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I favour the Liouvillian approach over the Cantorian approach, because although the diagonal argument will in principal principle let you write down the decimal expansion of a transcendental number, you will never see the whole expansion "all at once" (so to speak), and there will be nothing special about the truncated expansion you do see; it will just be some random decimal expansion, and any finite decimal expansion can be completed to be transcendental.

In the approach via Liouville, of course, you get to actually see the transcendental number.
And Liouville's argument is not so difficult; it boils down to the pigeonhole principle (which, by the sounds of what you've been teaching them, your 7th graders will have no trouble understanding, if they don't already know it).

I don't know an optimal reference, but if I was to pursue this path, I would fix my Liouville number first, and just focus on proving that that particular number is transcendental. (In other words, don't prove a general criterion and then check that your number satisfies; keep things more concrete by just directly proving the criterion for your chosen Liouville number.) I think that if you do this, and you just write down a putative polynomial equation with integer coeffs. that your Liouville number is supposed to satisfy, it won't be hard to argue your way to a contradiction. And because you have a concrete number, you can really work this through with your class, e.g. by beginning with a quadratic , actually plugging in your Liouville number, and staring at it and seeing why this couldn't give zero. I think this would make things quite intuitive and concrete.

Added: See Daniel Briggs's very nice answer for exactly this kind of explicit argument with a concrete Liouville number.

I favour the Liouvillian approach over the Cantorian approach, because although the diagonal argument will in principal let you write down the decimal expansion of a transcendental number, you will never see the whole expansion "all at once" (so to speak), and there will be nothing special about the truncated expansion you do see; it will just be some random decimal expansion, and any finite decimal expansion can be completed to be transcendental.

In the approach via Liouville, of course, you get to actually see the transcendental number.
And Liouville's argument is not so difficult; it boils down to the pigeonhole principle (which, by the sounds of what you've been teaching them, your 7th graders will have no trouble understanding, if they don't already know it).

I don't know an optimal reference, but if I was to pursue this path, I would fix my Liouville number first, and just focus on proving that that particular number is transcendental. (In other words, don't prove a general criterion and then check that your number satisfies; keep things more concrete by just directly proving the criterion for your chosen Liouville number.) I think that if you do this, and you just write down a putative polynomial equation with integer coeffs. that your Liouville number is supposed to satisfy, it won't be hard to argue your way to a contradiction. And because you have a concrete number, you can really work this through with your class, e.g. by beginning with a quadratic , actually plugging in your Liouville number, and staring at it and seeing why this couldn't give zero. I think this would make things quite intuitive and concrete.

Added: See Daniel Briggs's very nice answer for exactly this kind of explicit argument with a concrete Liouville number.

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I favour the Liouvillian approach over the Cantorian approach, because although the diagonal argument will in principal let you write down the decimal expansion of a transcendental number, you will never see the whole expansion "all at once" (so to speak), and there will be nothing special about the truncated expansion you do see; it will just be some random decimal expansion, and any finite decimal expansion can be completed to be transcendental.

In the approach via Liouville, of course, you get to actually see the transcendental number.
And Liouville's argument is not so difficult; it boils down to the pigeon hole principal pigeonhole principle (which, by the sounds of what you've been teaching them, your 7th graders will have no trouble understanding, if they don't already know it).

I don't know an optimal reference, but if I was to pursue this path, I would fix my Liouville number first, and just focus on proving that that particular number is transcendental. (In other words, don't prove a general criterion and then check that your number satisfies; keep things more concrete by just directly proving the criterion for you your chosen Liouville number.) I think that if you do this, and you just write down a putative polynomial equation with integer coeffs. that your Liouville number is supposed to satisfy, it won't be hard to argue your way to a contradiction. And because you have a concrete number, you can really work this through with your class, e.g. by beginning with a quadratic , actually plugging in your Liouville number, and staring at it and seeing why this couldn't give zero. I think this would make things quite intuitive and concrete.

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