Without evaluating the square root or using the calculator , show that $\displaystyle \sqrt{18}+\sqrt{3}$ is less than 6 .
Thanks ,,
Hi thereddevils,
$\displaystyle \sqrt{18}+\sqrt{3} < 6$
Subtract $\displaystyle \sqrt{3}$ from both sides.
$\displaystyle \sqrt{18}<6-\sqrt{3}$
Square both sides.
$\displaystyle 18<36-12\sqrt{3}+3$
$\displaystyle -21<-12\sqrt{3}$
Square both sides again.
$\displaystyle 441>432$
Hello, masters!
Sorry . . . you made a classic error.
(I'm familiar with it . . . I've made it enough times.)
The error is perhaps the most insidious one.. . $\displaystyle \sqrt{18}+\sqrt{3} \:<\: 6$
Subtract $\displaystyle \sqrt{3}$ from both sides:
. . $\displaystyle \sqrt{18}\:<\:6-\sqrt{3}$
Square both sides:
. . $\displaystyle 18\:<\:36-12\sqrt{3}+3$
. . $\displaystyle -21\:<\:-12\sqrt{3}$
Square both sides again. . ← Here!
. . $\displaystyle 441\:<\:432$ . . . . This is not true.
When you squared, you multiplied the left side by $\displaystyle -21$,
. . the right by $\displaystyle -12\sqrt{3}$
And you know what happens . . .
$\displaystyle
\left( {\sqrt {18} + \sqrt 3 < 6} \right) = \left( {\sqrt {2 \times 9} + \sqrt 3 < 6} \right) =
$
$\displaystyle
= \left( {3\sqrt 2 + \sqrt 3 < 6} \right) = \left( {3(\sqrt 2 + \frac{{\sqrt 3 }}
{3}) < 6} \right) =
$
$\displaystyle
= \left( {(\sqrt 2 + \frac{{\sqrt 3 }}
{3}) < 2} \right) = \left( {(\sqrt 2 + \frac{1}
{{\sqrt 3 }}) < 2} \right) =
$
$\displaystyle
= \left( {\frac{1}
{{\sqrt 3 }} < 2 - \sqrt 2 } \right) = \left( {\frac{1}
{{\sqrt 3 }} < 2(1 - \frac{{\sqrt 2 }}
{2})} \right)
$
Dividing by 2 and squaring both sides:
$\displaystyle
= \left( {\frac{1}
{{2\sqrt 3 }} < (1 - \frac{1}
{{\sqrt 2 }})} \right) = \left( {\frac{1}
{{12}} < (\frac{3}
{2} - \sqrt 2 )} \right) =
$
$\displaystyle
- \frac{{17}}
{{12}} < - \sqrt 2 ,\text{ - }\frac{{289}}
{{144}} < - 2,\text{ }\frac{{289}}
{{144}} > 2\text{ }
$
I explained the step before the last, which is not that obvious imo. Anything else needing explaining please let me know. I omitted the explanation of steps on purpose as I find that we learn better when we struggle to work out things ourselves. However, your solution is the best of all and that's the one that should be taken into account, again imho.
The last two steps need explanation, if we ignore the fact that at no point to you explain what you are attempting to do.
You are mis-using equality signs, if you must do it this way you should be using bi-implication "$\displaystyle \Leftrightarrow $"
If you handed this in to me for marking I don't think you would get more than 10% of the available marks for it as it is.
CB
You could use approximation!
4.2426^2>18 and 1.75^2>3
4.2426+1.75<6
there fore $\displaystyle \sqrt{18}+\sqrt{3}<6$
p.s. I assume that you know how to find approximate roots by averaging. I had a whole presentation on maple, but MHF will not let me upload. Maybe this'll help
p.s.s I only had to repeat twice for the approximation of root18 and once for root3. It only took a second!
p.s.s.s. If you would like to see my maple presentation of this concept, send me a pm and I'll send it to you.
[edit] Babylonian method
Graph charting the use of the Babylonian method for approximating the square root of 100 (10) using start values x0=50, x0=1, and x0=-5. Note that using a negative start value yields the negative root.
Perhaps the first algorithm used for approximating is known as the "Babylonian method", named after the Babylonians,[1] or "Heron's method", named after the first-century Greek mathematician Heron of Alexandria who gave the first explicit description of the method.[2] It can be derived from (but predates) Newton's method. This is a quadratically convergent algorithm, which means that the number of correct digits of the approximation roughly doubles with each iteration. It proceeds as follows:It can also be represented as:
- Start with an arbitrary positive start value x0 (the closer to the root, the better).
- Let xn+1 be the average of xn and S / xn (using the arithmetic mean to approximate the geometric mean).
- Repeat steps 2 and 3, until the desired accuracy is achieved.
This algorithm works equally well in the p-adic numbers, but cannot be used to identify real square roots with p-adic square roots; it is easy, for example, to construct a sequence of rational numbers by this method which converges to + 3 in the reals, but to − 3 in the 2-adics.