Using fractions sounds great, I think fractions have a human quality and are a more natural way to approach numbers. In writing, I think putting down 0.5 is slightly jarring compared to the better ½, at least in non-technical contexts.
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0⅛¼⅜½⅝¾⅞1
So the measurement, when taken, is 1⅜".
I've seen inch rulers with different fractions, e.g. one end might be eights, the other sixteenths, turn it over and there are tenths.
(I don't know how this works with digital measuring devices — I grew up in Britain so have never used inches professionally. Only in Warhammer and TV screens.)
To you, it’s obvious and natural that „half” is 1/2.
To me, it looks off and I have to „convert” it in my mind to 0.5, which is the way I think about it.
That said, as soon as I need to do any arithmetic on the number (including compare it to other numbers) I convert it from fraction to decimal.