8 Hours of Sleep

I often get forwards telling me how sleep-deprivation is dangerous and why 8 hours of sleep is mandatory. They specifically say how sleep-deprivation decreases your productivity. Obvious yes, but I wanted to take a mathematical look at the problem and remove all doubts.

Disclaimer: Questionable Assumptions, Flippant Arguments

I shall assume we take, on work days, an average of 6 hours for (in no particular order): Travelling, Bathing, Eating, Dressing, Cleaning, Ablutions, Chatting, Waiting (ex:- waiting for facebook to load photo albums, waiting for some &*^#%s reply on gchat), Searching (ex:- keys, spectacles, wallet), and tons of other worthless things. These are things we HAVE to do. Although I do know that the actual amount of time we collectively spend on these things could vary heavily, I think 6 hours as an average value is a reasonable assumption (obviously not valid on weekends).

This leaves us with 18 hours in a day for mental-health and work. I shall also assume that we should aim to maximize overall-benefit and that it is partially explained by the equation: $overallBenefit \propto output_{work} * mentalHealth$. Further, I take $output_{work} \propto f_1(x)$ and $mentalHealth \propto f_2(y)$, where $x$ is amount of work time and $y$ is amount of time spent for mental-health, both in hours, with the equation $x + y = 18$ holding good.

Lemma 1: $f_1(x) = 1-2^{-x}$

Proof: I decided, for $f_1$, an exponential function of x would be best and that polynomial and logarithmic functions of x did not make sense for some reason. And according to Abhiram (2009), 2 would be a wise choice for the radix. $2^x$ would be in $[1,\infty)$ when $\usepackage{amssymb} x \in \mathbb{R}_{\geq 0}$. I take the reciprocal of it to force its range to be $(0,1]$. Now, to make it an increasing function, I subtract it from 1. qed

Lemma 2: Lemma 1 is not total nonsense.

Proof: A plot of $1-2^{-x}$ looks like this.

This shows that there is a sharp increase in output initially and that it plateaus after a while. Kind of makes sense. qed

Lemma 3: $f_2(y) = log(y)$

Proof: We would once again need an increasing function of y. In general, spending twice the amount of time you are already spending on mental health cannot increase $f_2$ by a factor of 2. Thus we would need to attenuate y. Given this, and some other considerations, a logarithmic function was the only thing that made sense. qed

By Lemma 1, 2 and 3, we have

$overall-benefit = f(x) \propto log(18-x) * (1-2^{-x})$.

The plot of $f(x)$ versus $x$ looks like this.

Deduction: Maximum benefit is achieved by working for 6 hours, i.e. $x=6$ is the optimum.

This leaves us with 12 hours for mental-health. Of course, using all that for sleep would not be wise, because sleep is not the only thing that influences mental-health. General reading, playing, music, praying.... and a tons of other things could and probably SHOULD be done! Catching up with news, making small-talk..... Helping your computer-illiterate neighbours setup their video cam is also not a bad idea. It buys you good-will, which can sometimes enhance overall-benefit all by itself. I shall assume an average of 3.5 hours for all this. This, and half an hour everyday for latency (various things...) gives us 8 hours of sleep!

P.S. Read this manual sometime if you can, it is about how you get the maximum out of your sleep.