By Dr Manas Kumar Haldar
Everyone knows what a whole number (or integer) is. It is numbers like 1, 2, 3, etc. We also know that a fraction is a whole number divided by another whole number. The idea of fractions comes naturally to us, for example, half a glass of water or two thirds of students in a school. These are the fractions 1/2 and 2/3, respectively. Indeed, you might recall from your primary school lessons that the concept of fractions was introduced with diagrams of things we encounter in everyday life. You might also have encountered continued fractions in your high school. If you haven’t, here is an example of a continued fraction:
Notice that it looks like a staircase. At each step we have a whole number plus 1 divided by a whole number plus 1 divided by …. In the example above, there are a finite number of steps. Let us find the value of the continued fraction given in the example in terms of a fraction. 2+1/3 gives us 7/3. 1 divided by 7/3 gives 3/7. Add 3 to get 24/7. Now divide 1 by 24/7 to get 7/24 and add 2 to get the fraction 55/24.
We call the inverse process, getting a continued fraction from the fraction 55/24, synthesis for reasons given later. How do we carry out the inverse process? Very simple – divide by 55 by 24 to get 2+7/24. Now write 7/24 as 1 divided by 24/7 and continue the process. As far as I know, continued fractions were first used by the Indian mathematician Aryabhatta in the 6th century. The interest in continued fractions re-emerged in Europe only in the 15 century and “continued fraction” was coined by the Oxford mathematician John Wallis.
Continued fractions go much beyond fractions. They may have an infinite number of steps. So you can not express such beasts by fractions. Numbers which cannot be expressed as whole numbers or fractions are called irrational, perhaps because the Greeks, who believed that the universe can be described by whole numbers and fractions, had to accept such numbers grudgingly. Some of you will recall that the number pi used to calculate the area and circumference of a circle cannot be represented by a fraction. So it is an irrational number.
Continued fractions continue to fascinate mathematicians and scientists to this day. The more mathematically minded among you might like to read a very nice article on continued fractions, Chaos in Numberland. If you see this article, you may say, “Huh, this is all maths. What practical use has it anyway?” Well, unknown to those who are not electrical engineers, you are reaping the benefit of this simple school math everyday, for example, when you use mobile phones. Communication systems require electrical filters to accept the desired signal from a host of other signals. Modern filter synthesis requires expansion into continued fractions. The mathematics involved may be difficult for the general public. So I may be forgiven by erudite readers for giving a simple explanation. From your secondary school maths, you may know that a fraction may also be a ratio of algebraic expressions in terms of a quantity, say, f. Now f is the frequency of the signals. The fraction represents how the electrical output of the filter changes with the frequency, f. You can choose various fractions such that your filter will have very low output for undesired frequencies. Having chosen the fraction, you can expand it as a continued fraction. Now the response of a particular type of electric circuit made up of inductors and capacitors, called a ladder network, can be expressed in terms of continued fractions. By making your synthesized continued fraction the continued fraction of the ladder network, you get the values of the inductors and capacitors of the ladder. Voila, your filter design is done!
Filter design is still very much a topic of research. If you visit the School of Engineering at Swinburne Sarawak, you will find several students engaged in filter research to earn their doctor of philosophy degrees in electrical engineering. So, simple school math can be quite important in engineering.
Dr Manas Kumar Haldar is an associate professor with the School of Engineering, Computing and Science, at Swinburne University of Technology Sarawak Campus. He can be contacted firstname.lastname@example.org