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Depends on which kind of mathematician you ask for, can he utilize said math in engineering, for example, or does he only know pure math?

I am not a mathematician, but sometimes I get accused of being one; so given that no real mathematician have answered, I guess I can give it a shot.

Mathematician are in charge of building mathematical tools that are used by physicists, computer scientists, and many other subjects, including artist.

Why is math useful: mathematics are used in social science, physics, computer science and many other subject. Take a simple example from computer science: everyone is very excited about quantum computing, but what questions can be answered faster by a quantum computer than a classical computer? This is both a computer science question and also a math question. Many mathematicians are working on problem like these.

What is the difference between mathematican, computer scientists, physicist, and so on: although people from other subject also use advanced mathematical tools and work on similar questions as mathematicians (I guess why I was accused of being a mathematician), the difference is in their approach. Typically, for non-mathematicians (like me), proofs and math tools are means to an end. We often want to prove a very concrete problem (like are two reasonable ways to define the meaning of a program are equivalent), and usually we prefer the proof the takes the least amount of effort to get to the conclusion. Whereas mathematician often makes connection between different approaches, generalize, and just explore things that they feel is interesting. The mathematical approach often is slower but also gives deeper understandings: although it is common for many of their insights to be lost through time, it is also quite often for these exploration leading to important breakthrough in other fields.

What is the life of a mathematician like: like every other academic: teaching, research, writing grant to feed yourself, and sometimes traveling to discuss ideas and start new projects. I imagine OP is most interested in is mathematical research. I feel the most apt analogy is the creation of art: for an artist, they usually have a emotion trying to express, either something they see or feel. Then they do a couple sketch, see what detail/style works in expressing their ideas and what doesn't, then paint the painting. For mathematicians, they often have a question in mind, then they try some examples to see what steps closer to their goal and what leads to dead ends. Through these excersices they gain a intuition of what conditions are important for the desired conclusions, then they pain the full painting by finishing the proof.

These proofs can be exceptionally time consuming: even for computer scientists, they can easily take couple researcher a year of work to do a proof. Most of the sketches will be thrown away, either because they are too convoluted or because they don't lead to the correct conclusion. Usually, a proof by computer scientists like me can easily take 20-30 pages to explain properly, if not more; and the proof that were thrown away can double that quantity. I can only imagine proofs for mathematicians will be even more energy consuming.

I've had two software developer coworkers with math degrees

https://en.wikipedia.org/wiki/Quantitative_analysis_(finance)

Actuaries make a good living using mathematics for risk analysis.

I don't know a lot about other fields but stat people are hired a lot by research institutions. A good statistician can reduce the number of experiments you need to do, being able to test a drug/treatment with 7 people instead of 100 means a lot. They save a lot of money.

Also being able to make inference from past data, incomplete data, use correct math (there is always different ways to solve things) so they don't make mistakes.

And a lot of people with stat degree join either academia, or other fields that have actual problems and use their background to solve issues.

The analytical skills needed for an advanced math degree are transferable to wide variety of jobs. The tech companies I worked for actively sought out people with those skills, mostly for jobs that didn't require high level math.

Surprised no one has mentioned jobs like the NSA. They're thought to be the largest single employer of mathematicians.

A friend of mine works for a baseball team as a statistician making incredible amounts of money.

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There are a lot of different fields of math, and some of them are more useful than others.

Applied math is generally difficult but useful for ask kinds of things from finance to weather prediction.

Pure math is often more academic/paid for by universities.

And most universities realistically pay for math professors by having science and engineering classes take math courses, and by grants for research.

You could get hired as a consultant for a state-of-the-art zoo.

Teaching, winning grants and awards, paid analyst

Spooks (governmental, NGO or the companies who have convenient offices nearby) are always interested in hiring mathematicians.

Something about these answers isn't quite adding up. Are the commenters just trying to divide us?

Some comments seem like the product of some kind of operator. It's as though they have a quotient they're trying to fill for number of posts.

As the number of comments multiplies, I have to wonder if they think they are making a difference.

From my, admittedly limited, interaction with mathematicians in my life and a bit of extrapolation:

1. Academia: teach advanced mathematics and do research in mathematics for a university. There's still lots of unsolved problems in math and also plenty of overlap with computer science, which also has lots of research possibilities
2. Public sector: governments of all levels need at least statisticians, if not more specific mathematics skills depending on what they're trying to do (e.g. research, engineering, economics, etc.)
3. Private sector: lots of engineering companies employ a few mathematicians or at least physicists who are really good at math to make sure their next bridge/plane/ocean-boiler will actually work

There's a lot of overlap between all three but I roughly split them up based on where I'd expect the majority of jobs like that would be (e.g. I'm sure NASA employs a good deal of mathematicians, but so does Lockheed Martin and friends). Also a lot of people get a degree in mathematics and then specialize further with a masters and/or doctorate in computer science or physics, since both of those can be quite math-heavy and are better-funded fields.

Physicists get jobs everywhere. They require lots of math, but other info too.

Not really an answer as I don't know any specific math majors other than teaching?

By figuring it out.

I always thought of maths as a language, and physics as using that language to describe everything in the world. I often describe my physics degree as a degree in problem solving, so a maths degree would be similar but more abstract problem solving. So for jobs just about anything that is technical or requires abstract thought.

I have a friend working on a Master's in math. Get back to me in a year or two, and I'll let you know what he's doing.