Defining And Understanding Transcendental Numbers Engineering Essay

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"A transcendental number is a (possibly complex) number that is not the root of any integer polynomial, meaning that it is not an algebraic number of any degree. Every real transcendental number must also be irrational, since a rational number is, by definition, an algebraic number of degree one." - Quoted from Wolfram MathWorld

Or simply put, the set of transcendental numbers is uncountably infinite. Meaning that it cannot be expressed as a solution of axn+bx(n-1)+...+cx0=0 where all coefficients are integers and n is finite. For example, x=which is irrational, can be expressed as x2-2=0. This shows that the square root of 2 is nontranscendental, or algebraic.

As such to further understand the topic, we will be looking at 5 Known Transcendental Numbers below; pi (Ï€), sine 1, e, ln(a), and Chaitin's Constant.

Known Transcendental Numbers

, pi

What is π , pi ?

π , pi is Greek for the word perimeter. The name was suggested by a Welsh mathematician named William Jones in 1703 that the constant should be called, but, the term pi () was popularized by the Swiss mathematician Leonhard Euler in the early 18th century. It was by the Greek Philosopher Archimedes that the first rigorous approximation of π by circumscribing and inscribing -gons on a circle was obtained. It is also interesting to note that the Egyptians discovered that Pi can also be denoted by the commonly used simplified approximation fraction of 22/7 (Because Pi is an irrational number, meaning that it cannot be expressed as a fraction of two whole numbers). Now, the symbol pi (π) is symbol for the ratio of the circumference of a circle to its diameter. Pi is also sometimes known as Archimedes' constant or Ludolph's constant.

However, although pi can be simplified as 22/7, it is just an approximation. There exists quite a number of formulas to describe the exact value of pi, one of them is the Machin's Formula,

Transcendence of π

In 1882, Ferdinand von Lindemann proved that π is transcendental. As Pi cannot be expressed by the formula axn+bx(n-1)+...+cx0=0 it is transcendental. However, Lindemann proceeds to proof that pi is transcendental as follows :

(The following is quoted from "The Transcendence of pi", Mathematics in Tens, http://dialspace.dial.pipex.com/town/way/po28/maths/docs/pi.html)

If p satisfies an algebraic equation with coefficents in Q, so does ip (i=Ã--1). Let this equation be q 1( x) =0, with roots ip =a 1,... ,a n. Now eip +1=0 so

æ

è

e

a 1

 

+1

ö

ø

...

æ

è

e

a n

 

+1

ö

ø

=0

We now construct an algebraic equation with integer coefficients whose roots are the exponents of e in the expansion of the above product. For example, the exponents in pairs are a 1+a 2,a 1+a 3,... a n-1+a n. The a s satisfy a polynomial equation over Q so their elementary symmetric functions are rational. Hence the elementary symmetric functions of the sums of pairs are symmetric functions of the a s and are also rational. Thus the pairs are roots of the equation q 2( x) =0 with rational coefficients. Similarly sums of 3 a s are roots of q 3( x) =0, etc. Then the equation

    q 1( x) q 2( x) ... q n( x) =0

is a polynomial equation over Q whose roots are all sums of a s. Deleting zero roots from this, if any, we get

    q ( x) =0

    q ( x) =cxr+c1xr-1+... cr

and cr¹ 0 since we have deleted zero roots. The roots of this equation are the non-zero exponents of e in the product when expanded. Call these b 1,... b r. The original equation becomes

    eb 1+... eb r+e0+... e0=0

ie

    å eb i+k=0

where k is an integer >0     (¹ 0 since the term 1... 1 exists)

Now define

    f( x) =csxp-1[ q ( x) ] p/( p-1) !

where s=rp-1 and p will be determined later. Define

    F( x) =f( x) +f' ( x) +... +f( s+p) ( x) .

    d/dx[ e-xF( x) ] =-e-xf( x) as before.

Hence we have

    e-xF( x) -F( 0) =-ò0xe-yf( y) dy.

Putting y=l x we get

    F( x) -exF( 0) =-xò01e( 1-l ) xf( l x) dl .

Let x range over the b i and sum. Since å eb i+k=0 we get

    åj=1rF( b j) +kF( 0) =-åj=1rb jò01e( 1-l ) b jf( l b j) dl .

CLAIM:

    For large enough p the LHS is a non-zero integer.

åj=1rf( t) ( b j) =0     ( 0<t<p) by definition of f. Each derivative of order p or more has a factor p and a factor cs, since we must differentiate [ q ( x) ] p enough times to get ¹ 0. And f( t) ( b j) is a polynomial in b j of degree at most s. The sum is symmetric, and so is an integer provided each coefficient is divisible by cs, which it is. (symmetric functions are polynomials in coefficients = polynomials in ci/c of degree £Â s). Thus we have

    åj=1rf( t) ( b j) =pkt     t=p,... p+s.

Thus LHS=( integer) +kF( 0) .     What is F( 0) ?

    f( t) ( 0) =0     t=0,... ,p-2.

    f( p-1) ( 0) =cscrp     ( cr¹ 0)

    f( t) ( 0) =p (some integer)    t=p,p+1,... .

So the LHS is an integer multiple of p+cscrpk. This is not divisible by p if p>k,c,cr. So it is a non-zero integer. But the RHS ® 0 as p® ¥ and we get the usual contradiction.

(The text quoted from "The Transcendence of pi", Mathematics in Tens, http://dialspace.dial.pipex.com/town/way/po28/maths/docs/pi.html ends here.)

As described in the source of the quotation above & from various other sources, there are various proves that show pi is a transcendental number such as via proving that pi is an irrational number (accomplished by Lambert in 1761; Legendre in 1794; Hermite in 1873; Nagell in 1951; Niven in 1956; Struik in 1969; Königsberger in 1990; Schröder in 1993; Stevens in 1999; Borwein and Bailey in 2003).

Interesting Facts about Pi

Piphilology

Piphilology is known as the creation and use of mnemonic techniques to remember a span of digits of the mathematical constant π. An example of this would be the poem Cadaeic Cadenza written in 1996 by Mike Keith. The Cadaeic Cadenza is a short story of about 4000 words composed in Standard Pilish, in which the length (in letters) of successive words in the text "spells out" the digits of the number π.

An excerpt of the text is as follows:

One

   A Poem 

        A Raven 

Midnights so dreary, tired and weary, 

    Silently pondering volumes extolling all by-now obsolete lore. 

During my rather long nap - the weirdest tap! 

    An ominous vibrating sound disturbing my chamber's antedoor. 

        "This", I whispered quietly, "I ignore".

Perfectly, the intellect remembers: the ghostly fires, a glittering ember. 

    Inflamed by lightning's outbursts, windows cast penumbras upon this floor. 

Sorrowful, as one mistreated, unhappy thoughts I heeded: 

    That inimitable lesson in elegance - Lenore - 

        Is delighting, exciting...nevermore.

The excerpt is to be decoded using Standard Pilish (read below).

Pilish

Pilish is a method of writing a sentence in which the lengths of successive words represent the digits of the number π or can be described as "English that follows the successive digits of pi.". This "language" has been around since the early 1900's. Pilish is categorize into 2 types; Basic Pilish and Standard Pilish.

Basic Pilish

Each word of ɲ number of letters

The digit n if ɲ <10

The digit 0 if ɲ =10

Hence, the text as from Cadaeic Cadenza can be decoded as follows

Text:

One a poem a raven midnights so dreary tired and weary…

Decode:

3.1415926535

Standard Pilish

In response to the limitation of Basic Pilish permitting the usage of words with only 10 characters or less, the rules of Standard Pilish was born.

Each word of ɲ number of letters

The digit n if ɲ <10

The digit 0 if ɲ =10

Two consecutive digits if ɲ >10

(for example, a 15-letter word represents the digits 1,5)

If the rules for Pilish were to be put into words, it would be "To recover the digits of π from a text in Standard Pilish, write the number of letters in each word next to the word (except if the word has 10 letters, in which case write a 0).  Then read off all the digits in order from beginning to end to get the value of π."

Pi Day

Pi day is an appreciation day for the mathematical constant, pi. It is celebrated by Americans on the 14th March (Also known as "The World Pi Day") whereas the British celebrates it on the 22nd July. They respectively follow the common Pi values of 3.14, the American date format and 22/7, the British date format.

Pi Decimals Record

A French computer programmer, Fabrice Bellard in January 2010 has calculated pi to a world record of almost 2,700 billion decimal places on a desktop computer costing less than £2,000. This amazing feat took 131 days to complete resulting in 1,000GBs of data.

e

What is e?

'e' is a useful mathematical constant that is the basis of the system of natural logarithms.Its a numerical constant that is equal to 2.17828.The value of 'e' is in complex in many mathematicla formulas such as those describing a nonlinear increase or decrease such as growth or decay.'e' is also found in some problems of probability and also in in the study of the distribution of prime numbers.The mathematical constant 'e' utilized in many fields such as biology,business,demoographics,physics,and engineering fields.The number 'e' is widely used as the base in the exponential function and its inverse is the natural logarithm.'e' was also defined as:

The first reference to the mathematical constant 'e' were published in 1916 of a work on logarithms by John Napier. However, this did not contain the constant itself, but simply a list of natural logarithms calculated from the constant.It is assumed that the table was written by William Oughtred and the discovery of mathematical constant 'e'itself is credited to Jacob Bernoulli.Mathematical costant 'e' was first studied by the Swiss mathematician Leonhard Euler in the 1720s.It is also know as the Eulerian Number, or Napier's Constant. Euler was also the first to use the letter e for it in 1727.The number 'e' was proven by Euler that "e" is an irrational number ,unique real number and its not a ratio of intergers.As th result, the mathematical constant 'e' is transcendental its value cannot be given exactly as a finite or eventually repeating decimal.The numerical value of 'e' truncated to 20 decimal place is:

2.71828 18284 59045 23536

Besides that,mathematical constant 'e' was found in many applications such as the compound-interest problem,Bernoulli trial's,Derangements,Asymptotics. Jacob Bernoulli discovered the compound-interest problem constant by studying a question about the compound interest. The number e itself also has applications to probability theory, where it arises in a way not obviously related to exponential growth tis is known as the Bernoulli trial's. Moreover,the other application of 'e' is also discovered by Jacob Bernoulli along with Pierre Raymond deMontmort  is in the problem of derangements, also known as the hat check problem. The mathematical constant 'e' occurs naturally in connection with many problems involving asymptotics.An example of it is the Stirling's formula of the factorial function in which both numbers 'e' and pie enter:

A particular consequence of this is

.

Futhermore,the constant number 'e' was also found in calculus in which it is used to perform and integral calculus with exponential functions and logarithms.'e' is symbolically defined as

The properties of the mathematical constant 'e' in calculus was a motivation to the exponential function ex played a important part because is the unique nontrivial function which is its own derivative.

Ln(a)

What is In(a)?

The natural logarithm is the logarithm to the base e, where e is approximately equal to 2.718281828. The natural logarithm is generally written as ln(x), loge(x) or log(x).

In(a) is generally understood in various forms. Mathematicians, statisticians, engineers, biologists and some others recognize:

log(x)

ln(x)

loge(x)

However, it is generally written as "ln(x)" or "loge(x)".

Popular programs including C, C++, SAS, MATLAB, Fortran, and BASIC understand "log" or "LOG" as the natural logarithm, scientific calculators recognize the natural logarithm as In.

The natural logarithm was revealed by Nicholas Mercator in his work 'Logarithmotechnia' published in 1668. Though at a much earlier date, around 1619, the mathematician John Speidell had already discovered the natural logarithm. Initially it was called hyperbolic logarithm and sometimes refer to as the Napierian logarithm.

Chaitin's Constant

What is Chaitin's Constant ?

A Chaitin's constant, also called a Chaitin omega number, introduced by Chaitin (1975), is the halting probability of a universal prefix-free (self-delimiting) Turing machine. Every Chaitin constant is simultaneously computably enumerable (the limit of a computable, increasing, converging sequence of rationals), and algorithmically random (its binary expansion is an algorithmic random sequence), hence uncomputable.

A real number, represented by Ω (capital Omega) and also known as the Halting probability, whose digits are distributed so randomly that no attempt to find a rule for predicting them can ever be found. Discovered by Gregory Chaitin, Ω is definable but not computable. It has no pattern or structure to it whatsoever, but consists instead of an infinitely long string of 0's and 1's in which each digit is as unrelated to its predecessor as one coin toss is from the next. Although called a constant, it is not a constant in the sense that, for example, pi is, since its definition depends on the arbitrary choice of computation model and programming language. For each such model or language, Ω is the probability that a randomly produced string will represent a program that, when run, will eventually halt. To derive it, Chaitin considered all the possible programs that a hypothetical computer known as a Turing machine could run, and then looked for the probability that a program, chosen at random from among all the possible programs, will halt. The work took him nearly 20 years, but he eventually showed that this halting probability turns Turing's question of whether a program halts into a real number, somewhere between 0 and 1. Further, he showed that, just as there are no computable instructions for determining in advance whether a computer will halt, there are also no instructions for determining the digits of Ω. Ω is uncomputable and unknowable: we don't know its value for any programming language and we never will. This is extraordinary enough in itself, but Chaitin has found that Ω infects the whole of mathematics, placing fundamental limits on what we can know.

And Ω is just the beginning. There are more disturbing numbers - super-Omegas - whose degree of randomness is vastly greater even than that of Ω. If there were an omnipotent computer that could solve the halting problem and evaluate Ω, this mega-brain would have its own unknowable halting probability called Ω'. And if there were a still more God-like machine that could find Ω', its halting probability would be Ω". These higher Ωs, it has been recently discovered, are not meaningless abstractions. Ω', for instance, gives the probability that an infinite computation produces only a finite amount of output. Ω" is equivalent to the probability that, during an infinite computation, a computer will fail to produce an output - for example, get no result from a computation and move on to the next one - and that it will do this only a finite number of times. Ω and the Ω hierarchy are revealing to mathematicians an unsettling truth: the problems that we can hope ever to solve form a tiny archipelago in a vast ocean of undecidability.

Citations

, pi

"pi." Encyclopædia Britannica. Encyclopædia Britannica Online. Encyclopædia Britannica, 2010. Web. 13 June 2010 http://search.eb.com/eb/article-9059879

transcendental number. (2010). In Encyclopædia Britannica. Retrieved June 13, 2010, from Encyclopædia Britannica Online: http://search.eb.com/eb/article-9473503

"Lindemann, Ferdinand von." Encyclopædia Britannica. Encyclopædia Britannica Online. Encyclopædia Britannica, 2010. Web. 13 June 2010 <http://search.eb.com/eb/article-9048357>

Weisstein, Eric W. "Transcendental Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/TranscendentalNumber.html

Weisstein, Eric W. "Hermite-Lindemann Theorem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Hermite-LindemannTheorem.html

Weisstein, Eric W. "Pi." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Pi.html

Weisstein, Eric W. "Archimedes Algorithm." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ArchimedesAlgorithm.html

Weisstein, Eric W. "Circle Squaring." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CircleSquaring.html

Weisstein, Eric W. "Pi Formulas." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PiFormulas.html

Connor, S. (2010, January 8). The Big Question: How close have we come to knowing the precise value of pi? Retrieved June 13, 2010, from The Independent: http://www.independent.co.uk/news/science/the-big-question-how-close-have-we-come-to-knowing-the-precise-value-of-pi-1861197.html

http://mathforum.org/dr.math/faq/faq.pi.html

http://www.math.clemson.edu/~simms/neat/math/pi/piproof.html

"The Transcendence of pi", http://dialspace.dial.pipex.com/town/way/po28/maths/docs/pi.html

Mike Keith, "Near A Raven" from http://cadaeic.net/naraven.htm

Mike Keith, "Writing in Pilish" from http://www.cadaeic.net/pilish.htm

"Planet Math: Pi" from http://planetmath.org/encyclopedia/Pi.html

Your Piece of the π. "π is Transcendental" from http://library.thinkquest.org/C0110195/what/transcendental.html

Your Piece of π, "How Much Fun Can π Be?" from http://library.thinkquest.org/C0110195/main.html

Kaufmann. The Kaufmann Governance Post . "Pi, Irrational Numbers and Einstein: Brilliantly Mindful of Error" from http://thekaufmannpost.net/pi-irrational-numbers-and-einstein-brilliantly-mindful-of-error/

Cliff Pickover . "The 15 Most Famous Transcendental Numbers" from http://sprott.physics.wisc.edu/pickover/trans.html

Prepared by

Pi, π

by Daniel Ting, 1003R71396

Ln (a)

by Lionel Manuel, 1003R78219

e

by Pravin Jacob Chandy, 1003R79023

Chaitin's constant

by Joshua A/L Panir Selvam,

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