, 30 and so on. More terms of this infinite sequence can be obtained from The On-Line Encyclopedia of Integer Sequences at http://oeis.org/A027424. It may surprise the reader, but there is no simple formula for the size or order of the sets. Even the legendry mathematician Paul Erdős could provide only a rough estimate.
Some very complicated formulae have been obtained involving logarithms and other advanced functions. For practical purposes, a good estimate is given by the simple inequality . In mathematics, there are many such surprises. Even easy to understand problems such as the ‘size of the multiplication sets’ defy solutions. It is such problems, which keep some of the brightest mathematicians engaged!
[HAJIRA KHAN is a student of Indian School Salalah; and SAMEEN AHMED KHAN teaches at Department of Mathematics and Sciences, College of Arts and Applied Sciences, Dhofar University, Salalah, Sultanate of Oman. firstname.lastname@example.org]