Prove that 2 root 3 -1 …
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Sia ? 6 years, 5 months ago
Let us assume that {tex}2\sqrt 3 - 1{/tex} is a rational. number
Then, there exist positive co-primes a and b such that
{tex}2\sqrt 3 - 1 = \frac{a}{b}{/tex}
{tex}2\sqrt 3 = \frac{a}{b} + 1{/tex}
{tex}\begin{array}{l}2\sqrt3=\frac{\mathrm a+\mathrm b}{\mathrm b}\\\end{array}{/tex}
{tex}\sqrt 3 = \frac{{a + b}}{{2b}}{/tex}
Here {tex}\begin{array}{l}\frac{\mathrm a+\mathrm b}{\mathrm b}\\\end{array}{/tex} is a rational number ,so {tex}\sqrt3{/tex} is a rational number
This contradicts the fact that {tex}\sqrt 3{/tex} is an irrational number
Hence {tex}2\sqrt 3 - 1{/tex} is irrational
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