Category Archives: Middle Eastern mathematics

Six Times Seven is What Again?

Mathematics is complicated. I’m pretty sure most normal people would agree with me on that opinion. There are always rules to be understood and those special circumstances in which they may not apply, things to be memorized and foundations to gain before you can move onto the next complicated thing. At the foundation of all math, though, is the need to multiply.

We all remember growing up memorizing times tables. Gross. Taking those silly timed tests over and over again until we could finally (and usually lastly) get the 6’s, 7’s and 8’s. And once that foundation is built, we start doing bigger multiplication. Hundreds times thousands and the like. Even more gross. But there are so many ways, outside of our standard multiplication algorithm, that we can visualize multiplication! We learned of one such method in class, the Egyptian Doubling method. But there are many others as well; the Russian Peasant, Sieve Multiplication, Finger Multiplication and a new fun one, Line Multiplication.

The Russian Peasant method of multiplication is very similar to the Egyptian method. The reason these 2 methods are so similar is because they are both based off of a base 2 algorithm, or a binary system. In the Egyptian method, we can see that if we break a number into its binary components and then multiply those by our other number, we can get our product. In the Russian Peasant method, the same idea is applied in a different way.

85×18 done using peasant multiplication. Image: A. Bogomolny, Peasant Multiplication from Interactive Mathematics Miscellany and Puzzles.

Let us begin by trying 85 times 18. In this method, you start as the Egyptians did with 2 columns. 85 at the top of the left column and 18 at the top of the right. In the left column you begin dividing your number in half disregarding any remainders. So 85/2 would be 42. 42/2 would be 21. 21/2 would be 10 and so on until you are left with 1. On the right side, you then double your number for as many rows that were created in the left column. Finally, you will add up the right columned numbers associated with the odd left column numbers (disregard any evens). So in this case, you will disregard 42, 10 and 2 and add 18, 72, 288 and 1152. This gives you your answer of 1530 [1].

Now notice that in this method, the odd numbers play an important role as they are the ones that you keep while you disregard the evens. The odd numbers (or the numbers which you can break up as 2k+1) show you the binary decomposition. Wherever you have a one left over (aka an odd number), that is where you will place a 1 in your binary. So 85 would be 1010101. This is how you discover which numbers to add to get to your answer.

The Sieve method is commonly known as the lattice method and originated in the Middle East in the late 1300’s [2]. This method is most similar to our multiplication method of today because it uses a base 10 algorithm much like our own system.

58×231 done using the Lattice Method. Image: Javier Rosa, via Wikipedia

Let us multiply 58 by 213 by the Sieve method. First, create a grid where your first number is listed across the top and the second down the right side with each number getting a row or column. Next, place a diagonal through each square. Then multiply your row by your column. So the first square will by 5×2, the second, 8×2 and so on. Place the tens digit of your answer in the top half of your square and the ones digit in the bottom half. Now, add down your diagonals beginning with the bottom right corner, so in this example, 4. So the next number would be 8+2+5 or 15. Make sure you only write 5 and carry your 1 to the next diagonal. Finally, you read your answer by reading the numbers down the left side and across the bottom. So our answer is 12,354 [2].

8×7 done by Finger Multiplication. Image: G. Patrick Vennebush, via Math Jokes for Mathy Folks.

Remember how I said earlier that we all hate 6’s, 7’s and 8’s? There is a method of finger multiplication that is said to have come out of Italy and was widely used during the Medieval Period that was specifically designed for all times tables 5×5 to 10×10 [3]. To begin this method, understand that every raised finger is one more than 5. Let us do 8×7, one of the most brutal of the multiplication tables. On the left hand, you will raise 3 fingers (because 8 is 3 more than 5) and 2 fingers on the right (7 is 2 more than 5). You should have a total of 5 raised fingers and 5 closed fingers, 2 closed on the left and 3 on the right [6]. You will multiply all raised fingers by 10 and then add that to the number of closed fingers on the left multiplied to the number of closed fingers on the right. In this case we have (5×10)+(2×3) or our answer, 56. Pretty nifty huh?

The final method of multiplication I wanted to talk about is one that is commonly attributed, on the internet, to the Chinese or Japanese called Line Multiplication. But as I did my research a little bit more, there are no real references to this being a Chinese or Japanese method. One person found that the earliest reference to this method was a YouTube video in 2006 but no sources have been found to accurately date it [5].

22×13 using the Line Method. Image: Francis Su, via Mudd Math Fun Facts.

Let us do 22×13 for this method. Begin by drawing diagonal lines slanting up for the tens and ones digit of your first number (2 lines, a space and then another 2 lines). Then create diagonal lines slanting down (that intersect with your first lines) for the tens and ones digit of your second number (1 line, a space and then 3 lines). You will then circle where each set of lines intersect and count the number of intersecting points and place this number just under each group. In this case we have 2 on the left side, 6 and 2 down the middle column and 6 on the right side. Add these numbers moving down your column from right to left (carrying tens when needed). Here we get 286 as our answer [4].

The one thing that I love most about each of these methods is that each can apply to different types of learner. Some are more hands on, some are more visual and others are more mental. As I am going into teaching, I can implement each of these methods into my classroom so that each of my students is able to learn this foundational mathematical concept in their own learning style.

References

[1] A. Bogomolny, Peasant Multiplication from Interactive Mathematics Miscellany and Puzzles
http://www.cut-the-knot.org/Curriculum/Algebra/PeasantMultiplication.shtml, Accessed 30 January 2015

[2] Lattice multiplication. (n.d.). Retrieved January 30, 2015, from http://en.wikipedia.org/wiki/Lattice_multiplication

[3] West, L. (2011). An Introduction to Various Multiplication Strategies. 2-3. Retrieved January 30, 2015, from http://scimath.unl.edu/MIM/files/MATExamFiles/WestLynn_Final_070411_LA.pdf

[4] Su, Francis E., et al. “Visual Multiplication with Lines.” Math Fun Facts. <http://www.math.hmc.edu/funfacts&gt;.

[5] What is the origin of “how the Japanese multiply” / line multiplication? (2014, July 24). Retrieved January 31, 2015, from http://math.stackexchange.com/questions/877120/what-is-the-origin-of-how-the-japanese-multiply-line-multiplication

[6] Vennebush, G. (2011, May 28). Finger Multiplication. Retrieved January 31, 2015, from https://mathjokes4mathyfolks.wordpress.com/2011/05/28/finger-multiplication/

The Father of Algebra: Al-Khwarizmi or Diophantus?

The cover of Diophantus' book Arithmetica. Image: Public domain, via Wikimedia Commons.

The cover of Diophantus’ book Arithmetica. Image: Public domain, via Wikimedia Commons.

The history of algebra is very intriguing because of the many cultures that contributed to its origins. Although there were many ancient civilizations that studied algebra, there are two men that are best know for bringing algebra to our modern day: Al-Khwarizmi and Diophantus. The debate as to who is the “father” of our modern day algebra is still a subject of interest to which I hope to bring some light. I would like to share with you the lives of both of these mathematicians, their works and their legacy.

Diophantus:

Much of the life of the Greek mathematician Diophantus is unknown, but we do know that he lived in Egypt sometime after 150 BCE and before 350 CE. From what we have found it seems most likely that he lived during the 3rd century CE. We also have knowledge of his works that were popularized in the 17th and 18th centuries. Arithmetica, one of his greatest works, consists of 13 books of 130 algebraic problems. Out of these books, six were thought to be the only ones to have survived. However, in what’s known as Astan-i Quds, an Arabic manuscript, are thought to be the remaining books of Arithmetica. Even though these manuscripts have been found, some are not convinced of its veracity. The problems found in Arithemtica are known as Diophantine equations. These equations included polynomial equations, linear Diophantine equations, and Diophantine approximations among other Diophantine problems. Other works include Porisms, a collection of lemmas, and many works on polygonal and geometric, all of which helped expand mathematics.

Diophantine polynomial equations are polynomials with a number of unknowns for which only a rational solution is found. These equations usually had many solutions because of their many unknowns. Diophantus generally would only solve for one solution, instead of solving for all or them. A linear Diophantine equation is two sums of monomials of degree zero or more. To solve these equations one would have to use what is called Diophantine analysis. A Diophantine analysis would ask a series of questions, which would help find the solution.

Now the question is, what mark did this make on history? Although it is hard to know exactly who was influenced by Diophantus, we do have knowledge of many mathematicians who were influenced by his work. I would say the most famous work to have come from studying Diophantine equations was from Pierre de Fermat. De Fermat was studying Arithmetica when he scribbled “x^n+y^n=x^n where x, y, z, and n are non-zero integers, has no solution with n greater than 2.” This scribble is better known as Fermat’s Last Theorem, which later inspired algebraic number theorem. Other mathematicians that were inspired by his work are Andrew Wiles, who found proof of Fermats’ theorem, John Chortasmenos, a monk and mathematician, and Wilbur Knorr, a math historian. Above all else he was one of the first people to use symbols in mathematics. This is something we are all used to today in our mathematics from a young age.

Al-Khwarizmi:

A statue of al-Khwarizmi in Uzbekistan. Image: Jori Avlis, via Flickr.

A statue of al-Khwarizmi in Uzbekistan. Image: Jori Avlis, via Flickr.

Abu Abdallah Muhammad ibn Musa al-Khwarizmi, better known as al-Khwarizmi, was a Persian mathematician born in the latter part of the 8th century CE. One of his greatest works was Compendious Book on Calculation by Completion and Balancing. He also had books on arithmetic, astronomy, trigonometry, and geography to name a few. He also helped make the Indian numeric system part of western culture.

His most famous book, as mentioned earlier, is where we get the name algebra. The Arabic name of his book, Hisab al-jabr w’al-muqabala, Kitab al-Jabr wa-l-Muqabala, has the word Al-jabr, which means restoration. Al-jabr was the beginnings of the word algebra. When this book was translated into Latin it was called Liber Algebrae et Almucabola, which indicates clearly the source of algebra. This book expounded on the knowledge of quadratic functions among others. The book has hints of influence from past mathematicians, but the ties with Indian mathematics is most evident. One loss from the Indian mathematics was that of negative numbers. Because negative numbers were not used, equations with negative solutions were not studied. His book used squares, roots, and numbers to describe the equations. It also introduced the forcing of one side to be equal the other, which is what we would use today. This was the completing part. Balancing was done by subtracting the same amount from both sides of the equation. He also dealt with measuring areas and volumes. His work also included the concept of Algorithm, which is used in our everyday lives.

We now know what he taught, but, again, who or what was influenced by his works? In the 12th century, when his book was translated into Latin, Europe began to become familiar with his work. After a few centuries his work helped get Europe out of the dark ages.

Although I believe that both Diophantus and al-Khwarizmi contributed greatly to the math world, I think that al-Khwarizmi should be considered the father of algebra. This is due to the fact that his work is much closer to the algebra that is used today. His work was used for so long and was never lost. His worked helped Europe out of the dark ages, Diophantus did great work but al-Khwarizmi pushed the mathematical world in a great direction.

References

http://www.britannica.com/EBchecked/topic/164347/Diophantus-of-Alexandria#ref704023

http://www.onislam.net/english/reading-islam/research-studies/islamic-history/454243-al-khwarizmi-the-father-of-algebra.html

http://gulfnews.com/life-style/people/the-father-of-algebra-abu-jaafar-mohammad-ibn-mousa-al-khwarizmi-1.1233076

http://www.famousscientists.org/muhammad-ibn-musa-al-khwarizmi/