[Maths Class Notes] on Transcendental Numbers Pdf for Exam

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.