Irrational Numbers, help needed. (1 Viewer)

axwe7 · Dec 22, 2015

Why is it that 2 + {\sqrt {3}} is irrational? [Hint: Begin by saying; Suppose 'x = 2 + {\sqrt {3}}', then continue by subtracting 2 from both sides].

The hint was included in the question...

BTW, {\sqrt {3}} is irrational, I've already proven that.

InteGrand · Dec 22, 2015

axwe7 said:
Why is it that 2 + {\sqrt {3}} is irrational? [Hint: Begin by saying; Suppose 'x = 2 + {\sqrt {3}}', then continue by subtracting 2 from both sides].

The hint was included in the question...

BTW, {\sqrt {3}} is irrational, I've already proven that.

$Suppose by way of contradiction that $2+\sqrt{3}$ is rational. Then by definition of ``rational'', we have $2+\sqrt{3} =\frac{a}{b}$ for some integers $a,b$. Thus $\sqrt{3} = \frac{a}{b}-2 =\frac{a-2b}{b}$. The numerator and denominator here are both integers, so $\sqrt{3}$ is rational, which contradicts the fact that $\sqrt{3}$ is irrational. This means our original assumption is false, so $2+\sqrt{3}$ is irrational.$ \blacksquare$

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Irrational Numbers, help needed. (1 Viewer)

axwe7

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InteGrand

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