The Irrationality Of The Mathematical Constant E Mathematics Essay

The Ir balancenality Of The numerical Constant E Mathematics undertakeThis dissertation gives an account of the unlogicality of the numerical incessant. Starting with a look into the archives of absurd metrical composition game of which is a part of, dating back to the Ancient Greeks and finished to the theory female genitalia exactly wherefore is irrational.1. IntroductionIn this paper, I aim to look at some of the history and theory behind irrational takes ( in particular). It will take you through from learning the origins of irrational mos, to proving the irrationality of itself.The numeral constant is a very important and remark fit number it is some times referred to as Eulers number. It has some brisk applications in calculus, exponential growth/decay and also compound interest. One of the most(prenominal) fascinating things so far is taking the derivative of the exponential function defined. The derivative of is simply, i.e. it is its own rate of change.An i rrational number so-and-so be defined as any number that can non be written as a fragment that means to say any number that cannot be written in the phase.1.1 History of Irrational comeThe start conclusion of the existence of irrational numbers came a a few(prenominal) centuries BC, during the time when a commonplace group of mathematicians/philosophers/cultists c some(prenominal)ed Pythagoreans (after their leader and teacher Pythagoras) believed in the purity of expressions granted by numbers. They believed that anything geometric in the reality could be expressed as whole numbers and their ratios. It is believed a Pythagorean by the name Hippasus of Metapontum spy irrational numbers while investigating squargon topics of prime numbers he prove that he could not represent the square root of 2 as a fraction. Bringing his findings to his mentors (Pythagoras) forethought brought the death sentence upon himself. As story has it, Pythagoras (who believed in the absoluten ess 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 c wholeable to the algebra be intentd failing.It wasnt until Eudoxus (a student of Plato) that a strong mathematical ensnareation of irrational numbers was produced. His theory on proportion, taking into account irrational and rational ratio featured in Euclids Elements Book V.The sixteenth to nineteenth century saw negative, integral and decimal fraction fractions with the late notation being use by most mathematicians. The nineteenth century was particularly important in the history of irrational numbers as they had astronomically 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 numbe rs.An important advancement in the limpid 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. Mray, his definition being basically the same as Cantor suggested in the same stratum (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 commencement ceremony to prove the irrationality of and. It took another 23 years for the irrationality of to be proved, of which was accredited to Eulers colleague Lambert.The nineteenth century brought about a change in the way mathematicians viewed irrational numbers. In 1844 Joseph Liouville established the existence of otherworldly numbers, though it was 7 years later when he gave the basic decimal example such as his Liouville constant.Charles Hermite in 1973 was the first person to prove that was a transcendental number. victimisation Hemites conclusions Ferdinand von Lindemann was able to say the same for in 1882.1.2 History of the Mathematical ConstantThe number first arrived into mathematics in 1618, where a table in an appendix to work published by nates 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 extremely as totaled to bugger off been the work of an English mathematician, William Oughtred.Surprisingly the discovery of the constant itself came not from studying logarithms moreover from the study of compound interest. In 1683 Jacob Bernoulli examined continuous compound interest by trying to find the specialize of as tends to infinity. Bernoulli managed to show that the see of the compare 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 proportionateness 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 placesThe 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 ecan be defined by the detain(1)By the infinite series(2) excess case of the Euler formula(3)Where when,(4)2. The certaintys2.1 Proving the infinite series of eIn proof 2.2.2 we will use the point that(5)As this paper dedicated to, it would be useful to k r ight away where this equality comes from.The answer lies in the Maclaurin series (Taylor series expansion of a function centred at 0).(6)Let our, and we have that all derivatives of is equal to We forthwith have that.(7)We now let and we have equation (5).2.2 The irrationality of e and its powers.Continued fractions are nigh 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(6)Due to the complexity that can purloin in using the format in equation (6), mathematicians have adopted a more expedient notation of writing simple continued fractions. We have that can be expressed in the succeeding(a) bearing(7)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 exp ansion not be finite. utilise this tool we will now show the Eulers expansion forWe have(8) par (8) shows, we now invert the fractional part(9)Here we have, erst again we invert the fractional part(10)Hence, we continue in the same way to produce(11)So.(12)So.(13)So.(14)So.(15)So.(16)So.Using the figures above provides the following allow for(17)Observing equation (17) allows us to notice pattern and we can show this by re-writing in the following way(18)Clearly it seems that the sequence will clearly increase and never terminate. Similarly Euler shows this in other examples using.(19)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 frac tions 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 Eulers time mathematicians have found far more manageable and direct ways in proving the irrationality of.2.2.1 Proving the irrationality of e enchantment Euler was the first to establish a proof of the irrationality of using infinite continued fractions, we will use Fouriers (1815) idea of using infinite series to prove more directly. confirmationDefining the termsUsing the Maclaurin series expansion we have(20)Now lets define to be a partial sum of(21)For we first write the inequality(22)Equation (22) has to be positive as we adduced to be the partial sum of, which is the infinite sum.Now well find the upper limit of equation (22)(23)Taking out a factor of(24)Now as we are looking for an upper limit, we need an equation greater than equation (24)(25)We take note that the terms in the square bracket in equation (25) for the upper lim it is a geometric series with.Right hand Side (RHS) of equation (25)(26)(27)(28)(29)We have(30)Multiply through by(31)Now lets assume i.e. is rational.Using the substitution implies(32)Now by expanding the RHS gives us the following result(33)(34)We note the followingis an integer., this implies that divides into and hence is an integer.Each term within the square bracket is an integer we neck that can be divided by and upwards to and produce integer values.Therefore as all terms are integers, we have(35)where is an integer value.Observe that by choosing any we have and furthermore.Using equation (31) we now obtain the following result(36)(37)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 eaProof 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 completen irration al number, whose square is not.In order to show all integer powers (except zero) of are irrationals, we need a mo more calculus and an idea tracking back to Charles Hermite where the key is located in the following lemma.ProofLemma For some fixed, let(38)The function is a polynomial of the form, where the coefficientsare integers.For we haveThe derivatives and are integers for allProof (see appendix)Theorem 2 is irrational for any integer.ProofTake 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 deep enough that.State, (39)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(40)Now from observation we see tha t by multiplying equation (39) by and then eliminating the first term we end up with equation (40).(41)Equation (41) takes the form our required first order linear equation, which is solved in the following mannerFirst re-write in the standard form(42)Next we find the integrating factor to multiply to both sides of the equation(43)From equation (43) we now have the following equation(44)(45)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.(46)(47)We have that , so thereforeand hence(48)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(49)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(50)(51)Equation (51) states that cannot be an integer and hence contradicts Equation (48). Therefore we have that is proven to be irrational.3. Further WorkFollowing 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 Liouvi lles 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 silent seen today. Mathematicians knowis a transcendental number, but as of yet have not been able to prove this.4. findingOverall, 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 feeler from the first discovery of irrational numbers by the Pythagoreans of Ancient Greek, through to the work covered on Eulers number.ReferencesWebpage mental imagerysCook, Z. (2000), Irrational Numbers, The Guide to Life, The Universe and Everything, BBC Online. functional http//www.bbc.co.uk/ dna/h2g2/A455852, Accessed sixth January 2011.OConnor, J.J and Robertson, E.F. (1999), Theodorus of Cyrene Online. Available http//www-history.mcs.st-and.ac.uk/Biographies/Theodorus.html, Accessed sixth January 2011.OConnor, J.J and Robertson, E.F. (1999), Eudoxus of Cnidus Online. Available http//www-groups.dcs.st-and.ac.uk/history/Biographies/Eudoxus.html, Accessed sixth January 2011.OConnor, J.J and Robertson, E.F. (2001), The number e, Number Theory Online. Available http//www-history.mcs.st-and.ac.uk/HistTopics/e.html, Accessed 6th January 2011.Russel, D. (2002), Hippasus Expelled, Irrational Pythagoreans Online. Available http//math.about.com/library/blpyth.htm, Accessed 6th January 2011.Sondow, J and Weisstein, E.W. e. MathWorldA Wolfram Web Resource Online. Available http//mathworld.wolfram.com/e.html, Accessed 6th January 2011.Weisstein, E.W. Irrational Number, MathWorldA Wolfram Web Resource Online. Available http//mathworld.wolfram.com/IrrationalNumber.html, Accessed 6th January.Yee, A.J. (2010), e, Mathematical Constants Billions of Digits Online. Available http//www.numberworld.org/digits/E/, Accessed 6th January 2011.Zongju, L. Shuxue Lishi Diangu (Historical Stories in Mathematics), Chiu Chang Publishing Company Online. Available http//db.math.ust.hk/articles/calculus/e_calculus.htm, Accessed 6th January 2011.Arithmetic Sequences and Series, Arizona State University Online. Available http//fym.la.asu.edu/tturner/MAT_117_online/SequenceAndSeries/Geometric_Sequences.htm, Accessed 6th January 2011.Online PDF ResourcesCollins, DC. Continued Fractions, Online. Available http//www-math.mit.edu/phase2/UJM/vol1/COLLIN1.PDF, Accessed 6th January 2011.Conrad, K. (2005), Irrationality of, Online. Available http//www.math.uconn.edu/kconrad/math121/121piande.pdf, Accessed 6th January 2011.Field, B. (2010), Irrational and Transcendental Numbers, page 23 Online. Available http//maths.dur.ac.uk/Ug/projects/library/CM3/0910/CM3_BenField.pdf, Accessed 6th Januar y 2011.Sandifer, E. (2006), Who proved e is irrational?, How Euler Did it Online. Available http//www.maa.org/editorial/euler/How%20Euler%20Did%20It%2028%20e%20is%20irrational.pdf Accessed 6th January 2011.Online Video ResourcesAdams, C, Garrity, T and Burger, E. (2006), Pi verses e, The Mathematical Association of the States Online. Available http//www.youtube.com/watch?v=whpAX30vjoE, Accessed 6th January 2011Delaware, R. A Proof e is irrational, Proof, University of second Online. Available http//www.youtube.com/watch?v=FtIL7nGgDNM, Accessed 6th January 2011.Book ResourcesAigner, M and Ziegler, G.M. Proofs from THE BOOK, Third Edition, Berlin Springer, 2004.Dorrie, H and Translated by Antin, D. carbon Great Problems of Elementary Mathematics, THERE HISTORY AND SOLUTION, New York Dover Publications, Inc., 1965.Sandifer, C.E. The early mathematics of Leonhard Euler, USA The Mathematical Association of America (Incorporated), 2007.