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Thinking about numbers



A diagram of a circle, with the width labeled as diameter, and the perimeter labeled as circumference



What are numbers?  Are some numbers more real than others? More rational? Just … better?

No, of course not! You reply. It is absurd to make a moral or quasi-moral judgment about or classification of numbers. Numbers are just … numbers.  Numbers are simply sums or aggregates. Each is a very short way of saying something about a collection of things, events, or people, as in the perfectly comprehensible sentence, "a number of people were hurt in the accident." There is no need to make heavy water out of this.

But then, why do categories of numbers have the strange and sometimes stigmatizing names they do? Some numbers are called “irrational,” in contrast to the “rational.” Others are called “imaginary,” in contrast to the “real.” What is irrational about the former or unreal about the latter?

It is a tricky question. But it is worth unravelling this knot. It is worth so even if you took no pleasure in mathematics at school, or since, and even if you are content in an occupation that doesn’t require mathematical skill. It is worthwhile because, as I hope my next book will show, nothing less than freedom is at stake in getting this right.

There has of late been a great and important advance in the understanding of numbers. It is the fractal revolution, and it came about with the essential assistance of those oddly named ducks, the imaginary numbers. This advance changes our understanding of reality, and will in the course of time, I am confident, change our understanding of our relations with one another as well.

Let’s launch ourselves into this distinction between rational and irrational numbers.

A rational number is any number that can be represented as a simple fraction. The number 1 is rational. It can be represented readily as 1/1, or 2/2, or 3/3 or so forth. But the ratio of a circle’s circumference to its diameter, pi, also traditionally represented as π, is not a rational number. In fractional terms, it is approximately 22/7. But … not quite. Indeed it is ever not quite. It slips away from you whenever you try to pin it down.

If we represent numbers in decimal rather than fractional form we can express the strangeness of pi, its irrationality, in a different way....   
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  1. VincentFebruary 15, 2013 at 10:00 AM

    good,,, i want to share about irrational number

    http://www.math-worksheets.co.uk/048-tmd-what-are-irrational-numbers/

    ReplyDelete

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