Significant Numbers

As per https://en.wikipedia.org/wiki/Significant_figures it states:

In a number with a decimal point, trailing zeros (those to the right of the last non-zero digit) are significant.

I am not able to understand why leading in such case is also not important? Ex.

$0.0000340000$

Why the zeros before $3$ are not significant but after $4$ are - I can write the same number as $34\times10^{-6}$ (2 significant numbers) so why I need to state the number as $340000\times10^{-10}$ (6 significant numbers)?

Also what's the number of significant digits are for below number -

$00012000.000012000$

Is there a generic to determine significant numbers?

• $00012000.000012000$ is written to 14 digits of significance. You count every digit from the first non-zero digit in the number to the end, here it is from the first $1$ to the final zero: $000\color{red}12000.00001200\color{red}0$ – lioness99a Jul 13 '17 at 8:00
• Then if we assume the number without decimal 00012000000012000 how many significant numbers are there? – Programmer Jul 13 '17 at 8:03
• Still 14 digits – lioness99a Jul 13 '17 at 8:12
• Leading zeroes to the right of the decimal place are significant because they cannot be omitted: $0.000034 \ne 0.34$. Howver, there is another nuance to "significant digits": For example a recent quack study reported values of glyphosate found in breakfast cereals such as "$53.478$ ppb " (parts per $10^9$) when the test method had inherent errors of at least $\pm 80$ ppb , which means none of the data was of any significance, in any sense of the word. – DanielWainfleet Jul 13 '17 at 13:47

If we write a number as $0.0000340000$ (to $6$ significant digits), then we are saying that the true value lies in the range $$0.00003399995\leq n<0.00003400005$$

However, if we write it as $0.000034$ (to $2$ significant digits) then we are saying the true value lies in the range $$0.0000335\leq n<0.0000345$$

The higher the number of significant digits, the smaller we have reduced the range of the true value to be in

Hopefully now, you can see that the leading zeros make no difference to this range, so we don't bother counting them. We also note that you can add leading zeros indefintely without changing the value of the number:

$$12 = 012 = 0000012 = 000000012$$

Therefore, we can also see that the generic formula for calculating significant digits is to count the number of digits which occur after (and including) the first non-zero digit in the number, all the way to the last digit, regardless of whether it is a zero or not

It is a matter that concerns approximated quantities and the way to represent them with rational numbers, which being numbers are exact quantities. In this respect rational numbers are associated with a precision scale, through which one can ask what is the next or preceeding rational number of a given rational number in the given precision scale. That is, rational numbers with an associated precision scale is a subset of rational numbers in a one-to-one corrispondence with integers, where "the successive of a rational" makes sense and the difference between to consecutive rationals is not constant but is prescribed by the resolution of the precision scale.

When you say that a quantity has an approximated value given by a specific rational number without specifying the precision scale it is intended that the resolution is $10^n$, where $n$ is the position of the last digit of the decimal representation of the given rational number. That is, you are implicitly saying the true value is in an interval centered at that rational number and whose endpoints are halfway between the given rational number and the next and preceeding rational number in the implicit precision scale $[x-0.5 \cdot 10^n, x+0.5 \cdot 10^n[$.

You can see now how different are $3.14$ and $3.14000$ when they represent intervals in the model just said.

Now as to the significant digits the question is as follows.

In the implicit precision scale of $3.14000$ you can find also numbers like $3.14159$ (because in the range $[0,10[$ the resolution is $0.5\cdot 10^{-5}$), $314159$ (because in the range $[10^5,10^6[$ the resolution is $0.5\cdot 10^{-5+5}$), $3.14159\cdot10^{-123456}$ (because in the range $[10^{-123456},10^{-1213456+1}[$ the resolution is $0.5\cdot 10^{-5+123456}$), but you cannot find $3.141592$ (because that would mean that the resolution is $10^{-6}$, but that is impossible for we are implicitly assuming that in the range of $3.141592$ that is $[0,10[$ the resolution is $10^{-5}$) and you can say that the digit $2$ is a noisy digit or a non-significant digit.

Now you see that in this model the resolution (again the difference betweeen to consecutive rationals) varies depending on the decade range (where those rationals belong). What remains constant is how many rationals are in each decade range. In the examined case it is $10^6-1$, in general is $10^s-1$, where, you can guess, $s$ is the number of digits (in the decimal representation) from left-most non-zero to the right most, so those are digits that are significant for the derermination of the precision scale and the resolution. Zero digits at the left of the left most non-zero digit are not significant in this sense.

Again in what just said we are assuming that the rational number representing the approximated value of a quantity has been given without stating explicitly what is the precision scale. If the precision scale is given and so $s$ in known, also digits to the right of the $s$-th digit from the left-most non-zero digit are not significant because they are not allowed and must be discarded.

Hope this can be of any help.