Transcendental Number Explained with Meaning and Significance

In Mathematics, we can define a  transcendental number as a real number that is not algebraic as well as is not the solution of any single-variable polynomial equation whose coefficients are known to be all integers (basically whole numbers). 

Aendental numbers are generally irrational numbers. But keep in mind that there are some irrational numbers that are not transcendental.

Let’s See a Few Examples of Transcendental Numbers -

1. Pi, denoted by the symbol which is equal to the ratio of a circle's circumference to its diameter in a plane.

2. Exponential constant denoted by e, which is the base of the natural logarithm. 

Many of you would have probably heard of pi as well as e. But are there other famous transcendental numbers? Here are a few transcendental numbers.

1. pi = 3.1415 ...

2. e = 2.718 ...

3. Euler's constant, gamma which equals 0.577215.. Which equals  lim n -> infinity > (which equals 1 + 1/2 + 1/3 + 1/4 + ... + 1/n - ln(n)) (This is not proven to be transcendental, but generally this number is believed to be by mathematicians.)

4. Catalan's constant, G equals sum (-1)^k / (2k + 1 )² equals 1 - 1/9 + 1/25 - 1/49 + ... (Not proven to be a transcendental number, but generally believed to be a transcendental number by mathematicians.)

5. Liouville's number is equal to 0.110001000000000000000001000 ... which has a one in the first, second, sixth, twenty-fourth, etc. places, and this number has zeros elsewhere.

6. Chaitin's "constant", is the probability that a random algorithm halts. (Noam Elkies of Harvard notes that not only is this Chaitin's "constant" transcendental but it is also said to be incomputable.)

7. Chapernowne's number is equal to 0.12345678910111213141516171819202122232425 … and this number is constructed by concatenating the digits of the positive integers, that you can see in the pattern?)

8. Special values of the zeta function, for example, zeta (3). (At rational points, we can expect transcendental functions to give transcendental results).

More Transcendental Numbers

It took until the year 1873 for the first "non-constructed" number to be proved as transcendental when mathematician Charles Hermite proved that he was transcendental.

Then in the year 1882, Ferdinand von Lindemann proved that π was transcendental.

In fact, proving that a number is Transcendental is quite difficult, even though these transcendental numbers are known to be very common.

Transcendental Numbers are Common

Most Real Numbers are Transcendental. The Argument for This is:

• We can say that the Algebraic Numbers are "countable", we can make a list of such numbers.

• But the Real numbers are "Uncountable".

• And since a Real number is either Algebraic or a real number is said to be Transcendental, the Transcendentals must be "Uncountable".

• So there are many more Transcendentals than Algebraic.

Numbers Proven to be Transcendental

• ea if the exponent a is algebraic and if a is nonzero (by the Lindemann–Weierstrass theorem).

• π, known as pi and has a value of 3.14.. (by the Lindemann–Weierstrass theorem).

• eπ, which is known as Gelfond's constant, as well as e−π/2 equals ii (by the Gelfond–Schneider theorem).

• ab where a is algebraic but not 0 or 1, and b is irrational algebraic (by the Gelfond–Schneider theorem), in particular, that is: 2√2, the Gelfond–Schneider constant (or Hilbert number)

• sin a, cos a, tan a, cosec a, sec a, and cot a, as well as their hyperbolic counterparts, for any nonzero algebraic number a, expressed in radians (by the Lindemann–Weierstrass theorem).

• ln a, if a is algebraic and if the value of a is not equal to 0 or 1, for any branch of the logarithm function (by the Lindemann–Weierstrass theorem).

• logb a, if the value of a, as well as b is a positive integer, not both powers of the same integer (by the Gelfond–Schneider theorem).

The Distinction Between Algebraic Numbers and Transcendental Numbers May Also be Applied to Numbers, for Example, Numbers Like -

• The square root of √2 is called algebraic numbers because they satisfy polynomial equations with integer coefficients. 

• In this case, the Square root of √2 satisfies the equation x2 equals 2. All other numbers are called transcendental. As early as the 17th century, transcendental numbers were believed to exist, and pi denoted by the symbol π was the usual suspect. Perhaps Descartes had pi, denoted by symbol π in mind when he despaired of finding the relation between straight and curved lines.

• A brilliant, though flawed, attempt to prove that pi, denoted by the symbol π is transcendental was made by James Gregory in the year 1667. However, this problem was too difficult for 17th-century methods. The transcendence of pi, π was not successfully proved until the year 1882 when Carl Lindemann adapted a proof of the transcendence of e found by Charles Hermite in the year 1873.