A transcendental number is a real or complex number that is not algebraic, meaning that it cannot be the root of a non-zero polynomial equation with integer coefficients. The first transcendental numbers were discovered in the early 19th century, and their existence was not rigorously proven until 1844. The most famous transcendental numbers are e and π.

### What are examples of transcendental numbers?

There are many examples of transcendental numbers, but some of the most famous are e and π. These two numbers are so important that they even have their own special symbols!

e is the base of natural logarithms, meaning that it is the number that you get when you take the natural logarithm (ln) of any number. For example, ln(2) = 0.693… and ln(10) = 2.303… So, e must be somewhere between 2 and 3. In fact, it is exactly 2.71828…

π is the ratio of a circle's circumference to its diameter. It is also the number you get when you take the circumference of any circle and divide it by the diameter. So, if you have a circle with a circumference of 10 units and a diameter of 2 units, the ratio is 5 units, or π = 5.

There are many other transcendental numbers, but these are two of the most famous and important ones. Why are transcendental numbers important? Transcendental numbers are important because they can't be expressed as a rational number. This means that they're not repeating or terminating decimals, which makes them useful for things like measuring circles ( pi is a transcendental number).

Is Phi a transcendental number? The answer to this question is a bit complicated. Phi is an irrational number, which means that it cannot be expressed as a rational number (i.e. a number that can be expressed as a fraction). This means that it is impossible to find a rational number that is equal to Phi. However, it is possible to find an algebraic number that is equal to Phi. An algebraic number is a number that can be expressed as a root of a polynomial equation with integer coefficients. Therefore, Phi is not a transcendental number. What is the most mysterious number? The most mysterious number is the square root of -1. This number is also known as "i" and is represented by the symbol "√-1". It is a complex number that cannot be represented by a real number. It is often used in mathematical equations to represent an unknown quantity. Is pi irrational or transcendental? Both pi and e are irrational numbers, meaning they cannot be expressed as a rational number (a number that can be written as a fraction). Furthermore, pi is a transcendental number, meaning that it is not the root of any non-zero polynomial with rational coefficients. This means that it is impossible to solve for pi using any finite sequence of arithmetic operations (addition, subtraction, multiplication, division, etc.).