+1833 (a) Give an example of two irrational numbers which, when added, produce a rational number.
Now let's consider just the addition of radicals.
(b) Suppose that a and b are positive integers such that both sqrt a and sqrt b are irrational. For what values of and is rational? Prove your answer.
(c) Again assuming a and b positive integers such that both sqrt a and sqrt b are irrational, for what values of and is rational? Prove your answer.\((a) Give an example of two irrational numbers which, when added, produce a rational number. Now let's consider just the addition of radicals. (b) Suppose that $a$ and $b$ are positive integers such that both $\sqrt{a}$ and $\sqrt{b}$ are irrational. For what values of $a$ and $b$ is $\sqrt{a} - \sqrt{b}$ rational? Prove your answer. (c) Again assuming $a$ and $b$ positive integers such that both $\sqrt{a}$ and $\sqrt{b}$ are irrational, for what values of $a$ and $b$ is $\sqrt{a} + \sqrt{b}$ rational? Prove your answer.\)
+118609 Hi Mellie is is great to see you on the forum again :))
Both guests have given great answers. Thanks guests.
I will assume that you are only talking about irrational numbers where the irational bit is a surd.
(a) Give an example of two irrational numbers which, when added, produce a rational number.
\(\sqrt5 \quad and \quad(-\sqrt5)\\ 3+\sqrt2 \quad and \quad 3-\sqrt2\\ \)
Any two numbers where the one has has the negative radical of the other one.
That is not very mathematically precise but I expect you know what I mean.
Now let's consider just the addition of radicals.
(b) Suppose that a and b are positive integers such that both sqrt a and sqrt b are irrational. For what values of and is rational? Prove your answer. None I think.....
(c) Again assuming a and b positive integers such that both sqrt a and sqrt b are irrational, for what values of and is rational? Prove your answer.
Lets see.
First you should fully understand this proof.
Prove sqrt(2) is irrational.
I am not yet sure how to expand upon this to prove what you need. ......
I am still thinking......
If you message CPhill, Alan, Heureka or Gino3141 they may be able to better help you.
+118609 Hi Mellie is is great to see you on the forum again :))
Both guests have given great answers. Thanks guests.
I will assume that you are only talking about irrational numbers where the irational bit is a surd.
(a) Give an example of two irrational numbers which, when added, produce a rational number.
\(\sqrt5 \quad and \quad(-\sqrt5)\\ 3+\sqrt2 \quad and \quad 3-\sqrt2\\ \)
Any two numbers where the one has has the negative radical of the other one.
That is not very mathematically precise but I expect you know what I mean.
Now let's consider just the addition of radicals.
(b) Suppose that a and b are positive integers such that both sqrt a and sqrt b are irrational. For what values of and is rational? Prove your answer. None I think.....
(c) Again assuming a and b positive integers such that both sqrt a and sqrt b are irrational, for what values of and is rational? Prove your answer.
Lets see.
First you should fully understand this proof.
Prove sqrt(2) is irrational.
I am not yet sure how to expand upon this to prove what you need. ......
I am still thinking......
If you message CPhill, Alan, Heureka or Gino3141 they may be able to better help you.
+118609 ok I have been away and had a think
a and b are positive integers but they are not square numbers
so sqrt(a) and sqrt(b) are irrational numbers.
When will \(\sqrt{a}+\sqrt{b}=Q\) where Q is a rational number.
To me the answer seems to be an obvious 'never' but here is a 'proof' by contradiction.
Lets assume that such a number Q exists then
\(\sqrt{a}+\sqrt{b}=Q\\ \sqrt{a}=Q-\sqrt{b}\\ \mbox{Square both sides}\\ a=Q^2+b-2Q\sqrt{b}\\ a-b=Q^2-2Q\sqrt{b}\\ \mbox{Now } R^2 \mbox{ is rational and } 2b\sqrt{b}\mbox{ is irational }\\ \mbox{A rational - an irrational = an irational}\\ \mbox{therefore a-b is irrational}\\ \mbox{but a and b are integers so a-b is rational}\\ \mbox{Contradiction!! }\\ \mbox{Therefore there are no values of a and b such that }\sqrt a +\sqrt{b} \mbox{ is equal to a rational number!}\)
