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How can we establish algebraically if $\sqrt{2}\sqrt{3}$ is greater than or less than $\sqrt{2} + \sqrt{3}$?

I know I can plug the values into any calculator and compare the digits, but that is not very satisfying. I've tried to solve $$\sqrt{2}+\sqrt{3}+x=\sqrt{2}\sqrt{3} $$ to see if $x$ is positive or negative. But I'm just getting sums of square roots whose positive or negative values are not obvious.

Can it be done without the decimal expansion?

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Both quantities are positive so squaring preserves the inequality between them. Thus one checks if $2+2\sqrt{6}+3~\square~6$, which should be clear since $\sqrt{6}>1$. –  anon Dec 2 '12 at 4:30
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3 Answers 3

up vote 35 down vote accepted

Method 1: $\sqrt{2}+\sqrt{3}>\sqrt{2}+\sqrt{2}=2\sqrt{2}>\sqrt{3}\sqrt{2}$.

Method 2: $(\sqrt{2}\sqrt{3})^2=6<5+2<5+2\sqrt{6}=2+3+2\sqrt{2}\sqrt{3}=(\sqrt{2}+\sqrt{3})^2$, so $\sqrt{2}\sqrt{3}<\sqrt{2}+\sqrt{3}$.

Method 3: $\frac{196}{100}<2<\frac{225}{100}$ and $\frac{289}{100}<3<\frac{324}{100}$, so $\sqrt{2}\sqrt{3}<\frac{15}{10}\frac{18}{10}=\frac{270}{100}<\frac{14}{10}+\frac{17}{10}<\sqrt{2}+\sqrt{3}$.

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That is just what I wanted. Thanks! –  Jeffrey L Whitledge Dec 2 '12 at 4:43
    
+1 Your answer is spot on! –  amWhy Dec 2 '12 at 4:53
    
That's very good, Will Hunting. –  Grumpy Parsnip Dec 4 '12 at 2:56
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@Jim: Did you mean to write "That's very, Good Will Hunting." maybe? :-) –  Asaf Karagila Dec 30 '12 at 22:55
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@AsafKaragila yes that's what I was getting at. :) –  Grumpy Parsnip Jan 1 '13 at 23:10
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$$\sqrt{3}\sqrt{2}-\sqrt{3}-\sqrt{2}+1=(\sqrt{3}-1)(\sqrt{2}-1) < 1$$

The last inequality follows from the fact that $(\sqrt{3}-1)$ and $(\sqrt{2}-1)$ are in $(0,1)$.

Second solution

By AM-GM you have

$$2\sqrt[4]{6} \leq \sqrt{2}+\sqrt{3}$$

Combine this with $\sqrt[4]{6} < 2$, which is easy to prove, and you are done.

And a non-algebraic one, which is an overkill :)

Let $\theta$ be the angle so that $\cos(\theta)=-\frac{1}{2\sqrt{6}}$. Plot a point $A$ draw two rays with an angle of $\theta$ between them, and pick points $B$ respectively $C$ on these ray so that $AB=\sqrt{2}$ and $AC=\sqrt{3}$. By the cosine law

$$BC^2=2+3+2\sqrt{2}\sqrt{3}\frac{1}{2\sqrt{6}}=6$$

Thus the triangle $ABC$ has the edges of length $\sqrt{2}, \sqrt{3}$ and $\sqrt{6}$, and your inequality is exactly the triangle inequality.

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Another very straightforward way which works for many simple inequalities is to show that the inequality is logically equivalent to a tautology:

As we are only dealing with positive numbers, we have $$ \sqrt 2 + \sqrt 3 > \sqrt 2 \sqrt 3 \\ \Leftrightarrow \;(\sqrt 2 + \sqrt 3)^2 > 6 \\ \Leftrightarrow\; 2\sqrt 6 > 1.$$ The last inequality holds since $\sqrt x \geq 1$ for all $x\geq 1$.

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Dear Andy, This approach also has the advantage that we don't have to know in advance that the relationship is $>$, rather than $<$. Regards, –  Matt E Dec 30 '12 at 23:17
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