Steady state implies Leavers = Joiners.
If 100 people join each month then you'll have steady state when there are 100 leavers, i.e. when 50% of N is 100. N * 50% = 100 solves to N = 200.
In general with X% attrition and Y people joining you'll have a steady number of subscribers N using the formula
N * (1 - X) + Y = N
Which can be rearranged to N * X = Y.
Serious question: Is it simply a strong math phobia, or have people neglected the skill for so long that they've actually forgotten middle school level maths?
Sometimes half-assed is ok, if it's the the right half of the ass.[1] The wrong half becomes yet another type of technical debt to be refactored "someday".
(this is not an excuse to ignore algebra or blindly assume the easiest/most-fun method is correct/useful/sane)
[1] https://twitter.com/jimweirich/status/29662011323326464 via https://www.youtube.com/watch?v=sCZJblyT_XM
Of course there are a lot of softer tricks (like checking limiting cases) which would boost people's confidence, but for some reason this is only really taught in college level physics.
In (good) excel, most formulas you write have a clear input => output relationship. Solving a system of equations (no matter how simple) breaks this relationship down. I suspect a similar thing happens when programming.
In Ancient Egypt for example when they needed to explain things like the Pythagorean theorem they would give examples, because there was no concept of abstract variables yet. I think this is similar.
In the same way, many people are comfortable working with vectors, transforming them with matrices etc but they have never been taught to think of transforms in an abstract way. So if they want to check if a certain matrix actually performs the transformation they want, they apply it to a bunch of random vectors to see if it's reasonable instead of just checking what it does to the base vectors, because they have no real concept of linear transforms, and only vague ideas about dimensionality and degrees of freedom.
To some extent it IS phobia, because if you just want to rotate a vector 90 degrees you don't really need concepts of orthonormal transformations so you might be reluctant to look at it that way, but if on the other hand abstract algebra is very natural then the abstraction is trivial to do, it just makes things easier.
http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0471433349...
It's explicitly about shifting your thinking to a more abstract level. It's great if already have a working knowledge of matrices.
If you're ready to get very serious with mathematics, I like Algebra by Mac Lane and Birkhoff. (Note that this is a different book from A Survey of Modern Algebra by the same authors, which I'm sure is okay, but... a different book.)
If you're not ready to get very serious with mathematics... well, then why do you want to learn algebra?
PS I agree about Mac Lane, he's great
Well, I agree, but life is short. Why learn algebra rather than complex analysis or probability theory or functional analysis or differential geometry or mathematical logic or general relativity or quantum mechanics or any one of dozens of other topics from physics or mathematics that will almost surely seem more interesting and fun to a non-mathematician? (Not to mention non-mathematical topics...)
I agree there are tons of other things that people should study, like basic linguistics, relativity, quantum physics, optics etc. Geometry should be on top of the list, for as Plato said “let no one ignorant of geometry enter”.
But I don’t think time is an issue really, if you just spend an hour or so each evening reading you csn quickly get a basic understanding of lots of topics in not thst many years.
aka I found out that 500 is half of 1000.
I think it is tunnel-vision & familiarity: algebra requires different techniques than coding (API docs? REPL?)
I knew my code would give me the right answer and I lacked the confidence in math, where even if I did do it by hand, I would have written the code to check it. The less time-risk way won.
One bit of anecdata - I started using computers because I was terrible with maths and they did the numbers thing for me.
One could assume that "well if I keep on going my customer count will keep on going up... right?" This is showing that it won't.
Bonus pedantry: your proof just showed that _if_ there's a steady state, then that's the steady state. Turns out that you'll always get the steady state because of "maths" but if you replace 50% with 0% there's no fixed point.
I guess it's perhaps also worth pointing out then that the same equilibrium is reach from either side. If you start with a much higher volume of subscribers you'll reach the same equilibrium through attrition down to that number also.
You're summing a series of:
0.5^n * x
Y * X^(M - 1) < 1
so once
M > 1 - (ln Y / ln X)