# which is larger number? $\sqrt{7}-\sqrt{6}$ or $\sqrt{6}-\sqrt{5}$

Which is larger number? $\sqrt{7}-\sqrt{6}$ or $\sqrt{6}-\sqrt{5}$? Squaring both sides will give me something but I could not go any further.

• Use calculator to this end. Nov 11 '13 at 10:16
• Nov 11 '13 at 14:02
• Nov 11 '13 at 18:44
• Just need a bit of common sense: square roots rise slower the larger the number being square rooted Nov 14 '13 at 8:07

As $(\sqrt7+\sqrt5)^2=12+2\sqrt{35}$ and $(\sqrt6+\sqrt6)^2=12+2\sqrt{36}$

$$(\sqrt7+\sqrt5)^2<(\sqrt6+\sqrt6)^2$$

$$\implies \sqrt7+\sqrt5<\sqrt6+\sqrt6\text{ as } a^2>b^2\iff a>b\text{ for }a,b>0$$

$$\implies \sqrt7-\sqrt6<\sqrt6-\sqrt5$$

Hint. $$\sqrt{a+1}-\sqrt{a}=\frac{1}{\sqrt{a+1}+\sqrt{a}}$$

Consider the shape of the graph of the root function. It is monotonously rising but getting flatter all the time. Hence the differences between two values ($1$ apart on the x-axis) near the y-axis is greater than between two values ($1$ apart on the x-axis) further away from the y-axis. • Nice to have a plot. You do realize that this is exactly what T.Bongers's is saying :-) Nov 11 '13 at 17:12
• Yeah, his answer proves the second part of my second sentence. But I'm pretty sure my answer was here before his. Just the plot took me a little longer and was added later ;-)
– Alfe
Nov 12 '13 at 11:16

Consider the function $f(x) = \sqrt{x}$. Then the second derivative is $$f''(x) = -\frac{1}{4}x^{-3/2} < 0$$

Hence this function is concave down, so we see that $\sqrt{7} - \sqrt{6}$ is the smaller of the two numbers.

• If we wanted to go the calculus route (which is probably too advanced for OP), we could just note that the derivative of $\sqrt{x+1} - \sqrt{x}$ is $> 0$ for $x > 0$ Nov 11 '13 at 19:52

To take your idea of squaring both sides, you can do one thing to make that more successful. We are comparing $\sqrt 7-\sqrt 6$ with $\sqrt 6-\sqrt 5$ so $$\sqrt 7-\sqrt 6\stackrel{?}{\lt\gt}\sqrt 6-\sqrt 5\\\sqrt 7 + \sqrt 5\stackrel{?}{\lt\gt}2\sqrt 6\\ 12 + 2\sqrt{35} \stackrel{?}{\lt\gt}24$$ but since $\sqrt {35} \lt 6$ the left is less.

• My post is in the spirit of exploration, trying to find the answer. lab bhattacharjee's is a good example of how to write up what you discovered in this approach. Nov 11 '13 at 4:19
• +1 for $\stackrel{?}{\lt\gt}$ Nov 14 '13 at 7:07

Hint: \begin{align} \frac1{\sqrt7+\sqrt6}\le\frac1{\sqrt6+\sqrt5} \end{align}

You can also use the inequality between the arithmetic mean and the quadratic mean: $$\frac{x+y}{2} \leq \sqrt{\frac{x^2 + y^2}{2}}.$$ Setting $x = \sqrt{5}$ and $y = \sqrt{7}$ gives you $$\frac{\sqrt{5} + \sqrt{7}}{2} \leq \sqrt{6},$$ which is equivalent to $$\sqrt{7} - \sqrt{6} \leq \sqrt{6} - \sqrt{5}.$$

Brute force solution, Only Squarings:

Let's assume: $$\sqrt{7}-\sqrt{6} \geq \sqrt{6}-\sqrt{5}$$ after squaring $$13 -2 \sqrt{42} \geq 11 - 2 \sqrt{30}$$ after simplification: $$1 -\sqrt{42} \geq -\sqrt{30}$$ reversing: $$\sqrt{30} \geq \sqrt{42}-1$$ Squaring 2nd time: $$30 \geq 43 - 2 \sqrt{42}$$ Simplification: $$2 \sqrt{42} \geq 13$$ Last squaring: $$168 \geq 169$$ Contradiction, thefore $$\sqrt{7}-\sqrt{6} < \sqrt{6}-\sqrt{5}$$

$\sqrt{7} - \sqrt{6} = (\sqrt7 - \sqrt6 )\frac{\sqrt7 + \sqrt 6}{\sqrt7 + \sqrt 6} = \frac1{\sqrt7 + \sqrt 6}$

Similarily,
$\sqrt6 - \sqrt 5 = \frac1{\sqrt6 + \sqrt 5}$

Since $\sqrt 7 > \sqrt 5$,
$$\sqrt7 + \sqrt6 > \sqrt6 + \sqrt5$$ $$\implies \frac1{\sqrt7 + \sqrt6} < \frac1{ \sqrt6 + \sqrt5}$$ $$\implies \sqrt{7} - \sqrt{6} < \sqrt{6} - \sqrt{5}$$

So $\sqrt6 - \sqrt5$ is the greater one.

Let $A = (\sqrt{7}-\sqrt{6}) - (\sqrt{6}-\sqrt{5} )$

This comes to comes to $A = (\sqrt{7}+\sqrt{5})-2\sqrt{6}$

The square of the first is $12+2\sqrt{35}$, the square of the second is $12+2\sqrt{36}$

Since $35 \lt 36$, then $A \lt 0$, hence $\sqrt{6}-\sqrt{5}$ is bigger.

$7-6=6-5$

or, equivalently $(\sqrt7-\sqrt6)(\sqrt7+\sqrt6)=(\sqrt6-\sqrt5)(\sqrt6+\sqrt5)$

hence, $$\frac{\sqrt7-\sqrt6}{\sqrt6-\sqrt5}=\frac{\sqrt6+\sqrt5}{\sqrt7+\sqrt6}<1 \text{ }\text{ }\text{ } (\text{since } \sqrt7>\sqrt5)$$

$\therefore \sqrt6-\sqrt5>\sqrt7-\sqrt6$

$$\frac{f(5)+f(7)}{2} \lt f(\frac{5+7}{2})$$ for a concave function $f$, and so for $f(x)=\sqrt x$.

But even if you proceed on the way you have started you will get the result.

$$\begin{eqnarray} \sqrt{7}-\sqrt{6} &\lt& \sqrt{6}-\sqrt{5} &\mid& ^2 \\ 13-2\,\sqrt{6}\,\sqrt{7} &\lt& 11-2\,\sqrt{5}\,\sqrt{6} &\mid& -11+2\sqrt{6}\sqrt{7} \mid \div 2 \\ 1 &\lt& \sqrt{6}\,\sqrt{7}-\sqrt{5}\,\sqrt{6} & \mid& ^2 \\ 1&\lt& 72-12\,\sqrt{5}\,\sqrt{7} & \mid& +12\sqrt{5}\sqrt{7} \\ 12\,\sqrt{5}\,\sqrt{7} &\lt& 71 & \mid& ^2 \\ 12^2 \cdot 35 &\lt& 71^2 \end{eqnarray}$$ All squaring operations are reversible and don't change the inequality relation because the LHS and the RHS of the inequalities are positive. But $$12^2 \cdot 35 =12 \cdot 6 \cdot 2 \cdot 35=72 \cdot 70 = (71+1)(71-1)=71^2-1$$

So the last and therefore all inequalities are true.

Here's how to figure it out without any arithmetic:

1. As you know, the graph of the square function grows steeper and steeper: The difference between $(n+1)^2$ and $n^2$ grows ever larger as $n$ increases.

2. The square root function is the inverse of the square function, so the difference grows ever smaller as $n$ increases. Therefore, the difference between $\sqrt{7}$ and $\sqrt{6}$ is smaller than the difference between $\sqrt{6}$ and $\sqrt{5}$.

$\sqrt 6−\sqrt 5$ is greater than $\sqrt 7−\sqrt 6$ because the value of $\sqrt 6$ is 2.449, $\sqrt 5$ is 2.236 $\sqrt 7$ is 2.645 on subtracting $\sqrt 6−\sqrt 5$ you get 0.213 whereas on subtracting $\sqrt 7−\sqrt 6$ you get 0.196.

So, $\sqrt 6−\sqrt 5$ is greater.

$1-$ I will leave this part as it is; please read the comments below:

$$\lim_{n\rightarrow\infty} \sqrt{n+1}-\sqrt{n}=0$$

$2-$ let

$$f(n)=\sqrt{n+1}-\sqrt{n}$$

$$f^{'}(n)=\frac{1}{2}\left[\frac{\sqrt{n}-\sqrt{n+1}}{\sqrt{n(n+1)}}\right]$$ We see that $f^{'}(n)<0$ for all $n$. This shows that $f$ is monotonically decreasing.

As a result for any pair $n_1$ and $n_2$ s.t. $n_1>n_2$ we have $f(n_2)>f(n_1)$

Now take $n_1=6$ and $n_2=5$. this leads to $f(5)>f(6)\Rightarrow \sqrt{6}-\sqrt{5}>\sqrt{7}-\sqrt{6}$

I hope this help now.

• Point 1 is irrelevant to the demonstration, point 2 just generalises the statement without indication of how the more general statement is easier to prove that the particular one. Nov 11 '13 at 9:03
• @Marc van Leeuwen you are completely right. It is only the idea why one at first look could think thar $2$ should hold in a very simple way. Please don't get me wrong. I don't say that $1$ implies $2$. It gives the idea of $2$ together with the fact that $\sqrt{n+1}-\sqrt{n}$ is a simple function. Nov 11 '13 at 9:11
• Sharing ideas leading to the solution is in my opinion more important than everything. For example: Hint.... How did you find that hint? What lead you to give such a hint? Only these information will teach the owner of the question how to think for the next time. Otherwise every person becomes a cipher to be decoded. Nov 11 '13 at 12:44
• @MarcvanLeeuwen see the edit. Nov 12 '13 at 23:01