🌀 Why is the number π (pi) irrational?

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The number π (pi), a universal mathematical symbol, intrigues far beyond geometry. Present in various fields such as chemistry and medicine, it remains an enigma for mathematicians.

Pi is an irrational number, meaning it cannot be expressed as a simple fraction. Despite calculations reaching 105 trillion decimal places, its exact nature remains elusive. Rational numbers, unlike pi, can be represented as a division of two integers.

Proving the irrationality of pi is a research in itself. Mathematicians use proofs by contradiction, starting from the assumption that pi is rational to arrive at a contradiction. This method requires a deep understanding of mathematical analysis.

The proofs of pi's irrationality vary, some using trigonometric functions or infinite continued fractions. These approaches all rely on the principle of contradiction to establish that pi cannot be expressed as a fraction.

Pi is also a transcendental number, meaning it cannot be a solution to a polynomial equation with integer coefficients. This property reinforces its irrationality, as all transcendental numbers are irrational.

Despite its theoretical importance, in practice, an approximation of pi to a few decimal places is enough. Even NASA uses only 16 decimal places of pi for its space calculations, showing that extreme precision is not always necessary.

The study of pi continues to keep mathematicians busy, not only for its practical applications but also for the mysteries it holds. Its irrational and transcendental nature makes it an inexhaustible research topic.

What is an irrational number?

An irrational number is a real number that cannot be expressed as a fraction of two integers. Unlike rational numbers, irrational numbers have infinite non-repeating decimals.

Irrational numbers play a crucial role in mathematics, especially in geometry and analysis. Their existence was a major discovery in the history of mathematics, challenging notions of measurement and proportion.

Pi is one of the most famous examples of irrational numbers. It has profound implications for calculations and mathematical theories.

The study of irrational numbers continues to be an active area of research in mathematics, with implications in fields as diverse as cryptography and number theory.

Why is pi considered transcendental?

Pi is considered a transcendental number because it cannot be a solution to a polynomial equation with integer coefficients. This means it cannot be expressed as a root of an algebraic equation.

The transcendence of pi was proved in 1882 by Ferdinand von Lindemann. This proof solved the ancient problem of squaring the circle, showing that it is impossible to construct a square of the same area as a given circle using only a ruler and compass.

The transcendence of pi has important implications in mathematics and physics. It means that pi cannot be expressed exactly by an algebraic formula, which limits methods of calculation and approximation.

The discovery of pi's transcendence opened new avenues of research in mathematics, particularly in the study of transcendental numbers and their properties. It also has applications in fields such as number theory and cryptography.

kace Jul 28, 2026, 08:33 AM

The transcendence of π is very complicated to demonstrate, and its irrationality is not simple either.
An example of irrationality that is easy to demonstrate, and very interesting, is that of the square root of 2, which I will denote as √2.
Here too, we reason by contradiction: suppose that √2 is rational. There exists therefore a positive integer (p) and a strictly positive integer (q) such that:
√2 = p/q.
We can always choose p and q such that the fraction is irreducible (for example, if the initial fraction is 7p/7q, we divide by 7 on top and bottom to arrive at p/q). Thus, p and q are coprime (they have no common prime factor).
In particular, they cannot both be even, otherwise the fraction would not be irreducible.
Let us return now to √2 = p/q. By squaring both sides, we obtain 2 = p²/q², then, by multiplying by q² we obtain 2q² = p².
And since 2q² is even, p² is even as well. However, an integer and its square always have the same parity (the square of an even integer is even, and the square of an odd integer is odd). Therefore, since p² is even, p is also even.
We can therefore write p = 2p', where p' is an integer.
By substituting into the previous equality, we obtain 2q² = (2p')² = 4p'², then by simplifying by 2 we obtain q² = 2p'². Thus, q² is even, and therefore by the same reasoning, q is also even.
By assuming that √2 = p/q, we have therefore shown that p and q are both even. This contradicts the fact that we had chosen an irreducible fraction: p and q could not both be even!
Conclusion: our initial hypothesis is false. We can therefore not find integers p and q such that √2 = p/q. In other words, √2 is irrational. Q.E.D.

Kfé minet Jul 28, 2026, 12:44 PM

With this py we're going in circles, which is nothing irrational or transcendent... Unless you want to make it an object... of worship?