π: A Never-Ending Mystery - Why Its Decimal Expansion Defies Fractions
One of the most well-known proofs of π's irrationality is a proof by contradiction, often attributed to Johann Lambert. While a full, rigorous derivation involves more advanced calculus, the core idea can be understood conceptually:
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Assume for contradiction: Suppose that π is rational. This means that π can be expressed as a fraction π=ba, where a and b are integers and b=0.
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Utilize calculus and integration: The proof involves constructing specific integrals involving trigonometric functions and powers of x. These integrals are carefully chosen such that their values can be related to powers of πand factorials.
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Derive a contradiction: Through a series of steps involving integration by parts and careful analysis of the resulting expressions, it can be shown that if π=ba, then certain integral values must be integers. However, further analysis reveals that these same integral values must lie strictly between 0 and 1, which is a contradiction since an integer cannot be strictly between 0 and 1.
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Conclusion: Since the initial assumption that π is rational leads to a contradiction, this assumption must be false. Therefore, π is irrational.
In simpler terms, the proof demonstrates that if π were a fraction, it would lead to a situation where a non-zero integer is simultaneously smaller than 1, which is impossible. This contradiction definitively establishes that π cannot be expressed as an exact ratio of two integers, confirming its irrational nature. The decimal representation shown in the image is merely a finite approximation and does not represent the true, infinite, and non-repeating nature of π.