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
1. What is a transcendental number in Mathematics?
A transcendental number is a real or complex number that is not an algebraic number. This means it cannot be a solution (or a root) to any non-zero polynomial equation with rational coefficients. In simpler terms, you cannot create an equation like a_nx^n + ... + a_1x + a_0 = 0, where all 'a' coefficients are rational numbers, that has the transcendental number as its solution.
2. What are some common examples of transcendental numbers?
While most numbers students encounter are algebraic, some of the most famous numbers in mathematics are transcendental. Key examples include:
- Pi (π): The ratio of a circle's circumference to its diameter, approximately 3.14159.
- Euler's number (e): The base of the natural logarithm, approximately 2.71828.
- Liouville's constant: The first number explicitly proven to be transcendental.
- Chaitin's constant (Ω): A number representing the probability that a random program will halt.
3. What is the difference between an irrational number and a transcendental number?
The main difference lies in their relationship with polynomial equations. An irrational number is any number that cannot be expressed as a simple fraction (p/q), like √2 or π. A transcendental number is a specific type of irrational number that is also not a root of any polynomial with rational coefficients. Therefore, all transcendental numbers are irrational, but not all irrational numbers are transcendental. For instance, √2 is irrational but it is algebraic (it's the solution to x² - 2 = 0), so it is not transcendental.
4. Are all transcendental numbers also irrational?
Yes, every transcendental number is also an irrational number. The definition of a transcendental number is that it is not algebraic. Since all rational numbers are algebraic (for example, the fraction 5/7 is the solution to 7x - 5 = 0), a number that is not algebraic (i.e., transcendental) cannot be rational. Therefore, it must be irrational.
5. Why is a number like the square root of 2 (√2) not considered transcendental?
The square root of 2 (√2) is not transcendental because it is an algebraic number. It is the solution to the simple polynomial equation x² - 2 = 0. Since √2 is a root of this polynomial which has integer (and thus rational) coefficients, it fits the definition of an algebraic number, and therefore it cannot be a transcendental number.
6. What does it mean for π (pi) to be transcendental and what is its importance?
Stating that π is transcendental means it can never be the solution to a polynomial equation with rational coefficients. The primary importance of this fact, proven by Ferdinand von Lindemann in 1882, was that it definitively solved an ancient mathematical problem. It proved that "squaring the circle"—constructing a square with the same area as a given circle using only a compass and straightedge—is impossible.
7. How many transcendental numbers exist?
There is an uncountably infinite number of transcendental numbers. In fact, it has been proven that the set of transcendental numbers is much larger than the set of algebraic numbers. While we only know a few famous examples, almost all real numbers are transcendental. This is a counter-intuitive but fundamental concept in number theory.
8. What is a transcendental function and how does it relate to transcendental numbers?
A transcendental function is a function that cannot be expressed as a finite sequence of algebraic operations (addition, subtraction, multiplication, division, and raising to a power). Common examples include the trigonometric functions (sin(x), cos(x)), the exponential function (e^x), and the logarithmic function (log(x)). These functions 'transcend' algebra. The relationship is that applying a transcendental function to an algebraic number can produce a transcendental number. For example, the Lindemann-Weierstrass theorem states that if 'a' is a non-zero algebraic number, then e^a is a transcendental number.
