I was reading about the term, "Significant" digits in the book, "Numerical Mathematical Analysis " by J Scarborough.
The definition of the significant digits were given as:
A significant figure is any one of the digits $1, 2,3,... 9;$ and $0$ is a significant figure except when it is used to fix the decimal point or to fill the places of unknown or discarded digits.
Thus, in the number $0. 00263$ the significant figures are $2, 6, 3;$ the zeros are used merely to fix the decimal point and are therefore not significant. In the number $3809,$ however, all the digits, including the zero, are significant figures. In a number like $46300$ there is nothing in the number as written to show whether or not the zeros are significant figures. The ambiguity can be removed by writing the number in one of the forms $4.63\times10^4$ $4.630 \times10^4$, or $4.6300\times10^4$, the number of significant figures being indicated by the factor at the left.
If my understanding is correct, it means that say, there is a number $7878$ and it's written as $7800$ instead of $7878$ then, the number of significant digits in $7800$ is $2$ and they are $7$ and $8.$ The two zeros in $7800$ are ignored as they were used to fill the places of unknown or discarded digits, which were $7$ and $8$ in this case. (Is this a valid reasoning?)
Next, in the part, where it's written,
In a number like $46300$ there is nothing in the number as written to show whether or not the zeros are significant figures.
I think they called the number ambiguous as, $4,6,3$ are significant digits and the confusion is with the two zeros appended at the end.
The last two zeros in $46300$ are significant digits if $46300$ is equivalent to $46300.00$ or if, $46300$ is written in place of some number say, $46398$ (, i.e the last two digits in $46398$ which are $9$ and $8$ are discarded or is unknown and instead of it, we write, $46300$). But we aren't sure of whether any of the above two scenarios are taking place or not. (Is this inference correct?)
If this inference is correct then, why does they write,
The ambiguity can be removed by writing the number in one of the forms $4.63\times10^4$ $4.630 \times10^4$, or $4.6300\times10^4$, the number of significant figures being indicated by the factor at the left.
I don't understand how the ambiguity gets removed. Also, what's the number of significant digits in $4.63\times10^4$ $4.630 \times10^4$, or $4.6300\times10^4$. I have no idea about what it is ?
A number with a given number of significant figures usually comes from data that has more digits, but a limited accuracy, so it is rounded. For example, $46303.276\pm4.21$, where $4.21$ indicates a range of precision for the measurement. This could be written as $46300$, intending $4$ significant figures. However, written this way, as mentioned in the question, we don't know whether there are $3$, $4$, or $5$ significant digits.
Your understanding seems correct. The $0$'s take place of insignificant digits that have been discarded. However, usually, a number is rounded when doing this, so if the number were $7878$, and the probable error were such that only two digits were significant, I would have written it as $7900$.
Given $46300$, it is not clear whether $0$, $1$, or $2$ of the $0$'s are significant. The actual value before rounding may indicate the actual number of significant digits, and how to represent this number in scientific notation, but the section "if $46300$ is equivalent to $46300.00$ or if, $46300$ is written in place of some number say, $46398$" doesn't really make sense. If you have the number $46300.00$, it apparently has $7$ significant digits, so $46300$ would be intended to have $5$ significant digits, but that gets lost in the representation because we are back to the original problem.