Can someone explain to me what it means for floating point numbers to have "uneven" spacing on a number line? As I visualize this, all I see is evenly spacing out floating point numbers.
When I google for an explanation, most of them just say "floating point numbers have uneven spacing", but none of them explain why or how?
When I look at a ruler for example, everything is evenly spaced out so where does the "uneven" spacing thing come in and exactly mean visually?
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Floating point numbers are represented in the form $A \cdot 2^B$, where $A$ and $B$ are both integers. This means that the difference between a floating point number and the "next" (the one you get increasing $A$ by $1$) is $2^B$; thus the smaller $B$, the closer the spacing.
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Since there is a constant number of mantissa bits, you have the same number of floating point numbers as in the interval from $1$ to $2$ also in the interval from $2$ to $4$ and from $4$ to $8$, and generally in any interval $[2^n, 2^{n+1})$. As the interval length increases, so does the spacing between the floating point numbers.
The uneven spacing comes from the exponent. There are a fixed number of bits in the mantissa for a given storage format. For example, in IEEE $64$ bit storage there are $52$ bits for the mantissa, which says the LSB is $2^{-51}$ times the MSB. If the exponent is zero (after allowing for the offset) the difference between two neighboring floats will be $2^{-51}$. If the exponent is $20$, the difference between two neighboring floats will be $2^{-31}$ because the mantissa is multiplied by $2^{20}$. They are (approximately) separated by the same fractional amount, but by different absolute amounts.