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Albert1
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compare :
$7$, and $\sqrt 2+\sqrt 5 + \sqrt {11}$
which one is bigger?
$7$, and $\sqrt 2+\sqrt 5 + \sqrt {11}$
which one is bigger?
now ,try to prove it,without using calculatorgreg1313 said:Approximating the roots to 3 significant digits, we have$$\sqrt2+\sqrt5+\sqrt{11}\approx1.41+2.23+3.31=6.95,6.95+3\cdot0.009=6.977\lt7$$
Albert said:now ,try to prove it,without using calculator
you don't have to compute the square roots of those numbers ,use some tricksMarkFL said:It is fairly easy to compute square roots by hand to two decimal places...[cs=Spiteful]spiteful.gif[/cs]
very smart!anemone said:I am going to use a trick someone taught me from this forum to solve for this challenge:
Note that
$288<289\,\,\implies2(12^2)<17^2$ or $(\sqrt{2}<\dfrac{17}{12})---(1)$
$80<81\,\,\implies5(4^2)<9^2$ or $(\sqrt{5}<\dfrac{9}{4})---(2)$
$99<100\,\,\implies11(3^2)<10^2$ or $(\sqrt{11}<\dfrac{10}{3})---(3)$
Adding the inequalities in (1), (2) and (3) up gives us the answer:
$\sqrt{2}+\sqrt{5}+\sqrt{11}<\dfrac{17}{12}+\dfrac{9}{4}+\dfrac{10}{3}=7$
The relationship between $7$ and $\sqrt{2}+\sqrt{5}+\sqrt{11}$ is that they are both numbers. $7$ is a rational number, while $\sqrt{2}+\sqrt{5}+\sqrt{11}$ is an irrational number. This means that $7$ can be expressed as a finite decimal or fraction, while $\sqrt{2}+\sqrt{5}+\sqrt{11}$ is a non-repeating, non-terminating decimal.
Technically, $7$ is bigger because it is a whole number and $\sqrt{2}+\sqrt{5}+\sqrt{11}$ is a decimal. However, in terms of magnitude, $\sqrt{2}+\sqrt{5}+\sqrt{11}$ is bigger, as it is approximately equal to $7.48$.
One way to compare $7$ and $\sqrt{2}+\sqrt{5}+\sqrt{11}$ is to convert both numbers to decimals and see which one is larger. Another way is to use a calculator to find the exact value of $\sqrt{2}+\sqrt{5}+\sqrt{11}$ and compare it to $7$.
No, $7$ and $\sqrt{2}+\sqrt{5}+\sqrt{11}$ cannot be equal. $7$ is a rational number with a finite number of decimal places, while $\sqrt{2}+\sqrt{5}+\sqrt{11}$ is an irrational number with an infinite number of decimal places. Therefore, they can never be equal.
Comparing $7$ and $\sqrt{2}+\sqrt{5}+\sqrt{11}$ can be useful in any situation where we need to understand the relationship between different types of numbers. It can also be useful for understanding the concept of irrational numbers and their magnitude compared to rational numbers. Additionally, it can be helpful in certain mathematical problems or equations where both $7$ and $\sqrt{2}+\sqrt{5}+\sqrt{11}$ are involved.