Do You Know Transcendental Number?

Explanation
π and e are one of the best-known transcendental numbers. Transcendental numbers are real numbers that are not solutions to any algebraic equation with integer coefficients. Both π and e are irrational numbers and cannot be expressed as the ratio of two integers. They have infinite non-repeating decimal representations. Transcendental numbers have applications in various branches of mathematics and are of great importance in analysis and number theory.

Explanation
Transcendental numbers are a topic studied in mathematics. These numbers are real numbers that are not algebraic, meaning they cannot be the root of any non-zero polynomial equation with integer coefficients. They are considered "transcendental" because they transcend the realm of algebraic numbers. The study of transcendental numbers involves understanding their properties, proving their existence, and exploring their relationship with other mathematical concepts. Therefore, the correct answer is Mathematics.

Explanation
Transcendental numbers are a type of real number that are not algebraic, meaning they cannot be the solution to any polynomial equation with integer coefficients. Irrational numbers, on the other hand, are numbers that cannot be expressed as a fraction and have non-repeating decimal representations. Since transcendental numbers are a subset of irrational numbers, the correct answer is "Irrational".

Explanation
The name "transcendental" is derived from a Latin word. Latin is a classical language that has greatly influenced many modern languages, including English. The term "transcendental" refers to something that goes beyond ordinary limits or boundaries, and this concept was first introduced by philosophers in ancient Rome who used the Latin language to describe it. Therefore, it makes sense that the name "transcendental" is derived from a Latin word.

Explanation
The word "transcendental" was first used for the mathematical concept in a paper written by Leibniz in 1682. The context suggests that the question is asking for the year in which the term was first used, and the correct answer is 1682.

Explanation
The term "transcendental" was first used for the mathematical concept in Leibniz's 1682 paper.

Explanation
Euler is likely the first person to define transcendental numbers in the modern sense. Euler made significant contributions to the field of mathematics and is known for his work on many mathematical concepts, including the definition and study of transcendental numbers. His work laid the foundation for our understanding of these numbers and their properties, making him a likely candidate for being the first person to define them in the modern sense.

Explanation
Johann Heinrich Lambert conjectured that e and π were both transcendental numbers in his 1768 paper proving the number π is irrational, and proposed a tentative sketch of a proof of π's transcendence. This means that Lambert made a hypothesis or educated guess that both e and π are transcendental numbers, which are numbers that are not the root of any non-zero polynomial equation with integer coefficients. In his paper, he also provided a rough outline or plan for proving the transcendence of π.

Explanation
Joseph Liouville is credited with first proving the existence of transcendental numbers in 1844. Transcendental numbers are numbers that are not algebraic, meaning they cannot be the solutions to any polynomial equation with integer coefficients. Liouville's proof involved constructing a specific transcendental number using continued fractions. This breakthrough in mathematics provided a deeper understanding of the nature of numbers and their properties.

Explanation
Joseph Liouville first proved the existence of transcendental numbers in 1844.