1.5 Workshop 5 (Reflection)

In this workshop, you’ll receive feedback from your tutor on the proof you submitted for this block. Your tutor will give your group general feedback on the key areas for improvement and commendation that they noticed whilst marking your group’s proofs and they will also have a one-to-one conversation with you about your specific feedback. It is important that you understand the written feedback your tutor has given you to improve, especially if your proof has not been awarded a pass yet. So make sure you take this opportunity to ask about any part of the feedback you’re unsure of.

When you’re not discussing feedback with your tutor, work with your group on the summary task below. This task has been chosen to draw on a variety of concepts you’ve been exploring throughout this block and is deliberately open ended to give you more practice at forming conjectures of your own.

Task 1.21.

Summary Task

  1. 1.
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    For which positive integers nn is n\sqrt{n} rational?

  2. 2.
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    For which positive integers mm and nn is n1/mn^{1/m} rational?

EXTENSION: For which x,y∈ℕx,y\in\mathbb{N} do we have x,y∉ℚ\sqrt{x},\sqrt{y}\notin\mathbb{Q} and x⋅y∈ℚ\sqrt{x}\cdot\sqrt{y}\in\mathbb{Q}?

Prove any conjectures you make.

Questions to help you think

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    Which definitions are involved? Do you understand them?

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    Are there any specific examples that you can consider first?

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    Are there any relevant results you’ve learnt in this block or previous blocks that may help you?