In need of advice: How do I become a mathematician?

In summary: Much appreciated!In summary, the speaker is a senior in high school who has finished their applications and is now waiting for college decisions. They are worried about their future in math and seeking advice on what to do next. They are currently self-studying linear algebra and reading a book on understanding analysis, but feel lost and unsure about their path in mathematics. They are seeking guidance and reassurance from someone with experience in the field.
  • #1
I’m currently a senior in high school. All of my applications are finished and I am now in the waiting zone. At the same time I started to wonder what my future will be and frankly I was quite inspired by the discussion in 2006 about what does it take to become a mathematician.

I was pretty sure last year when I finished multi-calculus at my local university that I was into math and I wanted to study it. But now as I’m taking linear algebra, I’m not that sure anymore. I couldn’t really catch up with what the professor is saying (this could because we are learning very fast since we only have one class a week and we need to cover a lot of topics). I am basically self-studying for this course. Even though I understand the concepts (that’s what I thought so at least), I am sometimes struggled with understanding and solving many of the homework problems. I have always been told that mathematicians are for pure genius and the more I learn the more I start to be scared. What if I am not suitable for learning math? This is my biggest fear.

However, I do know that I love math and have a huge passion for it but simultaneously I’m afraid of something I can’t really articulate. I think I really need some enlightening advice to help me.

Another thing I wanted to seek for advice is what I should do next. I will probably finish linear algebra before graduation and I’m currently reading understanding analysis by Abbott, hoping to get a sense of what’s real proofs are like. I don’t have anyone who is related to mathematics (first generation in my family, people around me all hate math for reason I don’t know) so I’m quite lost here. To be honest, I have no idea of what’s coming next and what should I be reading at all. I don’t know what particular area I will be interested in because I don’t have enough knowledge to even choose. Therefore, I was just wondering if any advices could be given on what should I start reading and what should I do next to learn more about mathematics and about being a mathematician.

This is very very long and tedious I assume but you can’t possibly imagine how much appreciated I will be for whoever reads it and gives me advice. All the helps are valuable and really will give me a lot of guidances.
 
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  • #2
Honestly, I think you are overthinking this. Get to college, take a proof-based math course, and if you don't like it or don't do well, then do something else.
 
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  • #3
I agree with V50.

You're in high school, and you're taking university-level courses and finding that they're more challenging. This is because there is an academic bottleneck between high school and university. In high school, you're studying with a general cross section of people and are taught by instructors with specific training on how to best present you with the information. Once you get into university, your peers are now limited to a sample from a much smaller subset of those willing to pay extra to study this subject when they don't have to. Most have an interest and/or affinity for the subject. The instructors have much more subject-specific training, but much less teaching-specific training. As a result, a lot of students who have done well in high school get to university and can struggle with being average. A similar bottleneck happens in graduate school.

It's great that you love math. But you don't have to win a Fields medal to go on to a career in mathematics.

Once you get into university, pursue a mathematics degree. Join the undergraduate mathematics society. Find some way to get involved in research. Explore the different sub-fields and find/choose your niche. It's okay to take time in figuring this out.
 
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  • #4
Vanadium 50 said:
Honestly, I think you are overthinking this. Get to college, take a proof-based math course, and if you don't like it or don't do well, then do something else.
Thank you so much for your response. I think that's probably now I am waiting for the college decisions and I feel like everything is so uncertain. I felt quite lost because I don't have anyone in math to talk to about my anxiety. Thank you so much! Your words really help me.
 
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  • #5
Choppy said:
I agree with V50.

You're in high school, and you're taking university-level courses and finding that they're more challenging. This is because there is an academic bottleneck between high school and university. In high school, you're studying with a general cross section of people and are taught by instructors with specific training on how to best present you with the information. Once you get into university, your peers are now limited to a sample from a much smaller subset of those willing to pay extra to study this subject when they don't have to. Most have an interest and/or affinity for the subject. The instructors have much more subject-specific training, but much less teaching-specific training. As a result, a lot of students who have done well in high school get to university and can struggle with being average. A similar bottleneck happens in graduate school.

It's great that you love math. But you don't have to win a Fields medal to go on to a career in mathematics.

Once you get into university, pursue a mathematics degree. Join the undergraduate mathematics society. Find some way to get involved in research. Explore the different sub-fields and find/choose your niche. It's okay to take time in figuring this out.
I suppose you are absolutely right. It's just I heard so much story about how mathematician should be pure genius and I don't have a way to verify it and I probably got carried away with it. I felt comforted when seeing you said that I have time to figure out what I really want. Thank you so much! Your help is very much appreciated!
 
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  • #6
Eveeeeeelynzzz said:
I’m currently a senior in high school. All of my applications are finished and I am now in the waiting zone. At the same time I started to wonder what my future will be and frankly I was quite inspired by the discussion in 2006 about what does it take to become a mathematician.

I was pretty sure last year when I finished multi-calculus at my local university that I was into math and I wanted to study it. But now as I’m taking linear algebra, I’m not that sure anymore. I couldn’t really catch up with what the professor is saying (this could because we are learning very fast since we only have one class a week and we need to cover a lot of topics). I am basically self-studying for this course. Even though I understand the concepts (that’s what I thought so at least), I am sometimes struggled with understanding and solving many of the homework problems. I have always been told that mathematicians are for pure genius and the more I learn the more I start to be scared. What if I am not suitable for learning math? This is my biggest fear.

However, I do know that I love math and have a huge passion for it but simultaneously I’m afraid of something I can’t really articulate. I think I really need some enlightening advice to help me.

Another thing I wanted to seek for advice is what I should do next. I will probably finish linear algebra before graduation and I’m currently reading understanding analysis by Abbott, hoping to get a sense of what’s real proofs are like. I don’t have anyone who is related to mathematics (first generation in my family, people around me all hate math for reason I don’t know) so I’m quite lost here. To be honest, I have no idea of what’s coming next and what should I be reading at all. I don’t know what particular area I will be interested in because I don’t have enough knowledge to even choose. Therefore, I was just wondering if any advices could be given on what should I start reading and what should I do next to learn more about mathematics and about being a mathematician.

This is very very long and tedious I assume but you can’t possibly imagine how much appreciated I will be for whoever reads it and gives me advice. All the helps are valuable and really will give me a lot of guidances.
Hi,
What you going through it totally normal for any student in any major. Do not let one experience (in your case Linear Algebra course or the instructor) to let you down. Yes, you will see many up and downs like anything else in daily life and academia. Try to find ways to go around the problem.

In your particular case, find tutors, or online videos that explain Linear Algebra in a better way than your instructor. You are correct, these topics need time to digest and more importantly you need to know why the linear algebra is important. For example it is almost impossible to study Quantum Mechanics without basic understanding of Linear Algebra.

Most people hate a subject simply because they are not good at. They have never had proper environment (good teachers, good schools, ....) to teach them those subjects. I am a single person in my family or extended family that chose Physics as my major. I have gone through many problems similar to your experience but never gave up. I graduated from university, got Master and PhD degrees. I wish you all the best.
 
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  • #7
Eveeeeeelynzzz said:
I have always been told that mathematicians are for pure genius and the more I learn the more I start to be scared. What if I am not suitable for learning math? This is my biggest fear.
Other kids in your school probably say, "I hate math, it is for geniuses like @Eveeeeeelynzzz "

Now you know, you can learn something if you work at it. You don't have to "be genius" to do well.
 
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  • #8
DDSJ787878 said:
Hi,
What you going through it totally normal for any student in any major. Do not let one experience (in your case Linear Algebra course or the instructor) to let you down. Yes, you will see many up and downs like anything else in daily life and academia. Try to find ways to go around the problem. In your particular case, find tutors, or online videos that explain Linear Algebra in a better way than your instructor. You are correct, these topics need time to digest and more importantly you need to know why the linear algebra is important. For example it is almost impossible to study Quantum Mechanics without basic understanding of Linear Algebra. Most people hate a subject simply because they are not good at. They have never had proper environment (good teachers, good schools, ....) to teach them those subjects. I am a single person in my family or extended family that chose Physics as my major. I have gone through many problems similar to your experience but never gave up. I graduated from university, got Master and PhD degrees. I wish you all the best.
Thank you so much! Your response is very encouraging. I try to find some videos and the problem is that I understand the concept but I can’t figure out what the problems are asking for the most of time (this may seem inconsistent but it did happen to me). I never though of giving up. I guess I just got scared and have no one to talk to. Thanks anyway! It is really inspiring.
 
  • #9
gmax137 said:
Other kids in your school probably say, "I hate math, it is for geniuses like @Eveeeeeelynzzz "

Now you know, you can learn something if you work at it. You don't have to "be genius" to do well.
I guess I never think about it in this perspective. This is just so helpful and I really appreciate what you said!
 
  • #12
One more piece of free advice (warning: you usually get what you pay for!) In my humble experience, undergraduate education should be a broadening intellectual experience. Do not get so caught up in preconception about what you should be interested in that you miss the stuff that is uniquely what you do well. Always be ready to be surprised. Work hard (and play hard) and always ask the "stupid" (but well-considered) question. Remember you are paying these guys. And a good student asking good questions is valuable to a prof.
 
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  • #13
hutchphd said:
One more piece of free advice (warning: you usually get what you pay for!) In my humble experience, undergraduate education should be a broadening intellectual experience. Do not get so caught up in preconception about what you should be interested in that you miss the stuff that is uniquely what you do well. Always be ready to be surprised. Work hard (and play hard) and always ask the "stupid" (but well-considered) question. Remember you are paying these guys. And a good student asking good questions is valuable to a prof.
This makes so much sense. I really really appreciate it.
 
  • #14
Eveeeeeelynzzz said:
[snips]
This is very very long and tedious I assume but you can’t possibly imagine how much appreciated I will be for whoever reads it and gives me advice. All the helps are valuable and really will give me a lot of guidances.

Get good.



Watch to the end. It's only 4 minutes. Life lessons using a video game as a metaphor.

There probably isn't a "magic formula." The thing that works for you might not work for other people.

Find some aspect to study that "glimmers" for you. You find it interesting. It's engaging. Maybe not fun, exactly. But it grabs you and keeps you coming back to work on it. If you can get yourself on a track to work in something that you don't need to push yourself to work, you will very likely have a great career.

For me it was physics "posers." There is this thing called the Sir Isaac Newton Exam from university of Waterloo. https://uwaterloo.ca/sir-isaac-newton-exam/ There was also the Canadian Association of Physicists prize exam. And the math society prize exam. I got copies of old exams from my teacher. And I studied them, trying to do them, and learning the subjects I needed for the ones I could not do.

When I was in high school, we had grade 13. Only up to grade 12 in the schools here now. Grade 13 was intended for folks who were headed to university. I was busy doing grade 13 math and physics in early grade 11. Calculus, algebra, mechanics. This was because it was the only way I could do the problems on the prize exams.
 
  • #15
IMO, pure Math is pretty different from Calculus, Complex Analysis, and Linear Algebra. I'm not sure that you know yet what real Mathematicians do. I certainly didn't when I was in your position decades ago. Much of the math you have been studying is an essential tool in many physical sciences and engineering. It's a great thing for you to study and learn well, you will use it. However, it's just a bit too early for you to figure out how you will be using it years from now. Maybe you'll be an EE, ME, Physicist, Climate Scientist, or a Mathematician. Frankly, there aren't too many pure Mathematicians out there, there aren't a lot of teaching positions, and there are lots of other good careers that use math for more practical stuff, which you might find more interesting.

Don't worry just yet, you'll figure it out in time.
 
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  • #16
mathematics is what goes on in the minds of mathematicians. I used to think it was what is on the pages of math books. actually these books are just attempts to convey mathematics from the mind to the page. unforunately, the process of then conveying math from the page to the mind is very hard.

try to begin the process of thinking about math as much as possible, i.e. think about what the ideas should mean, not only what is being said. for this to work however you must work on acquiring a logical mental framework. somewhat paradoxically, before this is possible one must learn from books. to make this successful, it is best to read books by people who have good credentials as mathematicians.

the best beginning geometry book is that by euclid, and the best beginning algebra book is that by euler, but i do not know the best linear algebra or calculus book. maybe courant is the best calculus book, and as i said shields is a good candidate for a good beginning linear algebra book.

do some reading in books by real mathematicians. even if you do not understand them, you will get an idea of how they think and talk. And expect the reading of real math books to be very slow going. The most rewarding reading I have done was in Riemann's papers, which took me about one day per page. And that was after I had been a professional mathematician for many years.

as i remember from an excellent introduction to sleight of hand magic, the author said "if you can do one trick well, you are a magician". Similarly, if you can reproduce one proof correctly, you are doing proofs. so pick one you like, such as euclid's proof of the infinitude of primes, and master it.

good luck.
 
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  • #17
If this is useful, I was a mathematics professor in a state university for several decades and enjoyed a relatively fruitful career as a research and teaching mathematician. I was blessed with colleagues who filled gaps in my abilities and shared their expertise and strengths in collaborative efforts.

I can say for sure that to be a mathematician at this level, one does not need to be a genius nor anywhere near. The “genius” mathematicians are the few people like Riemann and Gauss and more recently Grothendieck, whose ideas other mathematicians spend their lives working on.

I felt I myself had decent geometric intuition, and an ability to draw analogies between different known results, which sometimes led me to insights that could give new results, of interest even to very good mathematicians. In comparison to other mathematicians I knew, I was not particularly “strong”. My weaknesses included lack of broad basic knowledge and mediocre computational ability, both a result of inadequate time and dedication spent in early training and study. Fortunately for me, these weaknesses were compensated for by my collaborators.

So there are many avenues to becoming a mathematician; you don’t have to be good at everything, much less be a genius. But it is very hard, absorbing work, and you must enjoy it or it isn’t worth the sacrifices it entails, e.g. financial ones as well as monk - like dedication.

Even if it is not in the cards for one to be a pure research mathematician, there are other rewarding careers in, e.g. national security or applied design, that use the skills one learns in mathematics.
 
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  • #18
I would like to add that, in my opinion, it is very important to solve problems. Hard problems, not just the routine ones.
 
  • #19
Eveeeeeelynzzz said:
Even though I understand the concepts (that’s what I thought so at least), I am sometimes struggled with understanding and solving many of the homework problems. I have always been told that mathematicians are for pure genius and the more I learn the more I start to be scared. What if I am not suitable for learning math? This is my biggest fear.
Guidance to groups from long ago was, study as much Mathematics as you can; and for possible alternatives in case no degree IN Mathematics, choose among fields which rely on heavy use of Mathematics, such as Computer Science, Engineering, Chemistry, or other natural sciences.
 
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  • #20
My original degree was in Mathematics, minor in Statistics. Now long retired, I still use problem solving techniques learned at university every day. The skills you learn following a mathematics curriculum can help you solve problems throughout your life and professional career.

Just this week I helped a student trying to analyze medical data published in a Washington Post science article. She tried to combine data points using only a factorial function. I explained combination functions use factorials in the numerator and denominator, but also certain requirements that must be applied to the function inputs in order to produce accurate results, directing her to the PF Mathematics sub-forum for expert advice.

Like several other PF members, I made a career as a software and IT engineer, applying skills I developed learning mathematics. Great fun and they paid me as well. Good luck at college.
 
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  • #21
In the spirit of martinbn's advice, I would suggest also the importance of working out lots of examples, not just memorizing proofs of theorems.
 
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  • #22
I wanted to be a classical singer. Botched audition, enrolled to math undergrad. Now I've almost finished math phd studies.

In short, don't overthink it. You want to become *insert profession*. Start by getting a relevant degree.
 
  • #23
nuuskur said:
I wanted to be a classical singer. Botched audition, enrolled to math undergrad. Now I've almost finished math phd studies.

In short, don't overthink it. You want to become *insert profession*. Start by getting a relevant degree.
The additional inference there is easy to miss. Study and check several areas of interest and choose what you believe will give you good potential; even if you choose more than just one.
 
  • #24
hutchphd said:
One more piece of free advice (warning: you usually get what you pay for!) In my humble experience, undergraduate education should be a broadening intellectual experience. Do not get so caught up in preconception about what you should be interested in that you miss the stuff that is uniquely what you do well. Always be ready to be surprised. Work hard (and play hard) and always ask the "stupid" (but well-considered) question. Remember you are paying these guys. And a good student asking good questions is valuable to a prof.
OP: You have received advice from multiple perspectives. Given your scenario, I second the one above. Your profile indicates that you are from Boston. I will assume that means Boston, MA (where I grew up) or at least one of the Bostons in the US. At many US universities, you don't need to declare a major until your sophomore year. And if you do need to declare a major earlier, you can readily switch in your sophomore year (or, with the proper choice of courses, even in your junior year). There are always exceptions, so keep this in mind when you are making your decision on which university to attend.

In the US, your freshman year courses will likely be general education requirements (which typically will include math of some sort) plus electives. Take advantage of this time to explore various fields. In high school, you've been exposed to the basics of humanities, science, and math (and in your instance advanced math). In some high schools, you will have received exposure to computer programming; in others, not. But you likely will not have been exposed to various branches of engineering.

So it's good that you have found something that you're interested in. But keep your options open.
 
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  • #25
My two cents worth is that Mathematics is a vast subject, compared to what you learned in high school a minute sliver of what is available.

The big four college courses:
- Calculus 1,2,3
- Linear Algebra
- Differential Equations
- Statistics

It is what everyone in STEM majors should cover in college, but they too represent mere slivers from a vast whole that is mathematics.

One decision you will first have to make after taking your core courses is what type of math you like:
- applied math
- pure math

This youtube video can give you a 10,000 feet overview of the various fields of Mathematics:



As others have said don't overthink things, don't have preconceptions, discover your limits, your likes and dislikes and don't skip more fundamental courses to get to the good stuff.

When I entered college, I had some preconceived notions of how things worked and was suprememly confident in my abilities to learn anything so I skipped over Differential Calculus and jumped into Integral Calculus.

Later, I repeated my mistake by skipping over several courses to take Abstract Topology. I was a junior physics major among senior math majors with no experience in doing proofs or real knowledge of Abstract Algebra. It was a disaster that the prof admired my tenacity and helped me squeak through with a C.

At the time I thought these shortcuts were smart moves. However, upon reflection years later I realized I missed some important foundational concepts and the chance to get better grades as my GPA suffered.

Bottomline, don't be that person follow your advisors course sequence advice and soak up all you can. Study hard, do lots of problems, collaborate with fellow students, decide what you like and don't like mathematically and enjoy your college life. Once you start working you will miss the freedom you once had.

Some guidance from Purdue Univ on your future as a mathematician and lists of some other sites you can check out:

https://www.purdue.edu/science/careers/what_can_i_do_with_a_major/Career Pages/mathematician.html

Indeed provides further information on a career as a mathematician:

https://www.indeed.com/career-advice/finding-a-job/how-to-become-mathematician
 
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  • #26
Eveeeeeelynzzz said:
I felt quite lost because I don't have anyone in math to talk to about my anxiety. Thank you so much! Your words really help me.

You do here. Although I am more into mathematical/theoretical physics these days (still a lot of math, I know), I majored in math (as well as computer science). The number one piece of advice I can give you, especially in modern times where computers are everywhere, is math is not about long-involved calculations. You do a bit of that, of course, and you get better with experience, but the key is math is really about concepts. That's right - concepts. Understanding what's going on is the most important thing. A friend who did biochemistry and a couple of masters asked me - how do you remember all those formulas and proofs. Some you do remember, like the quadratic formula, but you do not remember the proofs. If asked to prove it, you remember the key concept - completing the square. Eventually, you will come across what is called analysis. The proofs can be initially difficult to master - but I found a trick that helped. At the start of the proof, you say fix ε>o. The professor who taught me analysis told me the trick, and he was right - it always helped.

Thanks
Bill
 
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  • #27
Read books by Lara Alcock, specifically "How to study as a mathematics major" and "How to study for a mathematics degree". If you want to get a taste of proofs, check out Book of Proof by Hammack (free)
 
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  • #28
take as much algebra and analysis as possible during your undergrad. this will give you a strong foundation for everything else.
 
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  • #29
Eveeeeeelynzzz said:
Thank you so much! Your response is very encouraging. I try to find some videos and the problem is that I understand the concept but I can’t figure out what the problems are asking for the most of time (this may seem inconsistent but it did happen to me). I never though of giving up. I guess I just got scared and have no one to talk to. Thanks anyway! It is really inspiring.
You can pose your questions here at PF. We'll guide you along until you can fly on your own. But we'll be here if you need guidance further along on your journey.
 
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  • #30
There is a kind of anxiety that students have when learning new things. Often they convince themselves that they understand a concept only to get frustrated when using it to solve a problem.

The student will say: "I understand the concept, but I just can't apply it to the problem."

If so, you must change your thinking and not delude yourself. As you read passages in your book, ask yourself what they mean; write FAQ-like notes of your questions and what you've discovered.

Try teaching yourself by closing the book and imagining teaching someone the concept and notice where you hesitate, and soon you'll find the flaws in your learning that need to be patched.

Do lots of problems. It's the only way to get good at either math or physics.
 
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  • #31
also, read and work through your textbooks.
 
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  • #32
Eveeeeeelynzzz said:
I have no idea of what’s coming next and what should I be reading at all. I don’t know what particular area I will be interested in because I don’t have enough knowledge to even choose. Therefore, I was just wondering if any advices could be given on what should I start reading and what should I do next to learn more about mathematics and about being a mathematician.
You have probably taken more math courses than most entering math majors. Don't try to learn anything new. Review your previous course work with the intention of thinking that you will teach courses in those subjects. Many will tell you that you do not understand a subject until you teach it. Maybe you could try and tutor someone.

How did you do on the SAT math? Did your score indicate a good aptitude for math?

Every year there are about 30,000 math BS degrees are awarded and about 1200 Ph.D.s.

Are you familiar with the type of jobs available for a math major? see for example see https://www.geteducated.com/careers/jobs-for-math-majors/#/

When you say you are interested in math do you mean pure or applied? Pure math is a whole other ball game.

When you get to university you will be ahead and you will finally be able to talk to others who share your interest in math. Be warned that university may be more challenging than you expect. It is important to get off to a good start, starting by developing a sound foundation. Talk to the faculty about your background and any concerns you might have. Do not hesitate to retake any course you took in HS that you feel weak in. Remember your confidence is important especially when things become difficult.

As you firm up your goals develop a sensible plan. Keep in mind there are many who can help including faculty and PF. Good luck.
 
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  • #33
jedishrfu said:
There is a kind of anxiety that students have when learning new things. Often they convince themselves that they understand a concept only to get frustrated when using it to solve a problem.

The student will say: "I understand the concept, but I just can't apply it to the problem."

If so, you must change your thinking and not delude yourself. As you read passages in your book, ask yourself what they mean; write FAQ-like notes of your questions and what you've discovered.
"The first principle is that you must not fool yourself—and you are the easiest person to fool. So you have to be very careful about that. After you’ve not fooled yourself, it’s easy not to fool other scientists. You just have to be honest in a conventional way after that." - Richard Feyneman
 
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  • #34
gleem said:
You have probably taken more math courses than most entering math majors.

You certainly do. If it's not on your list I always suggest Virginia Tech - great school - easily in the 99 percentile for math, but for some reason is easy to get into - from Big Future 'Virginia Polytechnic Institute and State University is less selective with an acceptance rate of 70%. Students that get into Virginia Polytechnic Institute and State University have an SAT score between 1180–1390 or an ACT score of 26–32'When I read that my eyes bulge - that is far too easy for a school of that quality - maybe there is some self-selection going on - don't really know. Anyway since it is basically a waiting game here are two books I suggest:

Boaz: https://www.amazon.com/dp/0471198269/?tag=pfamazon01-20
Hubbard: https://matrixeditions.com/5thUnifiedApproach.html

Taken together will give you a great foundation and an excellent head start in both pure and applied math.

Thanks
Bill
 
  • #35
You can do it. Just remember that it's a long way to the top if you want to Mathematics. ;) (AC/DC)
 

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