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Transcendental Numbers

Last updated date: 23rd May 2024
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What is a Transcendental Number?

Number which is not algebraic, in a way that it is not the solution of an algebraic equation with rational-number coefficients is called a transcendental number. All transcendental numbers make for irrational numbers, but not all irrational numbers fall in the category of transcendental. For example, x² – 2 = 0 has the solutions x = ± √2; therefore, the √2, an irrational number, will not be a transcendental number but algebraic. Almost all real and complex numbers are transcendental, however, only a few have been proven to be transcendental.

Transcendental Numbers Examples

With the above description, you must be clear with the transcendental meaning in maths. Now what are the types of numbers that are transcendental? The numbers e and π are examples of transcendental numbers.

History of Transcendental Numbers

German mathematician—Ferdinand von Lindemann is chiefly remembered to have proved that the number π is transcendental.

Von Lindemann’s proof that π is transcendental had come to a possibility with the help of fundamental methods being developed by the French mathematician—Charles Hermite in the 1870s. In particular, under Hermite’s proof of the transcendence of e, the base for natural logarithms, was the 1st time that a number was exhibited to be transcendental. Lindemann met Hermite in Paris and learnt first hand of this popular outcome. Taking ahead Hermite’s work, Ferdinand von Lindemann published his proof in an article authorized as “Über die Zahl π '' 1882; (“Concerning the Number π”).

Algebraic Numbers

In mathematics, Algebraic numbers are all the numbers that are roots to polynomials with rational numbers as coefficients. The real numbers are classified into 2 types: rational numbers and irrational numbers. The rational numbers are those which can be expressed as the ratio of two integers, e.g., 0, 1, 1/2, and 5/3. The remaining are the irrational numbers, e.g., √2 and π.

Transcendental Irrational Numbers

Wondering how many transcendental numbers is irrational? In mathematics, transcendental numbers are a subset of irrational numbers. Also, remember that all transcendental numbers are irrational but not all irrational numbers are transcendental (despite the fact that algebraic numbers (rational numbers and non-transcendental irrational numbers) are countable. In addition, transcendental numbers (the complement of algebraic numbers to the real numbers) are not countable.

A key characteristic that differentiates transcendental numbers from other irrational numbers is that transcendental numbers are not the solutions to polynomials having rational coefficients.

The real numbers also are divided into two types: the (real) algebraic numbers and the (real) transcendental numbers. Algebraic numbers are numbers that are solutions to polynomials having rational coefficients. The real algebraic numbers involve rational numbers and also many other familiar irrational numbers, e.g., √2. The (real) transcendental numbers are the real numbers that are not algebraic, e.g., e and π.

How to Determine a Transcendental Irrational Number

It can be complicated, and conceivably impossible, to identify if or not a specific irrational number is transcendental. Some numbers disregard classification (algebraic, irrational, or transcendental) until present times. Two examples are the product of π and e (quantity P πe) and the sum of π and e (S πe). It is proven that both π and e are transcendental. It has also been exhibited that a minimum of one of the two quantities P πe and S πe are transcendental. But no one has meticulously proven that P π is transcendental, and no one has also stringently proved that S π is transcendental.

Proof That π is Transcendental

In order to prove that π is transcendental, we would require proving that it is not algebraic. If π were algebraic, πi would also be algebraic, and then by the Lindemann–Weierstrass theorem eπi = −1 (Euler's identity) will be transcendental, a contradiction. Hence, π is not algebraic, which implies that it is transcendental.

FAQs on Transcendental Numbers

Q1. π is Transcendental What Does This Mean?

Answer: When we say π is transcendental, it means that it does not satisfy any algebraic equation with respect to the rational coefficients. Proof that π is transcendentally established that the classical Greek construction problem of squaring the circle (constructing a square in reference to an area equal to that of a given circle) by straightedge and compass is insoluble.

Q2. What is a Transcendental Function?

Answer: In the same manner that a Transcendental Number is "not algebraic", thus a Transcendental Function is also "not algebraic". Formally to say, a transcendental function is a function which cannot be established in a finite number of steps from the elementary functions and their inverses. The Sine function sin(x) is an example of a Transcendental Function.

Q3. What is Meant by Liouville Numbers?

Answer: A Liouville Number is a unique kind of transcendental number that can be very closely estimated by rational numbers.

Precisely, a Liouville Number is a real number x, with a mathematical property that, for any positive integer n, there is integer’s p and q (with q>1) in a way:

0< |x – p/q |< 1/qn

Now you are aware that x is irrational, thus there will be a difference between x and any p/q always: thus, we obtain the "0<" part.

But the 2nd inequality reveals how little the difference is. In point of fact the inequality notifies "the number can be estimated infinitely close, but never quite reaching there". In fact Liouville made it possible to reveal that if a number has a quick converging series of rational estimations then it is transcendental.

Another amazing property is that for any positive integer n, there is an infinite number of pairs of integers (p,q) abiding by the above inequality.