Alan

Irrational

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4:5 is a ratio that can also be written as 4/5 or, as a decimal, 0.8

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How about (44 - 4)/4?

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I don't think a graphical solution is "cheating" - it seems perfectly valid to me.  However, if you want a non-graphical method then here is a possibility:

(That last d should be a δ, of course!)

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The gaps in the following graph show that x.floor(x) = M has a solution for some values of M, but not others:

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In general:  x^m*xⁿ = x^m+n

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See if this helps:

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One of these?

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Perhaps 2: an inside and an outside!

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If the first is meant to be 11*5 = 55 and the second is 0.0011*5 = 0.55 then the second is wrong!  It should be 0.0011*5 = 0.0055

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