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Prove that \(e\) is an irrational number
Solution
The proof that e is an irrational number is one of the most beautiful results in mathematics. Euler’s constant
𝑒
e, which appears in exponential and logarithmic functions, cannot be expressed as a simple fraction. Understanding why this is true not only strengthens your knowledge of number theory but also deepens your appreciation of how mathematics uncovers patterns and structure.
In this step-by-step explanation, we’ll follow the logic that shows why
𝑒
e cannot be written as a ratio of two integers. We’ll explore the series expansion of
𝑒
e, assume the opposite, and reach a contradiction — the core of mathematical reasoning.
To start, let’s expand the exponential function
\[\mathrm{e}^{x}=\sum_{k=0}^{\infty} \frac{x^{k}}{k !}\]
\[\Rightarrow e=\sum_{k=0}^{\infty} \frac{1}{k !}=1+\frac{1}{1 !}+\frac{1}{2 !}+\frac{1}{3 !}+\cdots\]
Assume that e is rational
\[e=\frac{m}{n}=1+\frac{1}{1 !}+\frac{1}{2 !}+\frac{1}{3 !}+\cdots \quad \text { for two coprime integers } m \text { and } n\]
\[\frac{m}{n}=1+\frac{1}{1 !}+\frac{1}{2 !}+\frac{1}{3 !}+\cdots+\frac{1}{n !}+\frac{1}{(n+1) !}+\frac{1}{(n+2) !}+\cdots\]
\[n ! \frac{m}{n}=n !+\frac{n !}{1 !}+\frac{n !}{2 !}+\frac{n !}{3 !}+\cdots+\frac{n !}{n !}+\frac{n !}{(n+1) !}+\frac{n !}{(n+2) !}+\cdots\]
\[\begin{aligned}
&\mathrm{A}=n ! \frac{m}{n}=(n-1) ! m \in \mathbb{N} \\
&\mathrm{B}=n !+\frac{n !}{1 !}+\frac{n !}{2 !}+\frac{n !}{3 !}+\cdots+\frac{n !}{n !} \in \mathbb{N} \\
&\mathrm{C}=\frac{1}{n+1}+\frac{1}{(n+1)(n+2)}+\frac{1}{(n+1)(n+2)(n+3)}+\cdots
\end{aligned}\]
\[\begin{gathered}
\text { For } n \geqslant 1 \\
\frac{1}{n+1} \leqslant \frac{1}{2} \\
\frac{1}{(m+1)(n+2)}<\frac{1}{2^{2}} \\
\frac{1}{(n+1)(n+2)(n+3)}<\frac{1}{2^{3}} \\
\vdots
\end{gathered}\]
\[\Rightarrow\mathrm{C}<\frac{1}{2}+\frac{1}{2^{2}}+\frac{1}{2^{3}}+\cdots=\left(\frac{1}{2}\right)\left(\frac{1}{1-\frac{1}{2}}\right)=1\]
\[0<c<1 \Rightarrow C \notin \mathbb{N}\]
\[\text { Therefore }\]
\[\huge \mathrm{e} \notin \mathbb{Q}\]
This proof not only confirms that
𝑒
e cannot be written as a fraction but also highlights how infinite series reveal deep truths about numbers. The concept of irrationality is central to understanding real numbers, calculus, and complex analysis.
The proof that e is irrational demonstrates how the combination of series, factorials, and logical reasoning uncovers the hidden structure of numbers. It’s a reminder that even simple mathematical constants like
𝑒
e carry profound and infinite complexity.
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