What is the Difference Between 'Identically Zero' and 'Zero' in Mathematics?

[member 428835]

hey pf!

when would you use "identically zero" as opposed to simple "zero". example: f is identically zero on interval a to b. or, f is zero on interval a to b.

why do we ever use identically? it seems superfluous...

thanks!

 

[micromass]

It's because the statement "$f$ is zero on the interval $[a,b]$", might be interpreted as there is a $c\in [a,b]$ such that $f(c) = 0$. I know that the proper language should be that "$f$ has a zero", rather than what I wrote. But writers want to be clear and write that it is identically zero to avoid misunderstandings.

 

[SteamKing]

'Identically' is sometimes used for emphasis.

Of course, sin$^{2}$θ + cos$^{2}$θ is identically 1.

 

[member 428835]

thanks to you both!

 

[CellCoree]

I understand your confusion about the use of "identically zero" versus "zero." In mathematics and science, we often use the term "identically" to emphasize that something is exactly the same in all cases or at all points in a given interval. In the context of functions, saying that f is "identically zero" on an interval means that the function has a value of zero at every single point within that interval. This is different from simply saying that the function is "zero" on that interval, which could mean that the function has a value of zero at some points but not necessarily all points within the interval.

Using "identically" can be useful in certain situations, such as when we want to emphasize the uniformity of a function or when we want to compare two functions that are both equal to zero but may have different behaviors at different points. It can also be helpful when discussing more complex mathematical concepts, such as the concept of a function being identically zero in a certain space.

Overall, the use of "identically" may seem superfluous at times, but in mathematics and science, precision and clarity are important. Using this term can help avoid any confusion or ambiguity in our statements and ensure that our ideas are accurately conveyed.