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What is the smallest number that can be stored in a 32 bit fixed point system of representation if the radix point is assumed to be in the middle?

My guess would be that 0000000000000000.0000000000000001$_2$ is the smallest number that could be represented which is 0.0000152587890625 in decimal, however I'm not sure if it is correct, as any negative number is smaller than a fraction of a positive number.

I'm a bit confused about how the fixed point system represents negative numbers (2's complement or sign magnitude) and if that would affect the smallest number that can be represented by the 32 bit fixed point system.

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  • $\begingroup$ Well, the smallest number will most likely be something of the form $.0\ldots01_2$, in binary, rather than in decimal. You have to check whether your integers are signed or unsigned to give a definitive answer. $\endgroup$ Jun 15, 2021 at 20:28
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    $\begingroup$ If your numbers can be negative, you need a sign bit, which only leaves 31 bits for the magnitude. So "in the middle" is undefined, because 31 is odd. Also, there is no universally agreed format for fixed-point numbers (see here), so it's basically up to you to decide on the format. $\endgroup$
    – TonyK
    Jun 15, 2021 at 20:42

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That seems to be the smallest non-zero number.

Of course, $0$ is smaller, but that's self-evident.

Usually we say that something like $-1,234,567,890$ is a large negative number rather than a small number. Something like $-0.000000001$ is small and negative, while $0.000000001$ is small and positive.

All of that to say I think you've met the intent of the question. The "radix in the middle" tells you how many bits are allocated to each side of the decimal. As to "two's complement" and the like, those don't really affect the precision of what you store in the number.

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  • $\begingroup$ Great point about the smallest negative number compared to a large negative number versus the smallest positive number. I didn't really think about that originally! Thank you so much $\endgroup$ Jun 15, 2021 at 20:45

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