The Math Myth

0.999... is equal to 1 only if we assign a particular semantics to the "..." notation. Namely if "..." means "the limit of the decimal number to the left, as the repetitions of the last digit grow ever larger", then 0.999... is an alternative notation for 1 since that limit is 1.

The actual number formed by repeating 9's an infinite number of times is not constructable. Whereas 1 is constructable. So they cannot be the same thing. That's because, philosophically, two objects must be identical in every property to be the same object, and constructability is a property.

As we add 9's, we are getting ever closer to 1, and the concept of a limit lets us "fast forward" to that value. If we agree that this "..." denotes the limit, rather than the non-constructable number implied by the notation's "face value", then the equality holds.

To actually regard 0.999... = 1 to hold without involving the limit shows an ignorance of (or denial of) the validity of induction. Because, look:

Base case: 0.9 is not equal to 1.

Inductive hypothesis: Adding another digit to a decimal fraction which is not equal to 1 produces a new decimal fraction which is also not equal to 1.

Therefore, by induction, no matter how many 9's we add, we do not get 1.

Induction is not somehow canceled by infinity; induction is how we understand that a property holds for infinity: a property such as "not equal to 1".

I think you're already doing the right thing in continually learning. No one can tell you whether or not studying ML will prove to be valuable with any certainty, so it might be a good bet to pursue it if it interests you and you have the spare time and especially if you're desperate to pivot your career towards it (then you're left with no options). I can't offer a definitive decisionmaking framework, except to say that if you want to monetize something, make sure you can (fairly obvious, but not always what happens with people in our industry).

Having a "different definition" of the reals doesn't make it correct, even if you tend to say it.

I would expect that most people with a passing knowledge of basic calculus would be able to eventually understand the argument that not only does 0.999... = 1, but that the real numbers are uncountable. It might take some convincing, but the truths are provable and very well understood across the world.

The OP is a special case of the old, big question of what to teach.

It is fair to say that there is an old and strong belief that a person who has studied broadly, and deeply through, say, college, in math, physical, biological, medical, social, and computer science, and the humanities will have a significant advantage in much of the rest of life. Lacking a better name, here I call such study a broad education.

To argue this belief in the context of the OP, the OP seems to claim that for 90% or so of people, it is enough for them to stop their math education, and by extension all their education, after the eighth grade. But in life it is fairly easy to tell the difference between the OP's eighth grade education and a broad education as I described it. So, there is a difference. Maybe the difference is significant and the broad education an advantage and worthwhile.

One point not mentioned very often is that, whatever 90% of the students do, the broad education was hoping that some of the students would find some really good uses of some of the education well past the eighth grade. The educators could have that hope even without knowing just what the good uses might be.

I studied a lot of math and physics heavily, but not entirely, because I hoped that they would help me make money. Well, early in my career within 100 miles of the Washington Monument, that hope was fully correct. I used what I had and was learning more as fast as I could drinking from a fire hose. Of course that work was mostly for US national security; there the math and physics were crucial.

Yes, it does appear that away from the work of US national security, the math and physics are less commonly used.

Still, in US commercial work, there are significant applications of the math and physics. Examples:

(A) How to operate an oil refinery. In simple terms, here is a list, with prices, of crude oil can buy and put into the refinery and a list, with prices, of refined products get out of the refinery, so a question is what to buy, produce, and sell to make the most money? First cut, the problem is linear programming, and for a while there was good money in selling IBM mainframe computers just for that work. Of course, past the first cut, the problem is in non-linear optimization.

A practical challenge is: It's a good guess that the first refinery management that did well seeing and exploiting this opportunity was well paid for their insight. Since much of the crucial core of that work was some college and/or grad school applied math and numerical analysis, knowing some math could have been an advantage for the management trying to understand and make good decisions.

(B) Take a big hammer and hit the ground and send an acoustic pulse through the ground. That pulse is commonly partially reflected at the boundaries of layers of rock, sand, etc. So, the acoustic signal that comes back is a convolution of the original. Doing a deconvolution, can map the underground layers and get some good hints of where to drill for oil. The deconvolution is basically some Fourier theory, and the fast way to do the computations is the fast Fourier transform (FFT). After Cooley, Tukey, etc. invented the FFT, such acoustic processing had an explosion that is still active. So, again, oil prospecting management needed to see, understand, and actively exploit the FFT. For that, some math was no doubt an advantage.

There are more commercial applications of math and physics. Some of the applications have been valuable already, and likely some more will be valuable in the future. So, in looking for what might be valuable in business, some math and physics stands to be an advantage.

So, in part, with a broad education we are fishing for advantages in the future. We are not sure just what subjects will lead to what advantages in the future, but we are quite sure that there will be powerful, valuable new work where, for successful exploitation, some studies will be important.

Or, the OP is concentrating on what the 90% of the people actually are using now. Well, in a sense the education wants to concentrate on what is new no one is doing yet.

Limits use a construction that's pretty similar to an infinitesimal. The epsilon-delta definition of a limit is no joke for students.