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Intuitively, what does the 1 mean in $\dfrac{a}{b} = \dfrac{1}{\frac{b}{a}}$?

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For intuition, I instantiate variables with fruits: $a$ as apple and $b$ as berry (pick whatever berry you like).

In terms of apples and/or berries, what does the red $\color{red}{1}$ below mean? What's the unit of $\color{red}{1}$?

$\dfrac{a}{b} \quad = \quad\dfrac{\color{red}{1}}{\dfrac{b}{a}}$.

Unmistakably, I know $\dfrac{a}{b}$ means $\dfrac{\text{apple}}{\text{1 berry}}$, and $b/a$ means $\dfrac{\text{berry}}{\text{1 apple}}$.

I ask only for intuition. Please pretermit formal arguments and proofs. E.g. please don't rely on rationalizing the denominator .

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2 comments

I have trouble to figure out what your intuition actually is. What is the unit of the denominator $2$ in "half a cake"? Anyway, the $1$ has to be dimensionless (having no unit) for the formula to work. It just means the number one. celtschk‭ 11 days ago

What is the division ring in your "intuitive" instantiation? Peter Taylor‭ 11 days ago

1 answer

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Your a and b are reversed in that equation because that equation corresponds to the tracks (0..1] and [1..) of this, that are symmetric.

The 1 / x plot is [at least] doubly symmetric. One of those two symmetries is y = x. That one's the symmetry axis you are asking about.

HTH.

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