\(
\newcommand{\cat}[1] {\mathrm{#1}}
\newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})}
\newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}}
\newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}}
\newcommand{\betaReduction}[0] {\rightarrow_{\beta}}
\newcommand{\betaEq}[0] {=_{\beta}}
\newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}}
\newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}}
\newcommand{\groupMul}[1] { \cdot_{\small{#1}}}
\newcommand{\inv}[1] {#1^{-1} }
\newcommand{\bm}[1] { \boldsymbol{#1} }
\require{physics}
\require{ams}
\)
Math and science::Analysis::Tao::05. The real numbers
The irrationality of \( \sqrt(2) \)
Consider a rational number fully reduced, \( \frac{p}{q}\) such that \(\frac{p^2}{q^2} = 2\). Investigate whether \(p\) or \(q\) are odd or even numbers leads to a contradiction which shows that \(2\) cannot be rational.
- \(p\) and \(q\) can't both be even, as [...].
- \(p\) and \(q\) must both be odd, as [...].
- \(p\) is even as [...].
These points lead to a contradiction.
