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Say I have a very large piece of paper, and I draw a number line on it. The integers are marked $10^{18}$ meters apart. The drawing instrument can make lines $10^{-7}$ meters thick. The book I'm reading says we can only distinguish between two numbers that are $10^{-25}$ apart(there isn't a unit given for this) on this number line. I think I understand why, but not with the reasoning given. My reasoning is divide the unit length, $10^{18}$, by the mark width, $10^{-7}$. Conceptually, this gives how many marks I can fit in between two integers, $10^{25}$. Thus, each mark represents an increment of $10^{-25}$. Their reasoning is the opposite, they divide $10^{-7}$ by $10^{18}$. I get that the result is the same, but I don't get it conceptually. What is the reasoning behind dividing the mark width by the unit length? Below is taken from the book:

Practical drawing is in fact extremely limited in accuracy. A fine drawing pen marks a line 0.1 millimetres thick. Even if we use a line 1 metre long as a unit length, since 0.1 mm = 0.0001 metres, we could not hope to be accurate to more than four decimal places. Using much larger paper and more refined instruments gives surprisingly little increase in accuracy in terms of the number of decimal places we can find. A light year is approximately $9.5 * 10^{15}$ metres. As an extreme case, suppose we consider a unit length $10^{18}$ metres long. If a light ray started out at one end at the same time that a baby was born at the other, the baby would have to live to be over 100 years old before seeing the light ray. At the lower extreme of vision, the wavelength of red light is approximately $7 * 10^{–7}$ metres, so a length of $10^{–7}$ metres is smaller than the wavelength of visible light. Hence an ordinary optical microscope cannot distinguish points which are $10^{–7}$ metres apart. On a line where the unit length is $10^{18}$ metres we cannot distinguish numbers which are less than $10^{–7} / 10^{18} = 10^{–25}$ apart. This means that we cannot achieve an accuracy of 25 decimal places by a drawing.

mbit
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When you divide $10^{18}$ by $10^{-7}$ and get $10^{25}$ you are computing the number of marks between integers. There is no unit as both of the numbers you divide are in meters. It would work the same if it were $10^{18}$ inches and $10^{-7}$ inches. When they divide the other way and get $10^{-25}$ they are computing the difference in numbers corresponding to neighboring marks. You are asking opposing questions and have opposite exponents to show for it.

Ross Millikan
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  • I get the part about us asking different questions, but I don't get why dividing the other way gives me the difference between neighboring marks. Could you clarify why? – mbit Sep 17 '20 at 23:07
  • If you think about a meter stick, there might be $100$ divisions (up to a fencepost error of $1$) of the meter. Each division then represents $\frac 1{100}=0.01$ meter. – Ross Millikan Sep 17 '20 at 23:40