Best Course Sequence For Math Majors
Summary
TLDRThis video script offers a comprehensive guide for math majors, outlining the typical sequence of courses from intermediate algebra to more advanced topics like topology. It recommends specific textbooks for self-study, including 'Intermediate Algebra' by Angel, 'College Algebra Essentials' by Julie Miller, and a calculus series. The speaker also shares insights on subjects like physics, differential equations, and abstract algebra, emphasizing the importance of proof writing. The script concludes with suggestions for supplementary courses in number theory and differential geometry, providing a roadmap for a well-rounded mathematical education.
Takeaways
- 📚 The video details the recommended course sequence for a math major, particularly for self-study.
- 🔢 Start with Intermediate Algebra, such as 'Intermediate Algebra for College Students' by Angel.
- 📖 After Intermediate Algebra, move on to College Algebra, with a recommended book being 'College Algebra Essentials' by Julie Miller.
- 📊 Learn statistics alongside College Algebra using 'Understandable Statistics' by Brase and Brase.
- 📚 Take Pre-Calculus and Trigonometry next, ideally with a book that covers both subjects like 'Precalculus' by Beecher, Penna, and Bittinger.
- 🧮 Proceed to the Calculus sequence (Calculus 1, 2, and 3), using a comprehensive book like 'Calculus' by James Stewart.
- 📘 After calculus, move on to Differential Equations, with suggested books being 'Differential Equations' by Shepley L. Ross or 'Differential Equations' by Edwards and Penny.
- 🧮 It's also beneficial to take Linear Algebra before or alongside Differential Equations, using a beginner-friendly book like 'Elementary Linear Algebra' by Larson and Edwards.
- 🔍 Learn proof writing with books like 'How to Read and Do Proofs' by Daniel Solow and 'Discrete Mathematics' by Susanna Epp.
- 🔢 Once proofs are mastered, move to more advanced subjects such as Complex Variables, Mathematical Statistics, and Partial Differential Equations, with respective books like 'Complex Variables' by Stephen D. Fisher and 'Mathematical Statistics' by Wackerly, Mendenhall, and Scheaffer.
- 📘 Explore higher-level courses such as Advanced Calculus (Real Analysis), Abstract Algebra, and Topology, with books like 'Advanced Calculus' by Patrick M. Fitzpatrick, 'Contemporary Abstract Algebra' by Joseph A. Gallian, and 'Topology' by James R. Munkres.
- 🔢 Optional courses like Number Theory, Differential Geometry, and Combinatorics can be pursued with books like 'Number Theory' by George E. Andrews and 'Combinatorial Theory' by Marshall Hall.
Q & A
What is the recommended starting point for a math major's course sequence according to the video?
-The recommended starting point for a math major's course sequence is Intermediate Algebra for college students, as it provides a solid foundation before moving on to more advanced topics.
Which book is suggested for self-study or supplementing a college-level algebra course?
-The book by Angel is suggested for self-study or supplementing a college-level algebra course, as it is modern and contains great examples and exercises.
What is the typical course that follows Intermediate Algebra in the recommended sequence?
-The typical course that follows Intermediate Algebra is College Algebra, which covers topics like polynomials, logarithms, rational functions, quadratic formulas, systems of equations, and inequalities.
Why might some math majors find the transition to statistics challenging?
-Some math majors might find the transition to statistics challenging because it requires a different set of skills and understanding, including basic math concepts and the ability to interpret data, which might not have been emphasized in their previous courses.
What book is recommended for learning statistics in the context of a math major's course sequence?
-The book 'Understandable Statistics' by Brass is recommended for learning statistics, as it provides intuitive explanations and is suitable for those who are new to the subject.
Why is it advised to take calculus before attempting physics for math majors?
-It is advised to take calculus before attempting physics because calculus provides the necessary mathematical foundation, such as differential and integral calculus, which are essential for understanding the physical concepts.
What book is suggested for learning differential equations after completing calculus?
-The book by Shepley L Ross is suggested for learning differential equations, as it covers standard topics expected in an undergraduate differential equations course.
Why is learning to write proofs considered important for a math major?
-Learning to write proofs is important for a math major because it is a fundamental skill in mathematics that allows students to rigorously establish the validity of mathematical statements and is a prerequisite for many advanced courses.
What is the recommended book for beginners to learn linear algebra?
-The book 'Elementary Linear Algebra' by lson and Edwards is recommended for beginners to learn linear algebra, as it starts with basic concepts like systems of linear equations, matrices, and determinants.
What are some of the more exotic or less commonly studied subjects mentioned in the video for math majors?
-Some of the more exotic or less commonly studied subjects mentioned in the video for math majors include Number Theory, Differential Geometry, and Combinatorial Theory.
What advice is given for those who are considering self-study in mathematics?
-For those considering self-study in mathematics, the advice given is to follow the course sequence provided in the video, use the recommended books for each subject, and ensure a strong foundation in algebra and proof writing before moving on to more advanced topics.
Outlines
📚 Essential Math Courses for Majors and Self-Study
This paragraph introduces the video's purpose: to guide viewers through the typical sequence of courses a math major would take, including recommendations for self-study. It emphasizes the importance of following a structured order and mentions the inclusion of both common and more unusual subjects. The paragraph also promises a list of books for further study, some well-known and others less so, to be presented later in the video.
🔢 Building Foundational Math Skills
The speaker discusses the foundational courses for math majors, starting with intermediate algebra using a book by Angel, which is praised for its modern approach and comprehensive content. The paragraph then moves on to college algebra, mentioning 'College Algebra Essentials' by Julie Miller, covering topics like polynomials, logarithms, and systems of equations. Statistics is also introduced as a subject that many math majors find challenging but can appreciate with the right resources, such as 'Understandable Statistics' by Brass. The importance of algebra as a stepping stone to more advanced topics is highlighted.
📈 Transitioning to Advanced Mathematics
This section delves into the transition from college algebra to more advanced courses like pre-calculus and trigonometry, which are typically taught together using a single textbook, exemplified by one authored by Beer, Panaah, and Benninger. The paragraph emphasizes the value of mastering algebra before tackling these subjects and suggests that pre-calculus might be a more natural progression due to its continuity from algebraic concepts. The sequence culminates in the calculus sequence, with a focus on a widely used textbook for calculus courses, which the speaker has experience teaching from.
🌐 Integrating Physics and Higher Mathematics
The speaker introduces physics as an important subject for math majors, recommending a classic textbook by Halliday and Resnick. The advice is to take calculus before physics to avoid the challenges the speaker faced by taking them concurrently. The paragraph also touches on differential equations, suggesting two textbooks that are standard in the field, and discusses the potential value of taking linear algebra before or concurrently with differential equations. The importance of learning to write proofs is highlighted as a prerequisite for more advanced courses.
📘 Exploring Proof Writing and Discrete Mathematics
The paragraph focuses on the importance of learning to write proofs, with recommendations for two books: one on discrete mathematics by Susanna S. Epp and another specifically on proof writing by Daniel Solow. The speaker shares personal experiences of taking discrete mathematics and proof writing concurrently, which enhanced their mathematical abilities. The paragraph also covers the topics typically included in a discrete mathematics course, such as counting, probability, and graph theory.
🎓 Advanced Math Courses and Specialized Subjects
This section outlines advanced math courses suitable for math majors, including linear algebra, complex variables, mathematical statistics, and partial differential equations, each with recommended textbooks. The speaker emphasizes the importance of a strong foundation in proof writing and calculus before tackling these subjects. The paragraph also touches on the challenges of learning from classic textbooks and the unique aspects of each subject, including the application of previously learned techniques in new contexts.
📚 Concluding with Challenging and Elective Math Courses
The final paragraph discusses more challenging and elective math courses such as abstract algebra, advanced calculus (real analysis), topology, number theory, differential geometry, and combinatorial theory. The speaker shares personal insights on the difficulty and importance of these subjects, the recommended prerequisites, and the value of proof writing in each. The paragraph concludes with a recommendation for a classic combinatorial theory book and a reminder of the resources provided for self-study and further exploration of mathematics.
Mindmap
Keywords
💡Math Major
💡Algebra
💡Calculus
💡Statistics
💡Trigonometry
💡Differential Equations
💡Linear Algebra
💡Proof Writing
💡Discrete Mathematics
💡Physics
💡Abstract Algebra
💡Real Analysis
💡Topology
Highlights
The video provides a recommended order for courses a math major should take.
Intermediate Algebra by Angel is suggested for self-study or supplementing college-level courses.
College Algebra Essentials by Julie Miller covers basic algebra topics, including polynomials and logarithms.
Understandable Statistics by Brass is recommended for learning statistics with intuitive explanations.
Pre-calculus and Trigonometry can often be studied together using a single textbook.
Calculus sequence (Calculus 1, 2, and 3) typically uses a single textbook for all three courses.
Physics is introduced as an important subject for math majors, with a recommendation to study it after calculus.
Differential Equations can be taken after calculus, with two standard textbooks mentioned.
Linear Algebra is crucial and should be studied with an understanding of proofs.
Discrete Mathematics and Proof Writing are recommended to be studied simultaneously for a strong foundation.
Complex Variables can be studied after learning calculus and proofs.
Mathematical Statistics or Statistical Theory is challenging and requires proof writing skills.
Partial Differential Equations builds on techniques from regular Differential Equations.
A second course in Linear Algebra, this time proof-based, is recommended after the initial course.
Abstract Algebra is introduced as a favorite subject of the speaker, with a book for beginners highlighted.
Advanced Calculus, also known as Real Analysis, is considered one of the most challenging courses for math majors.
Topology is recommended to be taken at the end with a strong foundation in proof writing.
Number Theory and Differential Geometry are optional but valuable advanced subjects for math majors.
Combinatorial Theory is an interesting subject that starts basic and accelerates quickly.
Transcripts
hello everyone in this video I'm going
to go through every single course that a
math major takes and we're going to go
through them in order in the order in
which you should take them now there are
many other possible orderings but I'm
going to give you the most typical and
perhaps what I think would be the most
recommended order to take these courses
in if you are doing self-study this is
going to give you the exact order in
which to study every single Le subject
that an undergraduate math major takes
also at the end of the video I will show
you a couple other more strange and
bizarre subjects these are subjects that
many people don't study they'll have
their math degree but they won't study
these subjects and so we'll look at some
of those more exotic subjects near the
end of this video I've also thrown in a
few books that are not as widely known
and also a few Classics which I'll show
you near the end of the video all right
let's go ahead and get started every
math major has to start somewhere and
this is a good place to start if you
have been away from mathematics for a
while this is intermediate algebra for
college students and this is the one by
Angel this is a great book that you can
use for self-study or to supplement your
current college level course it even
says for college students this has
pretty much everything you need to
become super good at algebra it's a
modern book so it's got great examples
great
exercises um yeah it's super awesome I
will leave a link in the description by
the way to all of these books in case
you want to check them out and it
doesn't have to be this book um but this
is a pretty good choice after you've had
some algebra you can jump to what's
called college algebra this is typically
offered at most colleges and
universities uh in the US this one is a
book that you can use for such a course
it's called college algebra Essentials
and it was written by Julie Miller um
this course covers basic stuff like
polinomial logarithms rational functions
um you know quadratic formula systems of
equations and inequalities you can see
some of the topics here but not super
hard or anything but it can be if you
haven't had math in a long time for me
when I took this course many many years
ago I did struggle quite a bit and I
thought it was a challenging course many
math Majors do not like statistics and I
can relate as a student I also wasn't a
huge fan it wasn't until I started
teaching statistics that I really
started to appreciate how beautiful the
subject actually is this is a great way
to learn statistics with a book like
this this one's called understandable
statistics by brass and Brass I've
actually taught statistics from this
book many many many many years ago um so
great book it's got really intuitive
explanations you can actually learn this
um at the same time that you're studying
college algebra so you could do both at
the same time but you probably want to
have some Intermediate Algebra before
jumping into stats just so you have some
math experience um you know there there
are some math things in statistics it's
really really basic math but having some
Algebra I think is a good a good
stepping stone after you take college
algebra there are two courses you can
take you can take pre-calculus or you
can take trigonometry typically those
courses are taught using the same book
this is an example of such a book this
is the one by beer panaah and beninger
this is like many other books most books
on pre-cal and trig are excellent uh
this one is no exception it's also a
great book these books are modern they
have really good examples they have
answers to the odd numbered exercises
usually and they have great exercises so
you can use these to learn pre-calculus
and tricks you can take two courses with
one book which is really really amazing
this is going to be a prerequisite for
learning calculus at most universities
uh here in the US so after college
algebra again it's pre-cal trig if you
have to take them separately I took them
at the same time but if you decide to
take them
separately people always say which
should you take first I think maybe
pre-cal is a more natural transition
because you do a lot of the things that
you've already done in algebra you know
you do a lot of the same things you do
logarithms and exponentials again
although you do do new things like
matrices hyperbolas lipes and conic
sections and stuff like that so I would
say preal ventri but you can certainly
take them at the same time once you have
some algebra and you've taken pre-calc
trig you're ready to tackle the calculus
sequence so Calculus 1 calculus 2 and
calculus 3 in the US most calculus
courses are taught using one book and
that book is used to teach Cal 1 calc 2
Cal 3 this is a book that yeah I
actually have used this to teach uh Cal
1 Cal 2 Cal 3 uh even some Cal one
honors and this is a great book I have
done a lot of the exercises from this
book it's awesome there's other really
good books out there the one by Stewart
is also really good but I chose this one
for this video because I think this one
might be I don't know I don't want to
say it's easier than the steart book but
it might be a little bit more gentle and
so if you're thinking about self-study U
this this could also be a really great
choice why is there a physics book in
this video it's because physics is
important and I thought it would be
great opportunity to introduce it so
this is physics uh for students of
science and engineering and this one is
by the famous holiday and Resnik right
so this is a classic book I have a few
additions of this book and it's been
used by tons of people over the years to
to learn physics so you can take physics
or learn physics on your own after
you've had some calculus okay so my
advice would be definitely take calculus
before you learn physics
I somehow was able to take Cal 1 and
physics one at the same time and it made
my journey very very challenging so I
don't recommend it make sure you take
calculus before you take physics most
schools don't even allow that anymore um
so yeah take calculus then learn physics
um as a math major you usually take two
physics courses I took three it was
mandatory for my degree at the time so I
took physics one physics 2 and physics 3
and I liked all of them it was very
challenging and and very eyee openening
next up is differential equations and
notice there are two books here I just
wanted to talk about both of these very
briefly because they're both excellent
books so differential equations is
something you can take after you've had
calculus so once you've had calculus you
could actually just jump into to um
differential equations so this is the
one by uh Shepley L Ross and this is a
standard book they both are fairly
standard uh this one is perhaps more
standard it has all the topics that you
would expect to see in a regular
differential equations course at the
undergraduate level I've taught
differential equations I've taught I've
taught it many many times and so this
content here let me just say is very
standard these are things that you'll
see in pretty much every differential
equations course um across the US in a
college so great book for self-study
great as great as a supplement you can
learn differential equations with this
book now some people think it's better
to take linear algebra before you take
differential equations and and I can see
how there is value in that in fact when
I've taught differential equations I do
teach some linear algebra in the class
that's in the book already and that's
the point a lot of the linear algebra
you need is already in the book but if
you wanted something with an extra
emphasis on linear algebra there's this
book here this is the one by Edwards and
Penny and I bought this book just for
fun and I thought it was very unique and
interesting compared to my other
differential equations books I would say
this one is extremely unique it's
probably the most unique one I have
at this level so if I compare this to
all the other de books I have which I
probably have like eight of them this
one is super different okay very very
different so yeah I recommend it also I
feel like it's a little bit harder so
you get some more interesting examples
and yeah I like it quite a bit learning
to write proofs is extremely important
and this is an appropriate time to begin
learning that so here we have two books
we have one on discret mathematics this
is the one by Susanna EP this this is
probably the easiest discrete math book
ever written it's I think it's probably
the best one for beginners and then we
have how to read and do proofs by Daniel
solo this is a proof writing book I've
had this for quite some time I haven't
talked about it much for some reason so
I thought I should include some books in
this video which I haven't talked about
too much um this is a book you can buy
for self-study and use it to learn to
write proofs highly recommend it what
happens is in colleges there's usually a
course you take it's called something
like introduction to proof writing or
introduction to higher mathematics it's
some type of transition course and
typically it will use a book like this
one here at the same time there's a
course called discrete mathematics
that's offered usually by computer
science departments and computer science
students take it typically however if
you're a math major and you have the
opportunity to take discrete mathematics
I highly recommend you do it is worth it
I was fortunate enough that I was a
double major at the time I was kind of
like undecided I was going to do
computer science and math so I got stuck
taking both and it was a blessing in
disguise learning discret math and
learning how to write proofs at the same
time made my proofs extra strong it
really really did both classes kind of
focused on different things and I was
able to just get a lot better at
mathematics in one semester uh by
learning both of these subjects so I
definitely recommend both of these and
again this is not typically a course
that math Majors will take but I highly
recommend it discret math by the way
covers counting probability
um recurrence relations set logic proofs
uh you do a little bit of graph Theory
so yeah it's pretty cool and this here
is just a book on proof writing so it's
got different types of proofs from
various areas of of mathematics but yeah
definitely worth learning both of these
now that you know how to write proofs a
really good place to start with that is
linear algebra now this is a beginner
book on linear algebra this one is
called Elementary linear algeb algebra
and it's the one by lson and Edwards
there's many other good linear algebra
books I just wanted to show you this one
because this one is good for beginners
so it starts with systems of linear
equations matrices determinants so all
the basic stuff that you would expect to
learn in a linear algebra course now the
proofs are light but they're still here
and so if you really want to learn
everything um you should know how to
write proof that's why I wanted to wait
until proof writing was introduced that
way you get the most from your
experience in linear algebra right
knowing how to write proofs is going to
make your experience much better this is
a great book it's got answers to some of
the exercises great exercises and great
examples so yeah linear algebra here we
have three different books for three
different math subjects and so we're
going to go through each and talk about
the PRX for each subject so first up we
have this one here this is complex
variables and I chose this book because
this is one that I haven't talked about
too much it's by Steph D fer in order to
learn complex variables you should know
how to write proofs and you should know
some calculus which you already know so
you should be good to go so if you've
made it this far in theory you can jump
into a book like this and learn complex
variables you'll find that when you're
studying comple comp Lex variables it's
kind of refreshing because a lot of the
things that you've learned and a lot of
the rules from calculus that you've
learned still apply in the complex
valued case for example the series tests
are very similar uh the rules for
differentiation are very similar the
rules for limits are similar there are
some differences and when you learn what
those differences are it's really kind
of exciting so yeah great course
shouldn't be too hard as long as your
teacher is not like super super hard I
had a friend who had a really hard
teacher and was really really hard for
him um I had a hard teacher but my
teacher was good so I think that was
that was the difference this is a book
that you can use to take courses in
college known as mathematical statistics
or statistical Theory uh I took um two
courses with this book it's extremely
challenging uh what makes this hard is
that one there's proofs so if you don't
know how to write proofs it's going to
be kind of challenging the proofs aren't
that hard and hindsight I should have
been able to handle them I mean they
were really easy proofs but I was just
learning and so I had a really really
hard time in fact I had such a hard time
that the first time I took this class I
withdrew after the first test um I was
taking Advanced calculus at the same
time and it was too much which we
haven't gotten to Advanced calculus yet
so but yeah very challenging course and
it's one that I know math Majors don't
look forward to but it's worth it uh I
think that it is quite an interesting
subject and then here we have partial
differential equations this is the book
by stuss so basically to learn this you
just need to know some calculus and it
helps to know how to write proofs you
know the more math you know the better
uh for sure and I think the teacher is
going to make uh a big difference in all
of these courses right in all of these
courses I'm sorry I just got to give it
a whiff here just calling me ah smells
smells amazing so pdes or partial
differential equations uh is a course
that you could take after you know
calculus and obviously after
differential equations one of the cool
things about this course by the way is
that a lot of the techniques that you
learned in differential equations you
use them again in this course okay let's
keep going three more math books except
this time the subjects are more
challenging let's start with linear
algebra again that's right this is the
second time we're seeing linear algebra
this time this is a proof-based course
you would be taking in college so
typically most colleges offer two
courses on linear algebra an intro
course and a harder course so this is a
famous book that was used many years ago
for that harder course at many good
schools it's called linear algebra it's
by Hoffman and cuns and this is my copy
of that book and it's a classic now
there are more modern books you can get
I usually recommend the book by
Friedberg but I thought for this video
let me just show you one of the classics
from my collection you could certainly
still use this book to learn and I could
smell it from where I am and I'm pretty
far away actually um and I can actually
smell the book wow wow it just releases
this intoxicating odor hopefully your
copy uh is As Nice uh as mine next up is
abstract algebra and I chose this book
because it's a wonderful book it's great
for beginners this is contemporary
abstract algebra so in abstract algebra
you're going to study uh different
algebraic structures you're going to
look at groups rings and Fields uh maybe
modules but probably not as an
undergraduate mostly uh group Theory and
maybe a little bit of ring Theory some
schools offer two courses on abstract
algebra uh as as an undergrad oh look
there's Paul Erdos oh oh oh sorry Paul
Erdos is a socially helpless Hungarian
who has thought about more mathematical
problems than anyone else in
history oh he is not socially helpless
or maybe he was yeah Paul Eros was a
very very
um unique unique person anyways abstract
algebra is a great subject it was
probably my favorite subject and
probably still is um I think it's
beautiful there's a lot of really cool
stuff you can learn in abstract algebra
and this is a really good book so
yeah Advanced calculus also known as
real analysis so this is typically
considered to be like the hardest course
for math Majors people really struggle
with this you you start with like
sequences and then it goes from there
basically it's all the calculus stuff
that you've already seen except you
prove it again so you might say you've
already seen it why is it so hard it's
just dealing with the inequalities and
you know getting used to this these
types of proofs takes um takes some work
right takes some work this is an
extremely challenging course for people
so this book by the way is pretty good
it's called Advanced calculus and it's
the one by Fitzpatrick so and you can
take two courses typically I took two
courses
uh with this book as an undergrad um I
got a in both but it was it was uh it
was very very
challenging no joke apology is
considered by many people to be an
extremely challenging subject and it is
and I wanted to leave it for now in the
video because you do need to see a lot
of mathematics before jumping into
topology in particular your proof
writing needs to be really really solid
you should be able to prove all of the
basic things surrounding sets quickly
and correctly or on the spot you know
just to verify that they're true you
should be very very comfortable uh
manipulating sets and set theoretic
proofs and that will help you
tremendously when you jump into
something like topology I mean the more
math you have the more proof writing you
have will help you I took topology my
last semester and I thought it was great
however many of my classmates did not
think it was great and the big
difference between me and my CL
classmates was that many of them were
not seniors they were Juniors and they
were taking topology at the same time as
advanced calculus I think in some cases
so very challenging to do that um
definitely take analysis first if you
can and leave topology to the end so you
can get the most out of it okay that
that's my advice this book by the way is
pretty good it's uh it's a legendary
book it's the one by Monk Reese the only
thing I don't like about this book is
that it doesn't have answers
but otherwise it's like the perfect
apology book here we have four more
books for three more subjects so these
subjects can kind of fit in in different
places throughout the sequence so for
example number Theory here you would
want to learn how to write proofs before
taking number Theory so here we have the
book by rosin and here we have the book
by long both of them are great number
theory is typically not a required
course for a math degree so I kind of
wanted to leave this to the end um but
it's still great if you can take it take
it um if you don't take it I don't think
it's the end of the world I think other
courses like Advanced calculus are much
more are much more important to take uh
as a math major you know if you're only
going to take a certain number of
courses uh definitely take analysis over
number Theory but great course you just
have to know how to write proofs and
then differential geometry definitely
you have to know how to write proofs and
I would say say take a course on
Advanced calculus before jumping into
differential geometry if your school
even offers it uh many colleges don't
even offer uh differential geometry um
or maybe they offer it as a grad level
course sometimes so again not something
that's required typically for a math
degree but I wanted to include it and
also just to show you this book CU it's
so pretty and good-looking it's the one
by oslander combinatorial Theory so this
would be a course called combinator at
your local College I took a course that
was a combined course it was called
combinatorics and graph Theory and I
used a discret math book kind of like
the one by ep um to take that course so
I I saw both some combinatorics and and
some graph Theory this is an old school
book this is the one by Hall and it's
just on combinatorial Theory which is
the theory of counting and you can see
how the book starts let me just show you
really quick it starts really basic um
talks about uh permutations of the very
beginning there a permutation is an
ordered selection of objects from a set
s yeah so it starts off pretty basic and
accelerates quite rapidly in this in
this classic book so other subjects that
you can kind of take that we kind of
missed throughout the sequence so that
is the best course sequence for math
Majors hopefully after watching this
video you kind of know what to expect if
you decide to study mathematics or if
you're already studying mathematics you
have some ideas for other books that you
can use to supplement your learning or
maybe you're just thinking about doing
self- study and now you actually know
what the courses are that math Majors
actually take I will leave links in the
description of this video to all of
these books also if you want to learn
math I have courses on udemy but use my
links please from my website math
sourcer tocom and if you found any value
in this content feel free to hit
subscribe until next time keep doing
math
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