Irrational Number/Examples/Square Root of 3
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Example of Irrational Number
$\sqrt 3$ is irrational.
Aiming for a contradiction, suppose $\sqrt 3 = \dfrac m n$ for integers $m$ and $n$ such that:
- $m \perp n$
where $\perp$ denotes coprimality.
- $m^2 = 3 n^2$
Thus $3 \divides m^2$ and so $3 \divides m$.
- $m = 3 k$
for some $k \in \Z$.
- $9 k^2 = 3 n^2$
and so $3 \divides n$.
But then we have $3 \divides m$ and $3 \divides n$
Hence $m$ and $n$ are not coprime after all.
From this contradiction the result follows.
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 1$: Real Numbers: Exercise $\S 1.20 \ (6)$