Transcendental Number Definition
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 -
Pi, denoted by the symbol which is equal to the ratio of a circle's circumference to its diameter in a plane.
Exponential constant denoted by e, which is the base of the natural logarithm.
Examples of Transcendental Numbers
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.
pi = 3.1415 ...
e = 2.718 ...
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.)
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.)
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.
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.)
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?)
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.
FAQs on Transcendental Number
1. What exactly is a transcendental number in mathematics?
A transcendental number is a real or complex number that is not algebraic. This means it cannot be a solution (or a root) to any non-zero polynomial equation with rational coefficients. The most famous examples of transcendental numbers are Pi (π) and Euler's number (e).
2. What is the main difference between transcendental and irrational numbers?
The main difference lies in their definitions. An irrational number is any number that cannot be written as a simple fraction (a/b), like √2 or π. A transcendental number is a number that cannot be the root of a polynomial with rational coefficients. While all transcendental numbers are irrational, not all irrational numbers are transcendental. For instance, √2 is irrational but not transcendental because it is the solution to the polynomial equation x² - 2 = 0.
3. What are some common examples of transcendental numbers?
While proving a number is transcendental can be very difficult, several well-known examples exist in mathematics. These include:
- Euler's number (e), the base of the natural logarithm, approximately 2.718.
- Pi (π), the ratio of a circle's circumference to its diameter, approximately 3.14159. Note that the fraction 22/7 is only a rational approximation of π, not its actual value.
- Liouville's constant, which was the first number to be proven transcendental.
- Gelfond's constant, eπ.
4. Is the square root of 2 (√2) a transcendental number?
No, the square root of 2 (√2) is not a transcendental number. While it is an irrational number, it is considered an algebraic number. This is because it is the root of a simple polynomial equation with integer coefficients: x² − 2 = 0. Since it solves a polynomial equation, it does not meet the definition of a transcendental number.
5. Why is pi (π) considered a transcendental number?
Pi (π) is considered transcendental because no polynomial equation with rational coefficients exists for which π is a root. This was famously proven by the German mathematician Ferdinand von Lindemann in 1882 using the Lindemann-Weierstrass theorem. The proof is highly complex, but its conclusion is a cornerstone of number theory. This proof also confirmed that it's impossible to "square the circle" using only a compass and straightedge.
6. Are all transcendental numbers irrational, or can some be rational?
Yes, all transcendental numbers are irrational. A rational number, by definition, can be expressed as a fraction p/q. Any such number is a solution to the simple polynomial equation qx - p = 0. Since every rational number is a root of a polynomial with integer coefficients, it is an algebraic number by definition. Because transcendental numbers cannot be algebraic, they can never be rational.
7. Is there a simple formula or test to identify if a number is transcendental?
No, there is no simple formula or straightforward test to determine if a number is transcendental. Identifying them is a very challenging task in mathematics. Proofs of transcendence for numbers like e and π are highly advanced and were major mathematical achievements. Numbers are generally proven to be transcendental on a case-by-case basis using powerful theorems like the Gelfond–Schneider theorem or the Lindemann–Weierstrass theorem.
