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It is given in the book by Malcolm Cameron, titled: Mathematics the Truth; that the circle $x^2+y^2=5$ has an infinite number of points with at least one coordinate irrational, but also an infinite number of rational coordinates too.

I cannot understand. Starting with the latter part as only $(\pm 1, \pm 2)$ seem to be only the $4$ points with rational coordinates; rather than an infinite number of rational coordinates.

Similarly for the irrational coordinates, as not clear how an infinite number of such (at least one) coordinates can be there.
Any such points have to anyway satisfy $\sqrt{a^2+b^2}=5, \ a,b\in \mathbb{R}$.

So, how to prove this statement.

I request at least suitable hints in the direction to pursue.

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jiten
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    Hint: make a line of rational slope passing through known rational point $(1,2)$. Similarly irrationals. – user10354138 Jul 24 '20 at 09:41
  • @user10354138 Thanks a lot. Please make an answer. Requesting to get more inputs as well as to avoid errors. – jiten Jul 24 '20 at 09:44
  • A strange question ... there infinitely many points on a circle, so how can you think that there would be finite points with rational and finite points with irrational coordinates? What are the coordinates of the other points then? :D – LL 3.14 Jul 24 '20 at 09:48

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Consider the point $P=(2,1)$ on the circle. Take any line through $P$ with rational gradient: $$y-1=m(x-2)$$ with $m\in\Bbb Q$. This meets the circle in two points (unless it's the tangent at $P$), namely $P$ and $Q$ say. Then $Q$ also has rational coordinates.

As an example take $m=2$. Then the line is $y=2x-3$. Substituting in $x^2+y^2=5$ gives $$x^2+(2x-3)^2=5x^2-12x+9=5.$$ This quadratic has roots $x=2$ and $x=2/5$ leading to $P=(2,1)$ and $Q=(2/5,-11/5)$.

Angina Seng
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  • Thanks, but it is not clear how it proves that there are an infinite number of at least one irrational coordinates. – jiten Jul 24 '20 at 09:47
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    That's easy: pick $x$ to be any irrational between $\pm\sqrt5$ and let $y=\sqrt{5-x^2}$. @jiten – Angina Seng Jul 24 '20 at 09:48