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**Mathematical Derivation of Significant Figure "Rules"**

Let's say I have a scale which can accurately read weights out to ten-thousandths of a gram so it might read 1.0005 g or 0.0005 grams ... why is it that the first reading has 5 significant figures and the second has only 1? Same instrument ... so how is it less precise just because the item weighs less? If I want to add two measurements together from the same scale ... say 0.0056 and 1.2345 --- why do I have to make it a two digit number? Why does that cause me to lose precision?

Please do not tell me how to apply the "rules" for significant figures -- I can read the tables in my chemistry/physics books just fine. I am asking how the rule is derived ... everywhere I've asked, I've had people saying "well leading zeroes aren't significant" ... I know this, and like a good monkey can apply the rules without a problem -- but I want to know why they aren't "significant".

I want an explanation centered around arithmetic of numbers in the decimal representation system -- something explaining why precision is lost because of leading zeros ... this is why I put it into the math section ... I figure this is more of a number theoretic question than anything else...

Thanks.