A day is not completely wasted if you learn something. Today I learnt something that I don't think they taught me at school.
Irrational numbers are those numbers like PI that can only be expressed with an infinite and unrepeating sequence of digits after the decimal point. PI, for example is 3.141592654 etc. etc. ad infinitum without repeat or pattern. Other numbers, such as 3.3333333..., are rational numbers because there is a pattern to the digits after the decimal. I always thought - and this is the bit I don't recall from school - that irrational numbers were so called because they seem to defy common sense. Not so. They are called irrational simply because they cannot be expressed as a ratio of two integers. The ratio (fraction) 22/7 is a pretty close approximation to PI, but it is not PI. In fact, according to Excel, it bottoms out at 3.14285714285714 (followed by an infite series of zeroes). There is no ratio of integers that is exactly equivalent to any irrational number.
Pythagoras was a world class thinker who reputedly first worked out that the square of the hypoteneuse of a right angled triangle is equal to the sum of the squares on the other two sides (Pythagoras' theorem). This might not seem important to the average Joe in C21 armed with his PC and calculator, but it's pretty important of you're trying to design a building, measure the height of a mountain or fence a farm. Pythagoras' followers were called Pythagoreans and they got in a bit of a tizz over irrational numbers when they first discovered them. Thitherto they had thought that all numbers formed an uninterrupted continuum, any point of which can be defined as an integer or the ratio of two integers. So, for example, the series from 1 to 5 can be subdivied into 1, 1½ (=3/2), 2, 2½ (=5/2) 3, 3½ (=7/2), 4, 4½ (=9/2), 5, and the series can be subdivided ad infinitum using ever smaller fractions to fill in the gaps between the integers. All well and good, but where do the irrational numbers fit in? The answer is somewhere between the fractions. PI fits in between 3 and 4 just below 22/7, but as there is an infinite number of fractions we appear to have a bigger infinite number if we also include irrational numbers. This was pretty devastating to the Pythagoreans who treated it as heresy, having imbued their number system with religious significance. They may even have tried initially to conceal the existence of irrational numbers because they had no place in their conception of reality. The suppression of inconvenient mathematical truths has been going on ever since. Copernicus knew it, Galileo knew it and even as late as 1897 the US State of Indiana enacted legislation that, inter alia, redefined PI as 16/4 (=3.2).
Mathematical and scientific heresies still exist today. If you disagree with me, try claiming that there is no proven causal link between smoking and cancer in humans or between carbon emissions and increases in mean world temperature. There isn't; there is powerfully persuasive correlational evidence but no proof of causality. Nevertheless, if you have a voice, and you say it, you'll be pilloried in the press.
Thanks to
I used this site to look up details of the Indiana PI Bill:
This is a link to the book by Michael Guillen in which I read the essay about irrational numbers:
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Bridges to Infinity: The Human Side of Mathematics
Michael Alec Bryant
18th November 2007
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