Best Course Sequence For Math Majors

Summary

This 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

00:00

📚 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.

05:00

🔢 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.

10:04

📈 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.

15:04

🌐 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.

20:06

📘 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.

Keywords

💡Math Major

A 'Math Major' refers to a student who has chosen mathematics as their primary field of study in college or university. In the context of the video, the script discusses the sequence of courses that a Math Major would typically take. It is central to the video's theme as it outlines the educational journey of someone specializing in mathematics.

💡Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It is foundational to higher mathematics. The script mentions 'intermediate algebra' and 'college algebra' as early courses for a Math Major, indicating the progression from basic algebraic concepts to more complex ones.

💡Calculus

Calculus is the study of change and motion, encompassing differential and integral calculus. The video describes a series of calculus courses (Calculus 1, 2, and 3) as pivotal in a Math Major's curriculum, following pre-calculus and trigonometry, and serving as a prerequisite for more advanced studies.

💡Statistics

Statistics is the discipline that deals with the collection, analysis, interpretation, presentation, and organization of data. The script notes that many Math Majors do not like statistics, but the presenter came to appreciate its beauty through teaching, highlighting the subject's importance and potential complexity.

💡Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. The video mentions trigonometry as a course that typically follows algebra and precedes calculus, emphasizing its role in the Math Major's sequence of courses.

💡Differential Equations

Differential Equations are equations that relate a function with its derivatives, used to describe rates of change. The script identifies differential equations as a course that comes after calculus, illustrating the progression from studying change in calculus to solving problems involving change in differential equations.

💡Linear Algebra

Linear Algebra is the study of vector spaces and linear equations. The video discusses two courses in linear algebra, indicating its importance in a Math Major's education. It first appears as an introductory course and later as a more advanced, proof-based course.

💡Proof Writing

Proof writing is the process of creating a logical demonstration of a mathematical statement. The script emphasizes the importance of learning to write proofs, mentioning it as a prerequisite for more advanced courses like discrete mathematics and linear algebra, and as a key skill for a Math Major.

💡Discrete Mathematics

Discrete Mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. The video describes discrete mathematics as a course covering topics like counting, probability, and graph theory, which is beneficial for Math Majors and computer science students alike.

💡Physics

Physics is the natural science that studies matter, its motion and behavior through space and time, and the related entities of energy and force. The script suggests that physics is an important subject for Math Majors, especially after having studied calculus, and it can be a challenging and eye-opening experience.

💡Abstract Algebra

Abstract Algebra, also known as modern algebra, is the study of algebraic structures such as groups, rings, and fields. The video mentions abstract algebra as a subject that the presenter particularly enjoyed and found beautiful, indicating its significance and the level of abstraction in a Math Major's studies.

💡Real Analysis

Real Analysis is the study of the real numbers and their properties from a rigorous, mathematical viewpoint. The script describes real analysis as a challenging course, often considered the most difficult for Math Majors, involving proving calculus concepts rigorously.

💡Topology

Topology is the branch of mathematics that deals with the properties of space that are preserved under continuous transformations. The video positions topology as a course best taken after real analysis, emphasizing its complexity and the importance of solid proof-writing skills.

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

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