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Transcendental Numbers

Mathematics is a subject that is all about the numbers. Numbers are seen everywhere not only in maths but also in our day to day life. There are different types of numbers studied in mathematics. The numbers do have different classifications.

On the basis of divisibility, the numbers can be classified into:
a) Even numbers (divisible by 2)
b) Odd numbers (not divisible by 2)
c) Prime numbers (divisible by 1 and itself only)
d) Composite numbers (divisible by at least one number other than 1 and itself).

On the basis of number line, the numbers can be classified into:
1) Natural Numbers (1, 2, ...)
2) Whole Numbers (0, 1, 2, ...)
3) Real Numbers (all number on number line)
4) Integers (...-2, -1, 0, 1, 2, ...)
5) Rational Numbers ($\frac{p}{q}$)
6) Irrational Numbers (not rational)
7) Complex Numbers (not on real number line).

The numbers can also be classified into two types:

Algebraic numbers
Transcendental numbers.

An algebraic number is one which is a root of a polynomial in one variable. On the other hand, the numbers that are not algebraic, are categorized as transcendental numbers. In this page, we shall learn about transcendental numbers. So, go ahead with us and gain knowledge about transcendental numbers.

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Definition

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A transcendental number can be defined as a number which is not an algebraic number. It may be any complex or real number which is not said to be algebraic. Lets recall algebraic number here. An algebraic number is defined as a number which is a root of some one variable nonzero polynomial having rational coefficients.  
The transcendental number cannot be a root of the equation containing a non-zero polynomial in one variable combined with rational coefficients.

For Example:  e and $\pi$ are the most famous transcendental numbers.
There are few known transcendental numbers, but these are not rare.  It is very difficult to prove for number if it is transcendental.  In fact, most of the complex and real numbers are claimed to be transcendental. This is because the algebraic numbers are said to be countable and we know that the set of complex and set of real numbers both are uncountable. Also, because all the rational numbers are algebraic numbers, we may say that all the real transcendental numbers are actually irrational.  But conversely, not every irrational number is said to be transcendental.

For Example: $\sqrt{2}$ is an irrational number. Since, it is a root of x$^{2}$ - 2 = 0, it is not a transcendental number.

Properties

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The main properties of transcendental numbers are illustrated below:

1) The transcendental numbers have the property that their set is said to be an uncountable set; i.e. this set contains infinite number of members and they are unable to be counted. Also, the real and complex numbers sets are uncountable, thus transcendental numbers set is also uncountable.

2) If some transcendental argument is applied to an algebraic function in one variable, then the result must be a transcendental in nature. 
For Example: A transcendental number $\pi$,
 can give numbers : $5\pi(\pi-8)$, $\sqrt{\pi}(4-3 \pi)$ etc.

3) The rational number cannot be transcendental. Since, we know that rational number is a number that is written in the form of $\frac{m}{n}$ in which both m and n are integers. Also, the equation nx - m = 0 has its root as $\frac{m}{n}$.

4) All the real transcendental numbers are necessary be irrational. Also, all irrational numbers are NOT essentially transcendental. 

5) The sum and product of two transcendental number may or may not be transcendental.
For Example: 
 e and (1 - e) are two transcendental numbers, but e + 1 - e = 1 which is not. Most usually, if x and y be two transcendental numbers, then either x + y or xy or both should be transcendental.

Continued Fractions

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The concept of continued fraction is extremely useful in the explanation of the theory of transcendental number. The continued fraction provides a method for constructing the transcendental numbers. We may say that a transcendental number can be written in the form of a continued fraction.
The general form of a continued fraction is given below :
C = 
$a_{0} $+ $\frac{b_{1}}{a_{1}+\frac{b_{2}}{a_{2}+ \frac{b_{3}}{a_{3}+...}}}$
If we substitute $b_{i}$ = 1, we get the simple continued fraction

C = $a_{0}$ + $\frac{1}{a_{1}+\frac{1}{a_{2}+ \frac{1}{a_{3}+...}}}$

Transcendental Functions

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Lets first learn about an algebraic function. A function is said to be an algebraic function, if it satisfies some polynomial equation having rational coefficients.

On the other hand, a transcendental function is a function that is obviously not algebraic. The function that cannot be explained in the terms of algebra is known as a transcendental function. In other words, a function is said to be a transcendental one if it does not satisfy some polynomial having rational coefficients. This function transcends, i.e. it goes beyond the limits.

For Example: Exponential functions, trigonometric functions, logarithmic functions.
Let us have a look at some particular examples of transcendental functions.
i) $f(a)$ = a$^{\pi}$
ii) $f(a)$ = $a^{a}$
iii) $f(a) $= $a^{\frac{1}{a}}$
iv) $f(x)$ = $sin x$ etc.

Examples

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However, it is extremely difficult to prove a number transcendental. Some most prominent examples of proven transcendental numbers are listed below:

1) $\pi$ = 3.14159... is a transcendental number.

2) e = 2.71828... is a transcendental number.

3) e$^{x}$ is transcendental, only when x $\neq$ 0 and x is an algebraic number.

4) sin x, cos x, tan x, csc x, sec x, cot x are transcendental if x is a nonzero algebraic number.

5) ln x is transcendental, if x is algebraic, also x $\neq$ 0, 1.

6) The continued fraction 
C = $1$ +$ \frac{1}{2+\frac{1}{3+ \frac{1}{4+...}}}$ is a transcendental number.

7) x y is a transcendental number, if x is algebraic and x $\neq$ 0 or 1, also y is irrational algebraic number.
For Example: 
$2^{\sqrt{2}}$.
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