Irrational Numbers

Prove that the $\sqrt{2} + \sqrt{3}$ is irrational.

#IrrationalNumbers #SquareRoot

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

This can be proved by contradiction. Suppose $\sqrt{2} + \sqrt{3}$ is rational. Then $\dfrac{1}{\sqrt{2} + \sqrt{3}} = \sqrt{3} - \sqrt{2}$ is rational as well. Now since the sum of rational numbers is always rational, we would then have that

$\dfrac{1}{2}((\sqrt{2} + \sqrt{3}) + (\sqrt{3} - \sqrt{2})) = \sqrt{3}$

is rational, which is not the case. So our original assumption must be false, i.e., $\sqrt{2} + \sqrt{3}$ is irrational.

Note that we can prove $\sqrt{3}$ is irrational by contradiction. Suppose that $\sqrt{3} = \dfrac{a}{b}$ for positive coprime integers $a,b.$ Squaring both sides gives us that $3b^{2} = a^{2},$ which by the Fundamental Theorem of Arithmetic implies that $3|a.$ But if $a = 3m$ then $3b^{2} = 9m^{2} \Longrightarrow b^{2} = 3m^{2},$ implying that $3|b.$ But we had assumed that $a,b$ were coprime, and thus our original assumption that $\sqrt{3}$ was rational is false.

Brian Charlesworth - 5 years, 9 months ago

Note that $\sqrt{2},\sqrt{3}$ are both algebraic integers, so $\sqrt{2}+\sqrt{3}$ is also an algebraic integer. However, if $\sqrt{2}+\sqrt{3}$ is rational then that means it is a rational integer, which is clearly false as $3 < \sqrt{2}+\sqrt{3} < 4$ .

Daniel Liu - 5 years, 9 months ago

Extension: if $\sqrt{a_1}+\sqrt{a_2}+\cdots +\sqrt{a_n}$ is rational, prove that it is integer.

In particular, prove that $a_1, a_2, \ldots , a_n$ are perfect squares.

Daniel Liu - 5 years, 9 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$