diff --git a/guide/english/mathematics/irrational-numbers/index.md b/guide/english/mathematics/irrational-numbers/index.md index 767ba26e265..898ccaa9d06 100644 --- a/guide/english/mathematics/irrational-numbers/index.md +++ b/guide/english/mathematics/irrational-numbers/index.md @@ -7,6 +7,21 @@ An irrational number is a real number that cannot be written as a fraction of in 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](https://en.wikipedia.org/wiki/Irrational_number#Ancient_Greece) 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](https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational#Lambert's_proof) that anyone was able to prove it to be irrational. Similarly, e had been used since the early [1600s](https://en.wikipedia.org/wiki/E_(mathematical_constant)#History) but was not proven to be irrational until [1737](https://en.wikipedia.org/wiki/Proof_that_e_is_irrational). 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! @@ -15,5 +30,22 @@ Given [Cantor's theorem](https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argum 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. -However, for most of your computing needs, there is no need to worry about this complexity! For languages like [JavaScript](https://stackoverflow.com/questions/1458633/how-to-deal-with-floating-point-number-precision-in-javascript) the math and calculations are done with [finite precision](https://en.wikipedia.org/wiki/Floating-point_arithmetic) 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](http://www.sagemath.org/), but if you do not need lots of accuracy this is completely unnecessary. +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](https://en.wikipedia.org/wiki/Decimal#Rational_numbers)), 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](https://en.wikipedia.org/wiki/Continued_fraction) 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](https://en.wikipedia.org/wiki/Square_root_of_2#History) 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](https://en.wikipedia.org/wiki/Hippasus) was drowned for claiming such heresy.) A proof that Euler's number `e` is irrational was not known [until 1737](https://en.wikipedia.org/wiki/Proof_that_e_is_irrational#Euler's_proof) and a proof that pi is irrational was first given [in 1761](https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational). 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)`](https://en.wikipedia.org/wiki/Irrational_number#Irrational_powers), 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](https://math.stackexchange.com/questions/11669/mathematical-difference-between-white-and-black-notes-in-a-piano), [geography](https://www.theguardian.com/science/alexs-adventures-in-numberland/2015/mar/14/pi-day-2015-pi-rivers-truth-grime), [physics](https://en.wikipedia.org/wiki/Einstein_field_equations), [biology](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3677036/), [pharmacokinetics](https://en.wikipedia.org/wiki/Pharmacokinetics#Metrics), [shockwaves](https://en.wikipedia.org/wiki/Burgers%27_equation), [heat](https://en.wikipedia.org/wiki/Heat_equation#Fundamental_solutions) and much, much more. + +However, for most of your computing needs, there is no need to worry about this complexity! For languages like [JavaScript](https://stackoverflow.com/questions/1458633/how-to-deal-with-floating-point-number-precision-in-javascript) the math and calculations are done with [finite precision](https://en.wikipedia.org/wiki/Floating-point_arithmetic) 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](http://www.sagemath.org/), but if you do not need lots of accuracy this is completely unnecessary.