3.4 Irrational Numbers

If an operation applied to a rational number (root taking, limit of an infinite sequence of rational numbers) leads to a number that is not a rational number, i.e. if it cannot be written as a ration of 2 integers, which is representable as a finite or periodic decimal fraction, then this number is considered as an irrational number. Here the term irrational has historical reasons, as a demarcation from the rational(numbers that are ratios) and has no secondary meaning irrational = unreasonable or unthinkable.

If one applies the operations mentioned above to irrational numbers, then this does not yield another type of number.

Rational numbers constitute a countable set - they can be ordered in such a way, that they constitute a countable sequence. The irrational numbers however do not constitute a countable set. In this sense there are more irrational as rational numbers.

3.4.1 Algebraic Numbers

The need to introduce numbers that are not rational, the Pythagoreans (Pythagoras, 570 - 510 b.C., mathematician and natural philosopher in the Greek colony metapont in southern Italy ) recognized during their reflections about the calculation of right triangles with the hypotenuse $Math content$ and the legs $Math content$ and $Math content$.

In the domain of integers there are only few solutions, the Pythagorean triples, that are often used in homework problems (3, 4, 5; 6, 8,10; 5, 12, 13; 8, 15, 17; 7, 24, 25; 9, 12, 15; 10, 24, 26; etc)

Theorem of Pythagoras: $Math content$

Example of an integer solution: $Math content$

Example of a rational solution:$Math content$

Example of an irrational solution: $Math content$

Numbers, that are in general obtained as solutions of polynomial equations with rational coefficients, i.e. that are their roots, are designated as algebraic numbers. The include both rational as well as irrational numbers. Rational numbers are thus rare special cases of irrational numbers

3.4.2 Transcendental Numbers

Irrational numbers, that are not a root of a polynomial with rational coefficients , are called transcendental numbers.

Here transcendental simply means going beyond the rational numbers and does not have any mystical background whatsoever.

The most common transcendental numbers are the circle number $Math content$ and the Euler number $Math content$ ( written in blocks of in the following(

$Math content$

$Math content$ It is a characteristic feature of transcendental numbers, that they are limits of infinitely often repeated operations (additions, multiplications, formation of continued fraction, root taking, etc.) (see below).

3.4.3 $Math content$ and the Quadrature of the Circle according to Archimedes

Using the example of the number $Math content$ it will be demonstrated, how this transcendental number of high practical importance can be obtained as the limit of a sequence. We follow the famous train of thought due to Archimedes.

Using the theorem of Pythagoras and the formula for the area of a triangles with baseline $Math content$ and height $Math content$ i.e. $Math content$ the mathematicians and surveyors of Egypt and antiquity were able to reduce the area of an arbitrary surface that is bounded by straight lines to that of a square of the same area, whose length is given by a square root, i.e mostly an irrational number; even today the unit of surface area for arbitrarily bounded surfaces is still the “square meter”.

The “Quadrature of the circle”, as paradigm of calculating the area of a surface, that is bounded by curved lines, stayed however unsolved for a long time. Archimedes

The famous inventor and mathematician Archimedes (287 - 221 b.C.), who lived in the greek colony Syracuse in Sicily found a royal road to this end, that was only developed further nearly 2000 years later, and that represents the start of working with convergent, infinite sequences and with limits.

His method, which starts with a polygon that is inscribed or circumscribed to a circle (Figure 3.2), will be demonstrated in short due to its historical significance. He uses the theorem of Pythagoras, the formula for the area of a right triangle and symmetry considerations. From the above it follows, that the baselines of the triangles constituting the polygons with $Math content$ corners are given as a simple function of $Math content$ when doubling $Math content$. The following diagrams visualize the procedure. The first regular polygon, a yellow square, is circumscribed around the circle filled in gray, a second one, without colour is inscribed.

The inscribed polygon has a smaller area then the circumscribed one; The true value for the circle lies between the two. It is immediately evident, that halving the angle of division to obtain an octagon, which is blue filled, will make the differences smaller, and that this goes on with further doubling of $Math content$ (a polygon with 16 corners also is shown in red). The sketch shows the first steps of the calculation for inscribed polygons with $Math content$ corners, with $Math content$.

The square, with which the calculation starts, consists of 4 equal right triangles, whose cathetuses for the unit circle under consideration have length $Math content$. According to the theorem of Pythagoras the hypotenuse of each triangle has the length $Math content$. The height $Math content$ is obtained via the theorem of Pythagoras using $Math content$ and the hypotenuse $Math content$ of the lower triangle. The distance $Math content$ is the difference to the radius $Math content$. The transition to the octagon again proceeds with the theorem of Pythagoras via $Math content$ and $Math content$. As the following calculation shows, this algorithm can be repeated in the same way in factors of $Math content$ towards a subdivision of the surface of the circle into ever smaller triangles. Thus this procedure results in recursion formulas, with which one can obtain the results of the $Math content$-th step from those of the $Math content$-th step. We give the results for the inscribed polygon with $Math content$ sides.

$Math content$

$Math content$

In the following the equations have been written out starting from the inscribed square $Math content$ to the polygon with $Math content$ corners. On realizes the iterated characters of the repeated root taking of the side length $Math content$ of the triangles making up the inscribed square.

$Math content$

These formulae fascinate via their aesthetic symmetry!

With a simple spreadsheet this calculation that was rather tedious for Archimedes can be done quite quickly up to a high number of corners.

Then one sees, how quickly the surface areas of the inscribed and circumscribed polygon approximate the number $Math content$ and the corresponding circumference approximates $Math content$. In Figure 3.4 they are shown for the square up to the polygon with $Math content$ corners (corresponding to $Math content$ to $Math content$). In addition the respective differences of the surface area from $Math content$ are given (logarithmic right hand scale).

Already for the $Math content$-th approximation (polygon with $Math content$ corners) the difference is only $Math content$. Archimedes himself started with a hexagon and took the calculation up to a polygon with $Math content$ corners and obtained his value for the circular number of $Math content$ (the symbol $Math content$ for this number was only introduced in the 18th century); we suggest that you retrace the calculation of Archimedes.