4b110d7bb8
* Corrected errors and expanded discussion Discussed rational vs. irrational numbers, gave examples, added discussion into the difficulty in studying them as well as real world examples to exhibit their vast importance and utility. * Update index.md
7.0 KiB
title
| title |
|---|
| Irrational Numbers |
Irrational Numbers
An irrational number is a real number that cannot be written as a fraction of integers, i.e., a number that is not rational. For example, pi, e and sqrt(2) are all irrational.
A common misconception is that a number that has an infinite decimal expansion must be irrational but this is completely false, for example, 1/3 = 0.333... is clearly rational but the decimal expansion has an infinite number of 3s. The correct characterization is that a real number that does not have a finite decimal expansion is irrational if the expansion does not eventually become a subsequence that repeats itself forever.
The number
1/2 = 0.5
is not irrational as the decimal expansion terminates (or repeats with 0s). The number
0.12543297051212121212...
which continues with 12 is not irrational as it eventually becomes a pattern that repeats forever. (In fact, one can show this decimal is equal to 41392880269/330000000000.) However, the number
0.101001000100001000001000001...
where we place a 1, then a 0, then another 1 and two 0s, then another 1 and three 0s, etc... always adding one more 0 between each consecutive pair of 1s never repeats itself, so this number is irrational.
The irrational numbers, despite having such a simple definition, are notoriously difficult objects. The square root of 2 was perhaps the first discovery that numbers that were not rational existed, shown around the year 500 BC, and while pi had been used since the early Egyptians it wasn't until 1761 that anyone was able to prove it to be irrational.
Similarly, e had been used since the early 1600s but was not proven to be irrational until 1737. So, while knowing a rational number is rational amounts to finding a repeating pattern in its decimal expansion or finding a fractional representation, showing something is not rational can takes decades or centuries!
Given Cantor's theorem it is known that most real numbers are irrational, so if you pick a number "at random" it is almost surely irrational, and yet it is likely impossible to prove so using the techniques available. Even worse, numbers such as pi + e, pi*e or pi^e are irrational, though it is hard to find anyone that believes any of them admits a fractional representation (or repeating pattern in their decimal expansion) that has not yet been discovered.
One of the problems in trying to prove things about irrational numbers is that they do not behave very well: unlike the rational numbers, the sum/product of two irrational numbers need to be irrational, e.g., pi + (-pi) = 0 and sqrt(2)*sqrt(2) = 2, so it is hard to prove general statements about the set of irrational numbers.
While proving a number n is rational is as simple as just finding integers a and b such that n = a/b (and being rational means such integers can always be computed), to show that a number m is irrational is not as simple because you have to show that no such fraction a/b equals m, not just fine one that isn't.
Similarly, using finite/repetition in decimal expansions is not as easy, as any rational number will eventually terminate or repeat, so to show a number is rational this way you need only compute the decimal expansion up to this point. However, showing a pattern never occurs cannot be done by simply showing a pattern has not appeared yet up to any finite point.
One other useful tool for studying irrational numbers is with continued fractions as every rational number has a finite continued fraction, whereas every irrational number has an infinite continued fraction. But once again, the same problem occurs. To show a number is rational using continued fractions is a finite process, simply compute the continued fraction until it stops. However, showing that a continued fraction has not stopped yet is not the same thing as proving is never stops.
Some of the more well known irrational numbers are
sqrt(2) = 1.41421356237...
e = 2.718281828459...
pi = 3.14159265359...
A proof that sqrt(2) is irrational was first given by Pythagoreans in ancient Greece, and was the first proof that there existed numbers that were not rational. (Rumour has it this discovery was so shocking at the time that Hippasus was drowned for claiming such heresy.) A proof that Euler's number e is irrational was not known until 1737 and a proof that pi is irrational was first given in 1761. Despite these proofs being given over 250 years ago, it is not yet known if pi + e or pi*e are irrational.
The sum of two irrational numbers can be irrational, e.g., sqrt(2) + sqrt(3), or rational, e.g., sqrt(2) + (-sqrt(2)) = 0. The product of irrational numbers can be irrational, e.g., pi*pi, or rational, e.g., sqrt(2)*sqrt(2) = 2. The power of two irrational numbers can be irrational, e.g., sqrt(2)^sqrt(2), or rational, e.g., [sqrt(2)^sqrt(2)]^sqrt(2) = sqrt(2)^2 = 2.
Despite irrational numbers being rather difficult to work with in numerous ways, they are an integral part of everyday life, with music, geography, physics, biology, pharmacokinetics, shockwaves, heat and much, much more.
However, for most of your computing needs, there is no need to worry about this complexity! For languages like JavaScript the math and calculations are done with finite precision and so irrational numbers are never available. You can build your own data structures to work with irrational numbers, such as what is done in SageMath, but if you do not need lots of accuracy this is completely unnecessary.