# Representation of $l_x, d_x, L_x, T_x$

There are some problems in my textbook where a life table is given, with some of the entries missing. Based on the given entries, I have to fill up the complete life table.

Suppose it is given $$l_0 = 880, q_0 = 0.005$$. Then, obviously, $$d_0 = q_0l_0 = 4.4$$.

So, here $$d_0$$ comes in fractional number. And further, many $$l_x$$ and $$d_x$$ come in fraction.

But $$l_x$$ denotes the 'number of persons' reaching age $$x$$ last birthday, and $$d_x$$ denotes the 'number of persons' dying between age $$x$$ and $$x+1$$ last birthday.

So, are the above fractional representation of $$l_x$$ and $$d_x$$ wrong, since their interpretation deal with number of persons? Should I always round them off to integer?

And also, what about $$L_x$$ and $$T_x$$? Because their interpretation usually deal with 'person-years', where fractional value is okay, but their interpretation deals with size of stationary population as well. So should their values, too, be rounded off to integers?

My doubt gets stronger because in the actual life tables that are available in internet, all the columns $$l_x, d_x, L_x, T_x$$ are integer-valued.

Strictly the numbers in $$l$$ and $$d$$ are expected values (not observed) given some notional $$l_0$$ according to some set of $$q_x$$ probabilities (taken as given for the purpose of the life table, though they're estimated from mortality data).
As such, the expected number of deaths at age $$x$$ (last birthday) is not strictly an integer, and every $$l_x$$ after $$l_0$$ shouldn't be an integer.
However, $$l_0$$ usually has more figures than the number of places of accuracy in $$q_x$$, so rounding $$d_x$$ to an integer will do essentially no harm (and is less confusing for people who want to deal with concrete concepts like "number of people alive at age $$x$$" rather than the expected number), except perhaps at very late ages (though note that $$q_x$$ is also less accurate there). Indeed, $$l_0$$ is deliberately chosen large for this purpose.
In short, there's nothing about non-integer values in the life table that is a problem. Rounding $$d_x$$ (and quantities which follow from it) to integers is common but completely unnecessary.