express an irrational as the sum of a rational and irrational number
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 19 hours ago
Klangen
1,65411334
asked Nov 3 '15 at 9:56
djna
1013
add a comment |
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 19 hours ago
Klangen
1,65411334
asked Nov 3 '15 at 9:56
djna
1013
2
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.
– lulu
Nov 3 '15 at 10:02
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.
– djna
Nov 3 '15 at 11:59
2
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.
– lulu
Nov 3 '15 at 12:11
add a comment |
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 19 hours ago
Klangen
1,65411334
asked Nov 3 '15 at 9:56
djna
1013
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 19 hours ago
Klangen
1,65411334
asked Nov 3 '15 at 9:56
djna
1013
edited 19 hours ago
Klangen
1,65411334
asked Nov 3 '15 at 9:56
djna
1013
edited 19 hours ago
Klangen
1,65411334
edited 19 hours ago
Klangen
1,65411334
edited 19 hours ago
Klangen
1,65411334
1,65411334
asked Nov 3 '15 at 9:56
djna
1013
asked Nov 3 '15 at 9:56
djna
1013
asked Nov 3 '15 at 9:56
djna
1013
1013
2
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.
– lulu
Nov 3 '15 at 10:02
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.
– djna
Nov 3 '15 at 11:59
2
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.
– lulu
Nov 3 '15 at 12:11
add a comment |
2
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.
– lulu
Nov 3 '15 at 10:02
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.
– djna
Nov 3 '15 at 11:59
2
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.
– lulu
Nov 3 '15 at 12:11
2
2
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.
– lulu
Nov 3 '15 at 10:02
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.
– lulu
Nov 3 '15 at 10:02
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.
– djna
Nov 3 '15 at 11:59
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.
– djna
Nov 3 '15 at 11:59
2
2
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.
– lulu
Nov 3 '15 at 12:11
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.
– lulu
Nov 3 '15 at 12:11
add a comment |
1 Answer
1
active
oldest
votes
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 20 hours ago
Klangen
1,65411334
add a comment |
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1 Answer
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oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
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 20 hours ago
Klangen
1,65411334
add a comment |
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 20 hours ago
Klangen
1,65411334
add a comment |
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 20 hours ago
Klangen
1,65411334
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 20 hours ago
Klangen
1,65411334
answered 20 hours ago
Klangen
1,65411334
answered 20 hours ago
Klangen
1,65411334
answered 20 hours ago
Klangen
1,65411334
1,65411334
add a comment |
add a comment |
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2
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.
– lulu
Nov 3 '15 at 10:02
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.
– djna
Nov 3 '15 at 11:59
2
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.
– lulu
Nov 3 '15 at 12:11