In part one of this article we discussed numbers in nature and their possible philosophical significance. In this part we consider some other unusual aspects of numbers.
Zero and Infinity
Unlike other real numbers zero and infinity do not have any quantitative attribute. Strictly speaking they are not numbers although mathematicians do consider zero as a finite number. The statement that I have five dollars makes perfect sense but ‘I have zero dollars’ does not. If asked to I can show the five dollars but cannot show the zero. So zero is just a concept like infinity. The concept of a thing is not the thing itself but only an attempt to describe what it is.
Zero and infinity figure prominently in philosophy, especially in Eastern philosophies. They are considered two sides of the same coin, as are all pairs of opposites. They are interlinked like the two ends of an infinite spiral and also contained in each other. We cannot go into the details of these philosophical arguments here and simply mention that the Ultimate Reality is described as ‘smaller than the smallest and larger than the largest’ (Vedanta) and ‘nothing is everything’ (Tao).
Irrational and Transcendental Numbers
Numbers that cannot be expressed as a ratio of two integers are called irrational. I suppose they are called irrational because of the belief that the only rational way of expressing a number is in terms of two other whole numbers. Their decimal expansions do not terminate nor become periodic. The most well known irrational number is the square root of two. These numbers are, by definition, indeterminate, although geometrically one can get the value of square root of two by measuring the hypotenuse of a right-angle triangle with height and base equal to one. However, as we shall see later, it brings in another problem – the error of measurement.
As one can guess from the name itself, the definition of transcendental numbers is a little complicated. (These are numbers that are not roots of integer polynomials so they are not algebraic numbers of any degree. ) All transcendental numbers are irrational. Frankly I do not know what they transcend other than an easy definition! Anyway, two are most common. One is associated with circle and is denoted by the Greek letter Pi; the other is denoted by e for exponential and is associated with logarithm. Another such number is the so-called golden ratio denoted by the Greek letter Phi, which is apparently favored by nature and can be traced even in human anatomy. Architectures using this ratio are aesthetically more pleasing. It also related to Fibonacci sequence discussed in part 1 of this article.
Approximations and Errors
Since the values of irrational and transcendental numbers cannot be determined precisely, one has to resort to approximation. Any approximation has some inherent error. This means that the use of these numbers will not yield a unique result. The circumference or the area of a circle cannot be determined precisely because it involves multiplication by Pi. The exponential growth of any variable quantity cannot be calculated exactly because of the use of e. The same is true for using the golden ratio. If we use geometrical methods of determining the values of the square root of two or the circumference of a circle, the approximation error translates into measurement error which can never be completely eliminated.
This takes us back to what was mentioned about zero. No matter what we do we can never get to zero. The same holds for infinity. This bears an analogy to the spiritual goal of reaching the Ultimate Reality. If this goal is nirvana, it cannot be attained while living. One can only strive to get as close as possible and the closest possible approach is called enlightenment.
Dharmbir Rai Sharma is a retired professor with electrical engineering and physics background. He obtained his M. S.degree in physics in India and Ph. D.in electrical engineering at Cornell University. He has taught in universities here and also in Brazil, where he spent sometime. He maintains a website http://www.cosmosebooks.com devoted mainly to philosophy and science.