the difference between t = 1 and ∆t = 1

Question

for example , when i want to see how fast is time , i see the longest hand in clock , the first time i start to observe , the clock hand is pointing at 12 , then after few moment the clock hand moved once , which mean i passed 1 second , and then moved once again , which mean in total i have passed 2 second.

From that statement , is that mean i have passed 2 second ( t = 2 ) , or the difference of my current time and the time when i start to observe is 2 ? ( ∆t = | t2 - t0 | = 2 )

so is there any difference between using t = 2 and ∆t = 2 mathematically ?

Answer 1

In this case, it is indeed possible to say that $\Delta t$ and $t$ are equivalent, but in general, you would use $\Delta t$ to describe the difference $t_1-t_0$, which is used if there is a process from $t=t_0$ to $t=t_1$. Some may also describe this as $t = t_0$ and $t' = t_1$.

After all, this is a notational problem, so you just need to be clear and consistent about which you use.

Answer 2

$t$ is an absolute time and $\Delta t$ is a difference of time, say $\Delta t=t-t_0$.

But when $t_0$ happens to be $0$, then $t$ and $\Delta t$ are undiscernible.

In practice this isn't a problem.