About Multiple Methods of Multiplication Why look at other ways to multiply when our usual algorithm works just fine? First, this provides alternatives for kids who struggle with the usual algorithm. Second, thinking about how these other algorithms work makes us think more deeply about numbers and operations. Third, it's kind of fun to figure out how people from other times and places using other numeration systems computed. The Egyptian method for multiplication, in particular, makes us appreciate the ease of calculation in the number system which comes to us from the Hindu Vedic tradition. Ancient Egyptian multiplication The Egyptian numeration system was radically different from the one we use today. Both that system and our current system represent numerals with powers of 10. Beyond that there is not much similarity. Our current system has a different symbol to represent each number from 0 through 9, and the placement of the symbol in the numeral determines whether that symbol represents the number of ones, or tens, or hundreds, etc. The Egyptian system had a different symbol for each power of ten, and the number of copies of the symbol indicated how many of that power of ten was in the numeral. This is an awkard numeration system in which to create an easy algorithm for multiplication. Multiplication is just repeated addition of a number. The multiplication algorithm which the Egyptians came up was remarkably efficient use of repeated addition to get around the limitations of a number system without place value. It is probably the first application of binary numbers. Vedic mathematics The history of mathematics in India is ancient. The Hindu Vedic tradition is an oral tradition of knowledge passed down in short verses, dating to before the invention of paper. The Vedas encompass a broad spectrum of knowledge, including the sutras (verses) pertaining to mathematics. In the early 20th century Swami Shri Bharati Krishna Tirthaji Maharaja claimed to have rediscovered a collection of 16 ancient mathematical sutras from the Vedas and published it in a book called Vedic Mathematics. Historians do not agree on whether or not these were truly part of the Vedic tradition. If these sutras date back to the Vedic era they were certainly part of an oral rather than a written tradition. However, they are a novel and useful approach to computation: they are flexible in application and easy to remember. They can often be applied in algebraic contexts as well as in simple arithmetic.. "Vertically and Crosswise" bridges the gap between arithmetic and algebra: the algorithm is very similar to the standard algorithm used in the US and also is similar to the "FOIL" (first, outer, inner, last) rule used for multiplying binomials in algebra. The oldest surviving math texts from India include a text treating math from a geometric perspective, the Sulbasutra written by Baudhayana around the 9th century BC, and the Aryabhatiya by Aryabhata, written around the 5th century AD. If the Vedic Sutras collected by Bharati Krishna have a historical source, they are probably older than either of these texts. They are in any case an interesting alternative approach. Our number system is derived from the Hindu numeration system which evolved from Vedic mathematics. The concept of zero and the Hindu-Arabic numerals found their way to Europe in the 12th century with Fibonacci, 5 centuries after the work by Hindu mathematician Brahmagupta. It is the place value system with zero which makes it possible for us to use our familiar multiplication algorithm and spares us from coping with the Egyptian algorithm or the abacus. Why have a picture of a lamp on this display? The lamp is a symbol for the Hindu festival of Diwali, an Indian festival of lights. One of many interpretations of Diwali is a celebration of the light of knowledge scattering the darkness of ignorance. Napier's Bones Napier's main contribution to mathematics, logarithms, arose from his desire to take the tedium out of computation. The difficult computations of his time were multiplication, division, and square and cube roots. Accurate computation was important for applications in navigation--the stars were the global positioning system of Renaissance Europe, and locating a position on the globe required careful computations involving trigonometry. The bones, of course, were an aid to quick and accurate multiplication. Using the bones for multiplication is similar to using the lattice, or Gelosia, method of multiplication. Materials preparation for this module The materials needed for this are a set of Napier's Bones and handouts for Egyptian multiplication. The Printable material includes a set of paper Napier's Bones which are intended to be cut out and glued onto 3/4 inch craft sticks (extra large popsicle sticks). I like to have three copies of each "bone". Running the activity The multiplication problems are not provided on this website. The person in charge of the display can make up multiplication problems and encourage the visitors to the display to try one of the other methods of working the problem. He or she will of course want to practice doing a few problems using each of these methods before kids show up. If there are difficulties producing problems, the problems given in Notes to the Volunteer can be used. References For general information on the history of mathematics, the History of Mathematics of St. Andrew's University's Mathematics and Statistics Department. The book Multicultural Mathematics by David Nelson, George Gheverghese Joseph, and Julian Williams contains, among other interesting topics in math history/culture, a description of ancient Egyptian multiplication. Two of the methods on this display were featured in an article in the December 2001 issue of Mathematics Teaching in the Middle School: "Multiplication from Lilavati to the Summa" by Rebecca Berg (vol 7, #4, p. 226ff). You can explore Egyptian arithmetic at these websites: Mathematics articles by Shelley Walsh You can find out more about Vedic mathematics at these websites: Nuggets from Vedic Mathematics by R. Gupta Vedic Mathematics This barely touches the surface. A web search will bring up many more sites about Vedic mathematics. The diwali lamp comes from this website: The India Clip Art Collection by Sumit Baghra A comprehensive biography of John Napier, with his contributions to mathematics, can be found at John Napier in History Topics, St. Andrew's. The article "The Great Mathematicians" by Herbert Westren Turnbull in The World of Mathematics, ed James Newman, contains another brief biography and summary of Napier's work. Acknowledgements Thank you to Juri Bhattacharyya for help with the printing and interpretation of the Sanskrit script on the display, and with Hindu cultural background. Written May 13, 2002 by Janet A. Dixon Revised Sep 9, 2004 Revised Jan 10, 2006