## Math GRE study books

Forum for the GRE subject test in mathematics.
mani_fold
Posts: 50
Joined: Wed Jan 08, 2020 10:49 pm

### Math GRE study books

I'm writing a little reflection in my profile post on math GRE. Thought I'd make a separate post with my favorite studying/training books and link to it in my profile reflection. Please feel free to add.

Background: I mostly do analysis and tutor calc/diffeq/linear so my background is weak in topology/geometry/algebra [hence my 71]. But I think I have good suggestions on those fronts even if I didn't exhaust those resources.

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• Calculus & ODEs Stewart is a good resource. Key sections are 7.5 and 11.7; make sure to do all of the problems in those sections once or twice. For ODEs, I like Schaum's Outline of Differential Equations by Bronson since it condenses the material and has a ton of problems.
• Analysis A Problem Book in Analysis, Aksoy & Khamsi; Understaning Analysis, Abbot.
• Algebra Easy problems in A First Course in Abstract Algebra by Fraleigh. Harder problems in Abstract algebra, Dummit \& Foote. For linear algebra, study Linear Algebra: Challenging Problems for Students by Zhang.
• Combinatorics Two types of combo should be studied. First is enumerative combinatorics; see the associated chapter in Problem Solving Strategies by Engel. The other type is graph theory, see Graph theory: a problem oriented approach by Marcus (great book).
• Probability For a gentle introduction, see A first course in probability by Ross. Some good problems, especially at the beginning over counting/discrete probability.
• Geometry & Topology For Topology, I like Elementary Topology Problem Book by Viro, Ivanov, et al.. For geometry, I like Euclidean Geometry in the Mathematical Olympiads by Chen (IMO god). Be warned: those problems are hard as ____, but a few hours spent in the first part attempting them pays off.
• Number Theory The number theory section of Problem Solving Strategies by Engel is a great resource. Very accessible and challenging. Also there's an old one by Minkowski(?) that I don't remember the title of.
• Statistics Don't rule out a random stats question on this exam, but again, don't try to master something new; some good general stuff can be found in Schaum's Outline of Statistics by Spiegel.
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~ G R E A T B O O K S ~
• Linear Algebra: Challenging Problems for Students, Zhang
• Problem Solving Strategies, Engel
• Elementary Topology Problem Book, Viro, Ivanov, et al..
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Some people will say YOu ShOUlDn'T sTuDY fOr tHe MAth GRE or that It'S A supER easY test aNyWAY, so don't be like those people ¯\_(ツ)_/¯

wth961209
Posts: 16
Joined: Sat Jan 11, 2020 12:28 am

### Re: Math GRE study books

https://www.math.ucla.edu/~chparkin/gre.html is another good source for review

Cyclicduck
Posts: 105
Joined: Sun Apr 14, 2019 9:55 pm

### Re: Math GRE study books

This is ridiculous...all those books are on a level far far past what's on the gre.

SmallRedBird
Posts: 6
Joined: Sun Mar 01, 2020 1:24 am

### Re: Math GRE study books

Cyclicduck wrote:
Sun Mar 08, 2020 11:05 am
This is ridiculous...all those books are on a level far far past what's on the gre.
Because everyone, including myself, is too scared to be anything less than prepared-for-anything for every topic regularly covered. But I sort of like this way of grinding tedious problems for sake of improving computational efficiency, especially for calculus/diff eq. I've heard the rumor of the test getting noticeably "more difficult" since the last publicly released practice test, so hopefully, it's not all as ridiculous as it seems.

Edit: https://math.uchicago.edu/~min/GRE/ I believe to be a great source

Cyclicduck
Posts: 105
Joined: Sun Apr 14, 2019 9:55 pm

### Re: Math GRE study books

SmallRedBird wrote:
Sun Mar 08, 2020 7:28 pm
Because everyone, including myself, is too scared to be anything less than prepared-for-anything for every topic regularly covered. But I sort of like this way of grinding tedious problems for sake of improving computational efficiency, especially for calculus/diff eq. I've heard the rumor of the test getting noticeably "more difficult" since the last publicly released practice test, so hopefully, it's not all as ridiculous as it seems.
I took it in April 2019, so I believe I have a fairly up-to-date sense of what it's like.
mani_fold wrote:
Sat Mar 07, 2020 6:14 pm
Calculus & ODEs Stewart is a good resource. Key sections are 7.5 and 11.7; make sure to do all of the problems in those sections once or twice. For ODEs, I like Schaum's Outline of Differential Equations by Bronson since it condenses the material and has a ton of problems.
Sure there are a lot of calculus problems, so it makes sense to practice them. But for ODEs there are like three things you need to know how to do. Separate and integrate, second order factorize a quadratic, and multiply by \$\$e^x\$\$ sometimes. You don't need to go overboard.
[*] Analysis A Problem Book in Analysis, Aksoy & Khamsi; Understaning Analysis, Abbot.

I don't know these books, but you only need to know the most basic principles of analysis, eg what a limit is.
[*] Algebra Easy problems in A First Course in Abstract Algebra by Fraleigh. Harder problems in Abstract algebra, Dummit \& Foote. For linear algebra, study Linear Algebra: Challenging Problems for Students by Zhang.
You definitely don't need Dummit and Foote. All you need is the definition of a group and a ring. If you know those definitions and are able to play around with inverses and maybe know the FT on finite abelian groups that's enough. Linear algebra I don't remember seeing anything more advanced than knowing what the rank of a map is, or basic properties like multiplicativity of determinants.
[*] Combinatorics Two types of combo should be studied. First is enumerative combinatorics; see the associated chapter in Problem Solving Strategies by Engel. The other type is graph theory, see Graph theory: a problem oriented approach by Marcus (great book).
Sure some of Engel is really easy but most of it is way too hard. The combinatorics in the GRE is stuff you'll find in a 4th grade math competition. Try practicing those, or some MathCounts (a middle school competition) if you want to overachieve. I don't know of any graph theory on the GRE at all but if there is any it is surely not above the middle school level.
[*] Probability For a gentle introduction, see A first course in probability by Ross. Some good problems, especially at the beginning over counting/discrete probability.
I don't know this book, but again it will more than suffice to look over some middle school competition problems for counting and probability.
[*] Geometry & Topology For Topology, I like Elementary Topology Problem Book by Viro, Ivanov, et al.. For geometry, I like Euclidean Geometry in the Mathematical Olympiads by Chen (IMO god). Be warned: those problems are hard as ____, but a few hours spent in the first part attempting them pays off.
I don't know the topology book, but I'm sure that you can get away with only knowing the definition of open sets, closed sets, and a topology on the GRE. Evan Chen's book is completely, completely irrelevant and overkill for the GRE. Like, seriously.
[*] Number Theory The number theory section of Problem Solving Strategies by Engel is a great resource. Very accessible and challenging. Also there's an old one by Minkowski(?) that I don't remember the title of.
I don't really think there is any number theory on the GRE. Maybe you need to know how to find the sum of the factors of an integer or something or how to take mods, but I don't think there's any theory needed. Much of Engel is way too hard.
[*] Statistics Don't rule out a random stats question on this exam, but again, don't try to master something new; some good general stuff can be found in Schaum's Outline of Statistics by Spiegel.
Why would you read a stats book? Maybe you need to remember the definition of median. You don't need anything not taught in elementary school.

These are probably good books in general but they aren't really geared for the GRE. I think practicing things like the AMC 10 (a high school contest under similar time restrictions) will be better for improving your test taking skills and score.

On the other hand, you probably should know everything listed in those books anyway. So I wouldn't advise against using them to patch up any holes; just not for the GRE.

mani_fold
Posts: 50
Joined: Wed Jan 08, 2020 10:49 pm

### Re: Math GRE study books

Cyclicduck wrote:
Sun Mar 08, 2020 11:09 pm
...
¯\_(ツ)_/¯

CoronalRain
Posts: 63
Joined: Sat Jan 26, 2019 1:02 am

### Re: Math GRE study books

Thanks, mani! I appreciate you taking the time to put this together, even though some may think certain books are overkill. Every little bit helps.