As a mathematics expert with a deep understanding of numerical properties, I am well-equipped to discuss the nature of various numbers. When we talk about the number Pi, denoted as π, we are referring to a mathematical constant that is defined as the ratio of a circle's circumference to its diameter. This constant is a fundamental aspect of geometry and has been a subject of fascination for mathematicians for centuries due to its unique properties.

Pi is an irrational number. This means that it cannot be expressed as a simple fraction, where the numerator and the denominator are integers. The statement that Pi is irrational is supported by the fact that its decimal representation is infinite and non-repeating. If a number were rational, its decimal expansion would either terminate or become periodic after a certain point, repeating a sequence of digits indefinitely.

The approximation of Pi as 22/7 is a common one, but it is indeed an approximation. While it provides a close estimate, it is not exact. The true value of Pi is more accurately represented by its decimal expansion, which goes on indefinitely without settling into a repeating pattern. This characteristic is what sets irrational numbers apart from rational ones.

One of the first rigorous proofs that Pi is irrational came from the work of mathematicians like Johann Lambert in the 18th century. They used the fact that if a number is rational, then it can be expressed as a fraction of two integers. However, they showed that if you assume Pi is rational and try to express it as such, you end up with a contradiction, proving that Pi cannot be rational.

In modern mathematics, Pi is also known to be a transcendental number, which means it is not the root of any non-zero polynomial equation with rational coefficients. This was proven by Ferdinand von Lindemann and Karl Weierstrass in the late 19th century. The transcendental nature of Pi adds another layer of complexity to its properties, distinguishing it even further from algebraic numbers, which are the roots of such polynomials.

The study of Pi extends beyond its irrationality. It has been calculated to trillions of digits with the help of computers, and its digits are tested for randomness. The digits of Pi are believed to be normally distributed, meaning that the frequency of each digit appearing is roughly the same over a large enough sample.

In conclusion, Pi is an irrational number with a rich history and many intriguing properties. Its infinite, non-repeating decimal expansion and its transcendental nature make it a fascinating subject for mathematicians and a cornerstone of mathematical constants.

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Example: -- (Pi) is a famous irrational number. We cannot write down a simple fraction that equals Pi. The popular approximation of 22/7 = 3.1428571428571... is close but not accurate. Another clue is that the decimal goes on forever without repeating.read more >>