This question came up in a physics class, however I feel that it is more maths-related.
State the amount of significant figures in $60s$, the number of seconds in one minute.
Significant figures are generally quite easy to understand. The amount of significant figures represents the digits in a number that are certainly correct (i.e. no error in measurement).
I answered $2$ significant figures, just like most people would, and then I checked the answers.
$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ $Infinite
What??? Why? How can $60$ have an infinite amount of significant figures? Clearly any trailing zeros after the decimal point have no impact on the error in the amount of seconds in a minute. Hence they are insignificant. So why is there an infinite amount of significant figures for $60$ seconds?
I believe this has a lot to do with the definition of significant figures and its interpretation in maths vs physics. However, I would like some clarification as to why this is the case.
A minute is exactly $60$ seconds by definition. Defined values and mathematical constants are often taken to have an infinite number of significant figures. This is so that no unwarranted truncation occurs during manipulation in combination with measured values (for which significant figures are more meaningful in defining precision).
Note that this does not apply to physical constants which have built-in measurement uncertainty, e.g. the numerical value of the electron charge in coulomb that you take to solve a problem does have a finite number of significant figures. But the definition of one metre as $100$ cm has an infinite number of significant figures. I hope you can see the distinction.