The ancient Babylonians studied polynomials in some depth. Of course, algebra had not been invented then, nor had graphs. However, the method that they used to analyse all kinds of mathematical patterns is one that fundamentally underlies polynomials.
At university, I learned that the Babylonians had tried to analyse the law of refraction using this method, but did not get accurate results. We now know that the refraction of light is based on trigonometric functions rather than polynomials (see Snell’s Law), but the Babylonians were not aware of that (as trigonometry had not been invented then either!).
It transpires that, if you write out a table of values for a polynomial (such as the quadratic expressions at left) and calculate the differences between the terms, and keep repeating the exercise, you will eventually obtain a row of identical numbers! This is true of all polynomials and is true of no other function. It is a special property of polynomials! This was the natural way that Babylonians analysed sequences of numbers.
What makes it particularly interesting is that the number of rows of differences required is the same as the degree of the polynomial (i.e. its highest power). In the examples shown, you can see that the analysis of quadratic expressions finishes at the second row of differences! If the analysis took three rows, we would know that we were dealing with a polynomial of degree three (a cubic), i.e. its leading term would be ax³.
I first encountered this form of analysis as a school student when I read Warwick Walter Sawyer‘s excellent book, The Search for Pattern. This, and other books that he wrote, revolutionised my love for and understanding of mathematics. I highly recommend that you try to obtain copies of his books and read them. Fortunately, The Search for Pattern can now be downloaded for FREE in a number of formats from The Internet Archive. I recommend that you read Chapter Eight: A Method of Discovery from page 166. The image at top left is from this chapter.
I hope this has whetted your appetite. It is time to read on …
You can use this ancient skill to solve difficult problems without algebra!
The material that I share in these three videos is rarely taught in schools or universities ... but it should be. You will earn how Babylonian mathematicians analysed number patterns 3,000 years ago and how we can use the same methods to better understand our polynomials today ... including our linear and quadratic equations! The method is astoundingly simple to understand, and very simple to learn and use. You only need to be about to count, add and subtract!
I first discovered this material when I was 15 years old, in a book by W W Sawyer called The Search for Pattern. This book revolutionised my understanding of mathematics and opened my eyes to so much more than that which was revealed through textbooks at school. Let me encourage you to read Sawyer's book yourself. You may obtain copies of it via BookFinder.com. I have also discovered that you can download a free copy of it from The Internet Archive (PDF file is 12.4 MB).
W W Sawyer was an English mathematician who wrote twelve books with the intention of explaining and popularising mathematics. His first such book, Mathematician's Delight (1943) sold over 500,000 copies and is probably still the most popular/successful mathematics book ever written. To learn more of this amazing mathematician and his work, let me encourage you to visit the W W Sawyer web site set up by the Navnirmiti Learning Foundation in Pune, India (Sawyer's family donated material to the Navnirmiti Foundation after his death in 2008).
Sawyer also had an enormous passion for education and you may read many of the articles that he wrote at the Marco Learning Systems site which is partly run by his family. If you value mathematics education I am sure that you can learn from this man and his work! You would do well to read about Morris Kline while you are there.
I have much to thank Walter Warwick Sawyer for. He writes with passion and in such an engaging way that you feel that you are joining him in an adventure. He certainly inspired me as a young mathematician.
In the meantime, please watch my video and learn about this wonderful analytical method!
If you asked me to list the most important, or exciting, mathematical discoveries that I made during my school years this would have to be in the top handful. I used to wonder how to create a formula to match a table of values ... and discovered part of the answer in W W Sawyer's book The Search for Pattern.
Interestingly, the process starts with an extraordinarily simple analysis method that is 3,000 years old! I shared about this Babylonian Differencing Technique in my previous video. It is so simple that it only requires that you be able to add and subtract numbers.
Once you have performed the analysis of your figures, you will then create your formula using some algebraic patterns that were developed (properly) only during the last few centuries. In this video I take you through the whole process (that is why the video is so long).
As a school student who enjoyed his mathematics, I cannot convey to you the sense of power and accomplishment that knowing this method provided. Suddenly, I gained new insights into a lot of other aspects of my algebra and graph work. I commend this video to you.
Please watch this full version (37:19) for a complete explanation:
If you have already seen the first video in the set (Solving Difficult Problems Without Algebra) and understand factorial notation and combinations [such as 5C3 = 5!/(3!2!)], then you may wish to watch this shorter (7:34) version:
This is the third and, for the time being, the last of my videos about the Babylonian Differencing Technique. In some ways this video is the most creative of the three. This is because I show how the differencing pattern can provide clues that allow us to deduce the formulae for sequences such as the Fibonacci Sequence.
Please watch and learn, and then go exploring with your new tool! You might also like to look up Padovan Numbers, Perrin Numbers, and Lucas Numbers and experiment with those number patterns. See if you can create formulae for their nth terms. This is where you can break free of standard text-book, crank-the-handle style questions and truly experiment and learn like a 'real' mathematician. There is no 'right' and 'wrong' ... the more you fiddle and play with the numbers and patterns, the more you will learn, and gradually insights will come to you.
This (the adventure of exploring in this way) was one of my big discoveries as a high school student and it goaded me to explore most of the mathematics topics that we encountered at school in a similar fashion. Never be content simply to answer the question that is asked in your text book. Ask yourself how it could be different ... what are you not being told ... what if you did something a little differently?
Please let me know if you decide to do your own exploring/research, and how this changes you and your relationship with mathematics. In the meantime, please enjoy the video:
By far the best educator I have been subject to in my schooling life. Apart from showing me with the utmost clarity the many concepts of mathematics, Graeme has inspired me to better myself in everything that I do in school and in my life. His dedication to his craft is truly admirable, and this level of dedication to helping the student achieve is something that is extremely hard to find. If you’d want anyone on your side during HSC mathematics, it’d be Graeme.
Harry B (student, 2012)