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I have an idea for demonstrating "why" based on Pythagoras but it needs a bit of fleshing out.

School maths isn't about proof.

I gather that it's typically some time in the second year of a university degree that mathematicians actually prove that 1+1=2.

I would say that in the first year of a Mathematics degree (in the UK) people will study algebraic structures like Rings and Fields at which point they would believe they can establish 1+1 =2 from quite elementary axioms.

It doesn't matter whether it was on the syllabus or not. It is an essential part of Mathematics but you balance this with keeping things interesting and not too frightening.

because ........ it cannot be rational

It is much quicker to show them that √2 is irrational

formulae for sums of series

My Pythagorean idea is to look at a segment of a circle with an inscribed (hypotenuse = r) and an escribed (adjacent = r) right-angled triangle.

But we have defined arc length s = πrθ,