express an irrational as the sum of a rational and irrational number
$begingroup$
Simple question, apologies. This is from some sample high school math questions, target is age 16 pupils. I don't think any great sophistication is expected.
$$
P + Q = sqrt {5}.
$$
$P$ is a rational number and $Q$ is an irrational number. Give possible values of $P$ and $Q$.
I can think of the trivial
$$
P = 0,
Q = sqrt {5}.
$$
I'm not even sure that the pupil would be expected to come up with
$$
P = 2, Q = (sqrt {5} - 2).
$$
I don't think $Q$ could then be simplified to another named irrational. Any ideas what an expected answer might be?
irrational-numbers
edited Jan 3 at 10:07
Klangen
1,69311334
asked Nov 3 '15 at 9:56
djnadjna
1013
$endgroup$
add a comment |
$begingroup$
Simple question, apologies. This is from some sample high school math questions, target is age 16 pupils. I don't think any great sophistication is expected.
$$
P + Q = sqrt {5}.
$$
$P$ is a rational number and $Q$ is an irrational number. Give possible values of $P$ and $Q$.
I can think of the trivial
$$
P = 0,
Q = sqrt {5}.
$$
I'm not even sure that the pupil would be expected to come up with
$$
P = 2, Q = (sqrt {5} - 2).
$$
I don't think $Q$ could then be simplified to another named irrational. Any ideas what an expected answer might be?
irrational-numbers
edited Jan 3 at 10:07
Klangen
1,69311334
asked Nov 3 '15 at 9:56
djnadjna
1013
$endgroup$
2
$begingroup$
Generalizing your own example, $P$ could be any rational! If, say, $P=frac mn$ then $Q=sqrt 5 - frac mn$. One should argue that such a $Q$ is necessarily irrational, but this is straightforward.
$endgroup$
– lulu
Nov 3 '15 at 10:02
$begingroup$
Just so. My puzzle is what the target student was expected to give as an answer. I wondered if there was some value of P that can yield a simpler value of Q.
$endgroup$
– djna
Nov 3 '15 at 11:59
2
$begingroup$
Well, I'd have thought that what I wrote was the intended answer. For what it's worth $-1+2phi=sqrt 5$ where $phi$ is the Golden Ratio. But I don't think that simplifying $Q$ was the point of the exercise.
$endgroup$
– lulu
Nov 3 '15 at 12:11
add a comment |
$begingroup$
Simple question, apologies. This is from some sample high school math questions, target is age 16 pupils. I don't think any great sophistication is expected.
$$
P + Q = sqrt {5}.
$$
$P$ is a rational number and $Q$ is an irrational number. Give possible values of $P$ and $Q$.
I can think of the trivial
$$
P = 0,
Q = sqrt {5}.
$$
I'm not even sure that the pupil would be expected to come up with
$$
P = 2, Q = (sqrt {5} - 2).
$$
I don't think $Q$ could then be simplified to another named irrational. Any ideas what an expected answer might be?
irrational-numbers
edited Jan 3 at 10:07
Klangen
1,69311334
asked Nov 3 '15 at 9:56
djnadjna
1013
$endgroup$
Simple question, apologies. This is from some sample high school math questions, target is age 16 pupils. I don't think any great sophistication is expected.
$$
P + Q = sqrt {5}.
$$
$P$ is a rational number and $Q$ is an irrational number. Give possible values of $P$ and $Q$.
I can think of the trivial
$$
P = 0,
Q = sqrt {5}.
$$
I'm not even sure that the pupil would be expected to come up with
$$
P = 2, Q = (sqrt {5} - 2).
$$
I don't think $Q$ could then be simplified to another named irrational. Any ideas what an expected answer might be?
irrational-numbers
irrational-numbers
edited Jan 3 at 10:07
Klangen
1,69311334
asked Nov 3 '15 at 9:56
djnadjna
1013
edited Jan 3 at 10:07
Klangen
1,69311334
asked Nov 3 '15 at 9:56
djnadjna
1013
edited Jan 3 at 10:07
Klangen
1,69311334
edited Jan 3 at 10:07
Klangen
1,69311334
edited Jan 3 at 10:07
Klangen
1,69311334
1,69311334
asked Nov 3 '15 at 9:56
djnadjna
1013
asked Nov 3 '15 at 9:56
djnadjna
1013
asked Nov 3 '15 at 9:56
djnadjna
1013
1013
2
$begingroup$
Generalizing your own example, $P$ could be any rational! If, say, $P=frac mn$ then $Q=sqrt 5 - frac mn$. One should argue that such a $Q$ is necessarily irrational, but this is straightforward.
$endgroup$
– lulu
Nov 3 '15 at 10:02
$begingroup$
Just so. My puzzle is what the target student was expected to give as an answer. I wondered if there was some value of P that can yield a simpler value of Q.
$endgroup$
– djna
Nov 3 '15 at 11:59
2
$begingroup$
Well, I'd have thought that what I wrote was the intended answer. For what it's worth $-1+2phi=sqrt 5$ where $phi$ is the Golden Ratio. But I don't think that simplifying $Q$ was the point of the exercise.
$endgroup$
– lulu
Nov 3 '15 at 12:11
add a comment |
2
$begingroup$
Generalizing your own example, $P$ could be any rational! If, say, $P=frac mn$ then $Q=sqrt 5 - frac mn$. One should argue that such a $Q$ is necessarily irrational, but this is straightforward.
$endgroup$
– lulu
Nov 3 '15 at 10:02
$begingroup$
Just so. My puzzle is what the target student was expected to give as an answer. I wondered if there was some value of P that can yield a simpler value of Q.
$endgroup$
– djna
Nov 3 '15 at 11:59
2
$begingroup$
Well, I'd have thought that what I wrote was the intended answer. For what it's worth $-1+2phi=sqrt 5$ where $phi$ is the Golden Ratio. But I don't think that simplifying $Q$ was the point of the exercise.
$endgroup$
– lulu
Nov 3 '15 at 12:11
2
2
$begingroup$
Generalizing your own example, $P$ could be any rational! If, say, $P=frac mn$ then $Q=sqrt 5 - frac mn$. One should argue that such a $Q$ is necessarily irrational, but this is straightforward.
$endgroup$
– lulu
Nov 3 '15 at 10:02
$begingroup$
Generalizing your own example, $P$ could be any rational! If, say, $P=frac mn$ then $Q=sqrt 5 - frac mn$. One should argue that such a $Q$ is necessarily irrational, but this is straightforward.
$endgroup$
– lulu
Nov 3 '15 at 10:02
$begingroup$
Just so. My puzzle is what the target student was expected to give as an answer. I wondered if there was some value of P that can yield a simpler value of Q.
$endgroup$
– djna
Nov 3 '15 at 11:59
$begingroup$
Just so. My puzzle is what the target student was expected to give as an answer. I wondered if there was some value of P that can yield a simpler value of Q.
$endgroup$
– djna
Nov 3 '15 at 11:59
2
2
$begingroup$
Well, I'd have thought that what I wrote was the intended answer. For what it's worth $-1+2phi=sqrt 5$ where $phi$ is the Golden Ratio. But I don't think that simplifying $Q$ was the point of the exercise.
$endgroup$
– lulu
Nov 3 '15 at 12:11
$begingroup$
Well, I'd have thought that what I wrote was the intended answer. For what it's worth $-1+2phi=sqrt 5$ where $phi$ is the Golden Ratio. But I don't think that simplifying $Q$ was the point of the exercise.
$endgroup$
– lulu
Nov 3 '15 at 12:11
add a comment |
1 Answer
1
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oldest
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$begingroup$
Let $a,binmathbb{Q}$ with $bneq 0$ and let $P=frac{a}{b}$. Then we have $Q=sqrt 5 - P$ such that
$$
P+Q=sqrt 5.
$$
answered Jan 3 at 9:42
KlangenKlangen
1,69311334
$endgroup$
add a comment |
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1 Answer
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$begingroup$
Let $a,binmathbb{Q}$ with $bneq 0$ and let $P=frac{a}{b}$. Then we have $Q=sqrt 5 - P$ such that
$$
P+Q=sqrt 5.
$$
answered Jan 3 at 9:42
KlangenKlangen
1,69311334
$endgroup$
add a comment |
$begingroup$
Let $a,binmathbb{Q}$ with $bneq 0$ and let $P=frac{a}{b}$. Then we have $Q=sqrt 5 - P$ such that
$$
P+Q=sqrt 5.
$$
answered Jan 3 at 9:42
KlangenKlangen
1,69311334
$endgroup$
add a comment |
$begingroup$
Let $a,binmathbb{Q}$ with $bneq 0$ and let $P=frac{a}{b}$. Then we have $Q=sqrt 5 - P$ such that
$$
P+Q=sqrt 5.
$$
answered Jan 3 at 9:42
KlangenKlangen
1,69311334
$endgroup$
Let $a,binmathbb{Q}$ with $bneq 0$ and let $P=frac{a}{b}$. Then we have $Q=sqrt 5 - P$ such that
$$
P+Q=sqrt 5.
$$
answered Jan 3 at 9:42
KlangenKlangen
1,69311334
answered Jan 3 at 9:42
KlangenKlangen
1,69311334
answered Jan 3 at 9:42
KlangenKlangen
1,69311334
answered Jan 3 at 9:42
KlangenKlangen
1,69311334
1,69311334
add a comment |
add a comment |
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2
$begingroup$
Generalizing your own example, $P$ could be any rational! If, say, $P=frac mn$ then $Q=sqrt 5 - frac mn$. One should argue that such a $Q$ is necessarily irrational, but this is straightforward.
$endgroup$
– lulu
Nov 3 '15 at 10:02
$begingroup$
Just so. My puzzle is what the target student was expected to give as an answer. I wondered if there was some value of P that can yield a simpler value of Q.
$endgroup$
– djna
Nov 3 '15 at 11:59
2
$begingroup$
Well, I'd have thought that what I wrote was the intended answer. For what it's worth $-1+2phi=sqrt 5$ where $phi$ is the Golden Ratio. But I don't think that simplifying $Q$ was the point of the exercise.
$endgroup$
– lulu
Nov 3 '15 at 12:11