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So, we represent numbers usually in a form of a sequence of digits where each one of them multiplies the power of a base:

$13.2 = 1 * 10^1 + 3 * 10^0 + 2 * 10^{-1}$

So that much is clear, perfectly. But what interests me is the "symmetry" between the left and right of the radix point which separates the integer and fractional part. Specifically, the "significant digits" or "where $0$s matter" to put it blatantly clear:

Clear example:

$00000050.02000000$ -> $50.02$

What is curious to me is the fact that after the radix point after the last non-zero digit, zeroes do not matter whereas on the left side it is the zeroes before the first non-zero digit that do not matter.

Does this symmetry come simply from the fact that the ratio does not change on the fractional side:

$2/100 = 20 / 1000 = 200/10000 $ etc. Is this the reason the added digits on the right do not matter?

I know it's trivial, but it kind of captured my attention since I like little details and they make me restless. Thank you for trying to assist in advance.

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    Thank you, J.M. That was the gist of it. :D2011-12-14

2 Answers 2

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Except for typographic conventions, the rule is the same on both sides of the point: A zero matters if and only if it comes between a nonzero digit and the decimal point.

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Zero digits do not count, ever. A zero digit in position $k$ corresponds to a term $0\cdot 10^k$. For instance, $502.03= 5 \cdot 10^2 + 0 \cdot 10^1 + 2 \cdot 10^0 + 0 \cdot 10^{-1} + 3 \cdot 10^{-2}$. There is no deeper reason.

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    @InterestedStudent, it's not the zero digits that count in your example, it's the *position* of the non-zero digits. See Jonas's first comment above.2011-12-14