Why do we teach floating point as representing numbers in the form “a × 2^b” instead of the easier-to-understand “a/b”, where b is a power of two (i.e. dyadic rationals)? I know they’re the same thing, but rational numbers are a lot easier to intuitively grasp, at least for me…
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I think the scientific notation analogy is helpful though, it explains the existence of the implicit 1 at the top of the mantissa. In scientific notation you can't have 0 on the left side of the point. So in base 2, you must have a 1 there. It would be redundant to write it.
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Which gives no understanding about the actual numbers floats can represent!