 # Simple school math and its importance to engineering

February 22, 2012

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.