In one of my problems, I needed to find a slope on a graph with different units in physics.
309 cm² - 0.0 cm² / 100 cm - 0.0 cm
The answer is A = 3.09 r² with A being area and r² being radius squared.
My main issue is with the significant figures.
0.0 is the smallest significant digit here. So logically, 0 should be the significant figure, since 0.0 is the smallest. But it is not.
Why is that so?
And what if the 0 was 1.0, would that make the answer A = 3.1 r²?
Thanks
The point of significant figures is to have an idea how many digits of the final result are actually meaningful to report. For example, you have no business reporting a particular result as $4.52055936$ if you cannot even be reasonably sure that the true value is between $4.51$ and $4.53.$
When you make several operations on numbers to get the final result, you have to keep track of significant digits at each step. When you take $309 - 0.0,$ the result is just $309$: you don't write it as $309.0,$ because doing so would imply that you know the digit after the decimal point, and the amount of variation that the actual first number might have from the stated value $309$ is too great to specify a digit after the decimal point. That is, in order for the result of $309 - 0.0$ to have a significant digit after the decimal point, both inputs have to have a significant digit after the decimal point.
Likewise with $100 - 0.0$ the result is $100,$ not $100.0.$ This is actually a bit ambiguous; I do not believe there is a universal convention regarding exactly how many of the digits of $100$ are significant. According to some conventions only the first digit (the $1$) would be significant. But if your particular convention is that all digits of a number written as an integer are significant, including trailing zeros, then $100$ has three significant digits.
You then divide a number of three significant digits by another number of three significant digits, and we optimistically allow you to keep three significant digits in the result: $309/100 = 3.09.$ (I say this is optimistic because every multiplication or division operation introduces the possibility of a larger percentage variation in the result.)
If you had $1.0$ instead of the first $0.0$ then you would have $309 - 1.0 = 308$ and $308/100 = 3.08.$ But if you had $-1.0$ instead of the first $0.0$ then you would have $309 - (-1.0) = 310$ and $310/100 = 3.10,$ assuming that all digits written in integer form are significant.