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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    $\begingroup$ I don't really know what is being asked, but the only reason the $0$s between nonzero digits matter is to show the place of the nonzero digits. $50.02$ could also be written as $5\cdot 10^1+2\cdot 10^{-2}$. The radix point itself is only important for showing the place of the nonzero digits, and is somewhat arbitrary. If you use "scientific notation" then there is little difference between representing $50.02$ vs. $0.000005002$ as far as what $0$s matter; all that matters is what comes between the first and last nonzero digit (if there is a last) in writing $5.002\times 10^k$. $\endgroup$ – Jonas Meyer Dec 14 '11 at 1:47
  • $\begingroup$ I understand the scientific notation, but it is just curious to me how the significant zeroes (those that increase or decrease exponents of non-zero digits in order to do the same as the scientific notation) are placed in this notation, like a mirror-symmetry. On the left side, the leftmost do not matter, on the right, the rightmost. That's what's curious to me and that is the end part of my question. If 0.6 is indeed the same as 0.600 is it because 6/10 == 600/1000? $\endgroup$ – InterestedStudent Dec 14 '11 at 1:52
  • $\begingroup$ ...i.e., the beauty of that neat digit the Indians invented is that it is a convenient placeholder... $\endgroup$ – J. M. isn't a mathematician Dec 14 '11 at 1:53
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    $\begingroup$ I still do not understand what is being asked. The symmetry is that adding $0$ doesn't change the number, no matter how big or small a power of $10$ you multiply it by first. $\endgroup$ – Jonas Meyer Dec 14 '11 at 1:55
  • $\begingroup$ Well, with $5000.03=5\times10^3+3\times10^{-2}$, I'm not sure I see any symmetry. The $3$ and $-2$ exponents indicate which of the digits in the infinite array of zeroes $\dots0000.00\dots$ should be replaced, and then one chops off unneeded zeroes accordingly. $\endgroup$ – J. M. isn't a mathematician Dec 14 '11 at 1:55

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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  • $\begingroup$ Zero digits are significant in this notation because: $503.03 $ is not $ 53.3$ But leftmost of the left side and rightmost of the right side, they do not. $\endgroup$ – InterestedStudent Dec 14 '11 at 1:56
  • $\begingroup$ @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. $\endgroup$ – lhf Dec 14 '11 at 1:58

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