There golden section it’s a irrational numberindicated with the Greek letter φwhich is worth approximately 1.618 and corresponds to the ratio that two lengths must have in order for the greater length and the sum of the two lengths to be in the same ratio. It is found spontaneously in nature and is often associated with aesthetic beauty, but from a purely mathematical point of view it is special because it is considered the most irrational of all irrational numbers. mathematicians indicate with the Greek letter φand which answers the following geometric question
If we need to divide a segment long 1 into two parts so that the entire segment is to the larger part as the larger part is to the smaller part. How long should the large part be? That is, how much should the relationship (the division) between the two parts be worth?
The solution to this problem is found by solving a second degree equation starting from the proportion “1:φ=φ:(1-φ)” and it is a number that is calculated with fractions and square roots (see figure above). If we write this number in decimal form we obtain
φ=1.6180339887…
where the dots indicate that the digits after the decimal point are infinite and follow one another in an irregular manner: it is in fact a irrational number which cannot be written with a finite or periodic decimal representation.
Let’s try to understand better,
the rational numbers it’s all those numbers that they can be written as a fractionwith integer numerator and denominator, without commas.
For example ¾ and ⅓ are rational numbers while the golden ratio is notit is said instead that it is irrational because, due to that square root of 5 that is used to calculate it (see figure above),
the Irrational numbers cannot be written as a fraction with integer numerator and denominator.
When we try to write a rational number in decimal form there are only two possibilities:
- the decimal number has a finite amount of digits after the decimal pointfor example ¾=0.75 has exactly two digits after the decimal point
- or the decimal number is periodicthat is, it has one or more digits that are always repeated the same, for example ⅓=0.3333… with the digit 3 repeating infinitely, so even if we can’t write all the digits, we know that they are all worth 3
On the contrary, the decimal form of an irrational numberlike the golden sectionpresents a infinite number of digits after the decimal point That they follow one another in an irregular mannerwithout periods, and it is not possible to know them all: at most try to calculate as many digits as possible, as happens for the most famous of irrational numbers, the pifor which there is a record of known figures which is updated from time to time.
The search for the digits that are after the decimal point of an irrational number can take more or less time and in the case of the golden section this search takes a lot of time, and this is why it is considered more irrational of pi and all other irrational numbers.
But how do you look for (and maybe find!) the digits of an irrational number like φ? One possibility is to use the continued fractionsa particular way of representing numbers that can be used to calculate approximations of irrational numbers. Let’s see what it’s about.
A continued fraction is made up of a number added to a fraction with numerator 1 and denominator which in turn contains a number added to another fraction whose denominator can in turn contain a number added to a fraction, and so on.
In the figure below we have reported the continued fractions relating to the numbers 5/3, 3/5 and 225/157.
The numbers we have considered are all rational numbers and their continued fractions are finite:
every rational number can be represented with a finite continued fraction.
As for the irrational numbershowever, every time we seem to have found the last small fraction at the bottom we realize that this in turn contains a sum between a number and a new fraction:
irrational numbers cannot be represented by finite continued fractions, but only by infinite continued fractions.
As regards the specific case of golden section its continued fraction contains only ones, as seen in the figure below.
This fraction is not just a choreographed way of writing the golden ratio, it can also be used to look up its approximate values. This can be done by truncating the continued fraction, or by replacing the infinite series of fractions from a certain level onwards with 1. In the figure below we did it at the third level obtaining the number as an approximation 3/2=1.5 which is not a great approximation given that only the figure “1” before the decimal point is equal to that of the golden ratio.
In the figure below you can see how, stopping at the 6th level, the approximation is 13/8 = 1.625 which shares only the numbers with the golden ratio 1.6.
Using this procedure, the first approximations of φ turn out to be in the order 1, 2, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, which approximate the number respectively to the digits 0, 0, 0, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 5, 5. These are approximations that they get better and better as we go deeper.
The same procedure can be used to approximate any irrational number starting from its algebraic fraction, gradually obtaining better and better approximations. The problem with the golden section is that this approximation process is very slowmuch slower than what happens for any other number, which is why it is considered the most irrational number of allbecause it is so far from being rational that it takes a lot of time to approximate it with its continued fraction, more than it takes for any other irrational number.
