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This dissertation gives an account of the irrationality of the mathematical constant. Starting with a look into the history of irrational numbers of which is a part of, dating back to the Ancient Greeks and through to the theory behind exactly why is irrational.
In this paper, I aim to look at some of the history and theory behind irrational numbers ( in particular). It will take you through from learning the origins of irrational numbers, to proving the irrationality of itself.
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The mathematical constant is a very important and remarkable number; it is sometimes referred to as Euler’s number. It has many vital applications in calculus, exponential growth/decay and also compound interest. One of the most fascinating things however is taking the derivative of the exponential function; defined. The derivative of is simply, i.e. it is its own rate of change.
An irrational number can be defined as any number that cannot be written as a fraction; that means to say any number that cannot be written in the form.
1.1 History of Irrational Numbers:
The first proof of the existence of irrational numbers came a few centuries BC, during the time when a prevalent group of mathematicians/philosophers/cultists called Pythagoreans (after their leader and teacher Pythagoras) believed in the purity of expressions granted by numbers. They believed that anything geometric in the Universe could be expressed as whole numbers and their ratios. It is believed a Pythagorean by the name Hippasus of Metapontum discovered irrational numbers while investigating square roots of prime numbers; he found that he could not represent the square root of 2 as a fraction. Bringing his findings to his mentor’s (Pythagoras) attention brought the death sentence upon himself. As story has it, Pythagoras (who believed in the absoluteness of numbers) had him drowned to death.
According to Plato (a prominent Greek philosopher and mathematician; 428/427 BC – 348/347 BC), the irrationality of the surds of whole numbers up to 17 was proved by Theodorus of Cyrene. It is understood that Theodorus stopped at the square root of 17 due to the algebra being used failing.
It wasn’t until Eudoxus (a student of Plato) that a strong mathematical foundation of irrational numbers was produced. His theory on proportion, taking into account irrational and rational ratio featured in Euclid’s Elements Book V.
The sixteenth to nineteenth century saw negative, integral and decimal fractions with the modern notation being used by most mathematicians. The nineteenth century was particularly important in the history of irrational numbers as they had largely been ignored since the time of Euclid. The resurgence in the scientific study of irrationals was brought upon by the need to complete the theory of complex numbers.
An important advancement in the logical foundation of calculus was the construction of the real numbers using set theory. The construction of the real numbers represented the joint efforts of many mathematicians; amongst them were Dedekind, Cantor and Weierstrass. Irrational numbers were finally defined in 1872 by H.C.R. Méray, his definition being basically the same as Cantor suggested in the same year (which made use of convergent sequences of real numbers).
Leonhard Euler paid particular attention to continued fractions and in 1737 was able to use them to be the first to prove the irrationality of and. It took another 23 years for the irrationality of to be proved, of which was accredited to Euler’s colleague Lambert.
The nineteenth century brought about a change in the way mathematicians viewed irrational numbers. In 1844 Joseph Liouville established the existence of transcendental numbers, though it was 7 years later when he gave the first decimal example such as his Liouville constant.Charles Hermite in 1973 was the first person to prove that was a transcendental number. Using Hemite’s conclusions Ferdinand von Lindemann was able to show the same for in 1882.
1.2 History of the Mathematical Constant:
The number first arrived into mathematics in 1618, where a table in an appendix to work published by John Napier and his work on logarithms were found to contain natural logarithms of various numbers. The table did not contain the constant itself only a list of natural logarithms calculated from the constant. Though the table had no name of an author, it is highly assumed to have been the work of an English mathematician, William Oughtred.
Surprisingly the “discovery” of the constant itself came not from studying logarithms but from the study of compound interest. In 1683 Jacob Bernoulli examined continuous compound interest by trying to find the limit of as tends to infinity. Bernoulli managed to show that the limit of the equation had to lie between 2 and 3, and hence could be considered to be the first approximation of.
1690 saw the constant first being used in a correspondence from Gottfried Leibniz to Christiaan Huygens; it was represented at the time by the letter. The notation of using the letter however came about due to Euler and made its first appearance in a letter he wrote to Goldbach in 17318. Euler published all the ideas surrounding in his work Introductio in Analysin infinitorum (1748). Within this work he approximated the value of to 18 decimal places;
The latest accurate account of is to 1,000,000,000,000 decimal places and was calculated by Shigeru Kondo & Alexander J. Yee in July 2010.
1.3 A few representations of e:
can be defined by the limit:
By the infinite series:
Special case of the Euler formula:
2. The Proofs:
2.1 Proving the infinite series of e:
In proof 2.2.2 we will use the fact that:
As this paper dedicated to, it would be useful to know where this equation comes from.
The answer lies in the Maclaurin series (Taylor series expansion of a function centred at 0).
Let our, and we have that all derivatives of is equal to We now have that.
We now let and we have equation (5).
2.2 The irrationality of e and its powers.
Continued fractions are closely related to irrational numbers and in 1937 Leonhard Euler used this link and was able to prove the irrationality of and. The most general form of a continued fraction takes the form:
Due to the complexity that can arise in using the format in equation (6), mathematicians have adopted a more convenient notation of writing simple continued fractions. We have that can be expressed in the following manner:
With the use of continued fractions it is relatively easy to show that the expansion of any rational number is finite. So it is obvious to note that all you would have to do to prove that a given number is irrational, would be to show its regular expansion not be finite.
Using this tool we will now show the Euler’s expansion for:
Equation (8) shows, we now invert the fractional part:
Here we have, once again we invert the fractional part:
Hence, we continue in the same way to produce:
Using the figures above provides the following result:
Observing equation (17) allows us to notice pattern and we can show this by re-writing in the following way:
Clearly it seems that the sequence will clearly increase and never terminate. Similarly Euler shows this in other examples using.
Equation (19) shows an arithmetic increase by 4 each time from the number 6 and onwards.
Noticeably equation (18) and (19) do not provide proof that is irrational and are merely just observations. However Euler uses his previous work on infinitesimal calculus, which then proves this sequence is infinite. The proof that Euler uses is very long and complicated as it involves transforming continued fractions into a ratio of power series, which in turns becomes a differential equation of that he can transform into the Ricatti equation he needs.
Since Euler’s time mathematicians have found far more manageable and direct ways in proving the irrationality of.
2.2.1 Proving the irrationality of e:
While Euler was the first to establish a proof of the irrationality of using infinite continued fractions, we will use Fourier’s (1815) idea of using infinite series to prove more directly.
Defining the terms:
Using the Maclaurin series expansion we have:
Now let’s define to be a partial sum of:
For we first write the inequality:
Equation (22) has to be positive as we stated to be the partial sum of, which is the infinite sum.
Now we’ll find the upper limit of equation (22):
Taking out a factor of:
Now as we are looking for an upper limit, we need an equation greater than equation (24):
We take note that the terms in the square bracket in equation (25) for the upper limit is a geometric series with.
Right hand Side (RHS) of equation (25):
Multiply through by:
Now let’s assume i.e. is rational.
Using the substitution implies:
Now by expanding the RHS gives us the following result:
We note the following:
is an integer.
, this implies that divides into and hence is an integer.
Each term within the square bracket is an integer; we know that can be divided by and upwards to and produce integer values.
Therefore as all terms are integers, we have:
where is an integer value.
Observe that by choosing any we have and furthermore.
Using equation (31) we now obtain the following result:
Equation (37) implies is not an integer.
This is a contradiction to the result obtained in 1) and so therefore is proven to be irrational.
2.2.2 Proving the irrationality of ea:
Proof 2.1 successfully shows how is irrational however, the proof is not strong enough to show the irrationality of. Using an example, we have the as a known irrational number, whose square is not.
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In order to show all integer powers (except zero) of are irrationals, we need a bit more calculus and an idea tracking back to Charles Hermite; where the key is located in the following lemma.
Lemma: For some fixed, let:
The function is a polynomial of the form, where the coefficientsare integers.
For we have
The derivatives and are integers for all
Proof: (see appendix)
Theorem 2: is irrational for any integer.
Take to be rational, where is a non-zero rational number. Let with non-zero integers and. being rational implies that is rational. This is a contradiction to theorem 2 and hence is irrational.
Assume where are integers, and let be large enough that.
where is the function of the lemma.
Note that can also be written in the form of an infinite sum as we see that any higher derivatives where for vanishes.
We now want to obtain a first order linear equation using equation (39). We start by differentiating:
Now from observation we see that by multiplying equation (39) by and then eliminating the first term we end up with equation (40).
Equation (41) takes the form our required first order linear equation, which is solved in the following manner:
First re-write in the standard form:
Next we find the integrating factor µ to multiply to both sides of the equation:
From equation (43) we now have the following equation:
Note the limit runs as stated in of the lemma.
We now manipulate equation (45) by multiplying by so that we can apply of the lemma.
We have that , so thereforeand hence:
As is just a polynomial containing integer values multiplying derivatives of, we can state using of the lemma that is an integer.
Part of the lemma states . With this we can now estimate the range that lies within.
Firstly we know that is a positive value and hence. For the upper limit we have:
Note that to find the upper limit we eliminate the integral and substitute the upper bounds for and.
From before we have and also that we took n large enough so that, which can be re-written , which implies the following:
Equation (51) states that cannot be an integer and hence contradicts Equation (48). Therefore we have that is proven to be irrational.
3. Further Work:
Following on and further proving the irrationality of, would be to prove that is a transcendental number.
Irrational numbers can be split into two categories algebraic and transcendental; hence transcendental numbers are numbers that are not algebraic. Algebraic numbers are defined as any number that can be written as the root of an equation of the form. A minimal polynomial is achieved when is the smallest degree possible for a given. The square root of 2 is an example of an irrational number, but also it is an algebraic number of degree 2, of which the minimal polynomial is simply.
Euler in the late 18th century was the first person to define transcendental numbers, but the proof of their existence only came around in the papers of Liouville’s in 1844 and 1851.
The number was the first important mathematical constant to be proven transcendental and was done so by Charles Hermite in 1873. The techniques Hermite used influenced many future mathematical works including the first proof of being transcendental by Ferdinand von Lindemann; also used in the creation of the Lindemann-Weierstrass theorem.
Further work on transcendental numbers involving can be still seen today. Mathematicians knowis a transcendental number, but as of yet have not been able to prove this.
Overall, the main objective of this paper was to give an account of the irrationality of. This has been achieved and with it we have been able to see the progress from the first discovery of irrational numbers by the Pythagoreans of Ancient Greek, through to the work covered on Euler’s number.
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