12 ms·

Mathematics for Computer Science

395 points by moks 12y ago

UUMMUU 12y ago
I know it's not related to this but if anyone has not watched the Linear Algebra lectures by Gilbert Strang of MIT. They are definitely worth a look as a strong foundation in Linear Algebra never hurt anyone. (http://web.mit.edu/18.06/www/videos.shtml http://web.mit.edu/18.06/www/videos.shtml) Also available on iTunes U.
- tariqr 12y ago
  [Demo] There is also collection of Gilbert Strang lectures here: http://classmill.com/jennathompson/introduction-to-algebra http://classmill.com/jennathompson/introduction-to-algebra - You'd have to login to watch the whole series, but lets you mark lectures as "done", ask questions, and track progress etc..
- Niksko 12y ago
  Thanks for the resource. I never really paid attention when I took linear algebra, and I've always regretted it since it seems to rear it's head everywhere.
  - UUMMUU 12y ago
    I was the same way. I went through it thinking it was very abstract and had no real application. It wasn't until after that I realized it was basically used (in some way/shape/form) everywhere.
farresito 12y ago
The reading material is fantastic. If anyone wants a more recent version of it, here's the link: https://courses.csail.mit.edu/6.042/spring15/mcs.pdf https://courses.csail.mit.edu/6.042/spring15/mcs.pdf
- alphazerozero 12y ago
  Are there companion solutions for the problem sets available anywhere?
  - farresito 12y ago
    Take a look here: https://stellar.mit.edu/S/course/6/sp15/6.042/ https://stellar.mit.edu/S/course/6/sp15/6.042/
    - enjoylife 12y ago
      Seems like you need a MIT certificate.
      - farresito 12y ago
        That's unfortunate. Oh well...
- discardorama 12y ago
  That's not just "reading material" ... it's a frigging 1000-page book! :-D
  - farresito 12y ago
    It all started with a 300 page reading notes, which, in my opinion, is pretty long already, but, boy, 1000 pages is a lot indeed :).
  - mjcohen 12y ago
    Easier to search than the videos.
- ralmidani 12y ago
  Thanks for that link! It looks like it was just updated today.
- dxbydt 12y ago
xigency 12y ago
This seems extremely useful. My school did not have a consolidated math class like this, although computer science and software engineering students were required to take various courses in math, particularly discrete and combinatorial math and linear algebra. I do know that there were many students who double majored in Math / Computer Science although there was a restriction on minoring in mathematics as a CS student because of the large number of prerequisites. Still, it seems useful to combine proofs, induction, graph theory and related domains into one course. I did opt for taking a large number of math classes and some fairly high-level electives, and I have to say the most interesting area I came across was the use of generating functions for solving recurrence relations and the study of the Catalan numbers. For those of you unfamiliar with the Catalan numbers, it is the sequence of the number of binary trees that can be created with a certain number of elements. Another area they come up in (among a large number of problems) is creating paths by Manhattan distance. The Catalan numbers also count the number of expressions with matching pairs of parentheses.
- _asummers 12y ago
  Catalan numbers show up all over the place and are a really useful sequence. My combinatorics course in undergrad had a "Catalan Problem of the Week" where you would have to show that the Catalan sequence solves a particular problem.
vowelless 12y ago
This is a similar course with some overlap: http://www.cs.cmu.edu/~15251/ http://www.cs.cmu.edu/~15251/
- aroman 12y ago
  I haven't looked at the MIT course in detail, but doesn't it seem like Concepts would be more similar to this than 251?
hkailahi 12y ago
MIT OpenCourseWare has been a great tool for me as an undergrad. Since the start of year, I've watched the CS and Econ uploads concurrently with the courses I've taken. It's not that the lectures are better, but getting a second chance to see the topics under slightly different contexts has been pretty effective for study. I've had the same experience with Khan Academy in my lower division courses.
dnissley 12y ago
I've been trying to self-study mathematics since around the new year. I initially tried following this course, but with only a background of high school math through precalculus I found this to be overwhelming. So after a bit of research I decided to start working my way through "How to Prove It" by Velleman, which has introduced me to many of the concepts that seem to be pre-requisites for this course: set theory, logic, proof strategies, etc.
- cdelahousse 12y ago
  Learning those topics will not be a waste of time. You'll see them come up time and time again on your path towards more CS topics.
- bargl 12y ago
  I did something similar last year with Statistics and Probability. The best place I've found for a linear course for online math is https://www.khanacademy.org/ https://www.khanacademy.org/ I was able to start at the beginning and get all the necessary math to actually step forward to some of the more complex courses. I'd suggest starting there.
  - dnissley 12y ago
    I did start there actually. As much as I love Khan Academy there are lots of holes and the quality/thoroughness of video explanations really seem to drop off the further along you go. To me they were most useful when I just needed a refresher on the precise operations to go through in order to solve a particular kind of problem, but already understood it. It's hard to beat a textbook for learning material that is completely new to me.
- cgag 12y ago
  I'm working through that as well, it's excellent.
- jonnybgood 12y ago
  This may also help you if you're going the autodidactic route: http://hbpms.blogspot.com/ http://hbpms.blogspot.com/
cdelahousse 12y ago
If some of you feel a bit overwhelmed with the material, you may be well served by watching the first few lecture's of the following playlist: https://www.youtube.com/channel/UCG96LXNYz9x7eTqSRtQ2R9A https://www.youtube.com/channel/UCG96LXNYz9x7eTqSRtQ2R9A It is Carleton University's freshman/first year Discrete Structures/Math course (COMP1805). They cover a lot of the same material, but at a more leisure pace and assume a lot less background.
- cdelahousse 12y ago
  Looking over the second part of the MIT course, I see that a lot of the same material is covered in this free textbook. http://cglab.ca/~michiel/DiscreteStructures/ http://cglab.ca/~michiel/DiscreteStructures/
cdelahousse 12y ago
Oh man. This course brings back so many memories of suffering through this material. But I can't emphasize how important this stuff is to know if you want to know anything significant in Computer Science. What helped me ace the the exam was doing a hundreds of problems. I couldn't answer them most of the time, so I'd need an answer key nearby to get unstuck. Drill, drill, drill. Just like you'd expect to drill for calculus, you can drill for this class. What I found helpful was Rosen's "Discrete Mathematics and Its Applications". The material wasn't the best, but the problems are excellent. You can buy an older edition of the book and its solution manual for next to nothing on used book sites like abebooks.com
- georgehm 12y ago
  > I couldn't answer them most of the time, so I'd need an answer key nearby to get unstuck. This happens with a lot of books out there. It would have been great if the authors of MCS gave solutions for at least some select questions.
carlsednaoui 12y ago
I'm looking to gain the math foundation needed do inch my way towards data science. Could anyone on HN recommend a good path to take? For simplicity's sake, imagine my math level is 0. Also, curious to know if anyone has good book suggestions or places with practice problems. Edit: This is the best list I've been able to find so far http://datasciencemasters.org/#math http://datasciencemasters.org/#math
- hevelvarik 12y ago
  I am self-teaching myself mathematics for a few years now and will suggest the following: Algebra and Trig, a subset of which is known as pre-calc. Don't short yourself on this because it is basic and you don't want to be struggling with it at the same you are struggling with the higher level stuff. I cannot recommend Sheldon Axler's Algebra and Trigonmetry highly enough. After much searching I found that and have been through it 3 times. Next Calculus: I recommend Gilbert Strang's text book which will take you through what in uni is called Caculus 3, 1 being intro, 2 being differential and 3 being integral. I in the midst of this book and so can't speak to further but my plan is to move onto Strang's linear Algebra. After that, where to go depends on particular interest but leaving any of that out, in my opinion, will just handicap your further studies. Moder education does a lot of things wrong but the standard math sequence in use everywhere isn't one of them imo.
  - carlsednaoui 12y ago
    Thanks so much for the recommendations, hevelvarik. Just to confirm, are these the right books? - http://www.amazon.com/Algebra-Trigonometry-Sheldon-Axler/dp/047047081X http://www.amazon.com/Algebra-Trigonometry-Sheldon-Axler/dp/... - https://open.umn.edu/opentextbooks/BookDetail.aspx?bookId=10 https://open.umn.edu/opentextbooks/BookDetail.aspx?bookId=10 / http://ocw.mit.edu/ans7870/resources/Strang/Edited/Calculus/Calculus.pdf http://ocw.mit.edu/ans7870/resources/Strang/Edited/Calculus/...
    - hevelvarik 12y ago
      Sure :) They are.
abhididdigi 12y ago
Being a CS student in my Bachelors I was able to spend sometime on Number Theory. But the lectures here blew mind.
daveed 12y ago
I just want to put a plug in for the video lecture portion of this. Tom Leighton is an excellent lecturer. There is plenty of electronic textbooks/courses, but I really do think he explains the course material in an engaging and lucid way.
deevus 12y ago
Having done an equivalent course in Australia I can say: despite being a struggle at the time to get my head around the content, in hindsight knowing this stuff is invaluable as you continue with your degree. Discrete Mathematics has been invaluable in Formal Languages (Turing machines), Algorithms and Data Security.
nimish 12y ago
Concrete Mathematics is also a good background.
- qntty 12y ago
  One of my favorite math books ever, but definitely not for the faint of heart. It's starts off slow but gets hard fast.
JAFTEM 12y ago
I believe this is a pre-req to CLRS too.
ilaksh 12y ago
Math is just manually running code in your head. Which is just an ignorant and wasteful obsoleted vanity, like remembering and performing a series of notes.
- oskarkv 12y ago
  Is thinking obsolete too?

395 points by moks 12y ago

I know it's not related to this but if anyone has not watched the Linear Algebra lectures by Gilbert Strang of MIT. They are definitely worth a look as a strong foundation in Linear Algebra never hurt anyone. (http://web.mit.edu/18.06/www/videos.shtml http://web.mit.edu/18.06/www/videos.shtml) Also available on iTunes U.

tariqr 12y ago
[Demo] There is also collection of Gilbert Strang lectures here: http://classmill.com/jennathompson/introduction-to-algebra http://classmill.com/jennathompson/introduction-to-algebra - You'd have to login to watch the whole series, but lets you mark lectures as "done", ask questions, and track progress etc..
Niksko 12y ago
Thanks for the resource. I never really paid attention when I took linear algebra, and I've always regretted it since it seems to rear it's head everywhere.
- UUMMUU 12y ago
  I was the same way. I went through it thinking it was very abstract and had no real application. It wasn't until after that I realized it was basically used (in some way/shape/form) everywhere.

farresito 12y ago

The reading material is fantastic. If anyone wants a more recent version of it, here's the link: https://courses.csail.mit.edu/6.042/spring15/mcs.pdf https://courses.csail.mit.edu/6.042/spring15/mcs.pdf

alphazerozero 12y ago
Are there companion solutions for the problem sets available anywhere?
- farresito 12y ago
  Take a look here: https://stellar.mit.edu/S/course/6/sp15/6.042/ https://stellar.mit.edu/S/course/6/sp15/6.042/
  - enjoylife 12y ago
    Seems like you need a MIT certificate.
    - farresito 12y ago
      That's unfortunate. Oh well...
discardorama 12y ago
That's not just "reading material" ... it's a frigging 1000-page book! :-D
- farresito 12y ago
  It all started with a 300 page reading notes, which, in my opinion, is pretty long already, but, boy, 1000 pages is a lot indeed :).
- mjcohen 12y ago
  Easier to search than the videos.
ralmidani 12y ago
Thanks for that link! It looks like it was just updated today.
dxbydt 12y ago

xigency 12y ago

This seems extremely useful. My school did not have a consolidated math class like this, although computer science and software engineering students were required to take various courses in math, particularly discrete and combinatorial math and linear algebra. I do know that there were many students who double majored in Math / Computer Science although there was a restriction on minoring in mathematics as a CS student because of the large number of prerequisites. Still, it seems useful to combine proofs, induction, graph theory and related domains into one course. I did opt for taking a large number of math classes and some fairly high-level electives, and I have to say the most interesting area I came across was the use of generating functions for solving recurrence relations and the study of the Catalan numbers. For those of you unfamiliar with the Catalan numbers, it is the sequence of the number of binary trees that can be created with a certain number of elements. Another area they come up in (among a large number of problems) is creating paths by Manhattan distance. The Catalan numbers also count the number of expressions with matching pairs of parentheses.

_asummers 12y ago
Catalan numbers show up all over the place and are a really useful sequence. My combinatorics course in undergrad had a "Catalan Problem of the Week" where you would have to show that the Catalan sequence solves a particular problem.

vowelless 12y ago

This is a similar course with some overlap: http://www.cs.cmu.edu/~15251/ http://www.cs.cmu.edu/~15251/

aroman 12y ago
I haven't looked at the MIT course in detail, but doesn't it seem like Concepts would be more similar to this than 251?

hkailahi 12y ago

MIT OpenCourseWare has been a great tool for me as an undergrad. Since the start of year, I've watched the CS and Econ uploads concurrently with the courses I've taken. It's not that the lectures are better, but getting a second chance to see the topics under slightly different contexts has been pretty effective for study. I've had the same experience with Khan Academy in my lower division courses.

dnissley 12y ago

I've been trying to self-study mathematics since around the new year. I initially tried following this course, but with only a background of high school math through precalculus I found this to be overwhelming. So after a bit of research I decided to start working my way through "How to Prove It" by Velleman, which has introduced me to many of the concepts that seem to be pre-requisites for this course: set theory, logic, proof strategies, etc.

cdelahousse 12y ago
Learning those topics will not be a waste of time. You'll see them come up time and time again on your path towards more CS topics.
bargl 12y ago
I did something similar last year with Statistics and Probability. The best place I've found for a linear course for online math is https://www.khanacademy.org/ https://www.khanacademy.org/ I was able to start at the beginning and get all the necessary math to actually step forward to some of the more complex courses. I'd suggest starting there.
- dnissley 12y ago
  I did start there actually. As much as I love Khan Academy there are lots of holes and the quality/thoroughness of video explanations really seem to drop off the further along you go. To me they were most useful when I just needed a refresher on the precise operations to go through in order to solve a particular kind of problem, but already understood it. It's hard to beat a textbook for learning material that is completely new to me.
cgag 12y ago
I'm working through that as well, it's excellent.
jonnybgood 12y ago
This may also help you if you're going the autodidactic route: http://hbpms.blogspot.com/ http://hbpms.blogspot.com/

cdelahousse 12y ago

If some of you feel a bit overwhelmed with the material, you may be well served by watching the first few lecture's of the following playlist: https://www.youtube.com/channel/UCG96LXNYz9x7eTqSRtQ2R9A https://www.youtube.com/channel/UCG96LXNYz9x7eTqSRtQ2R9A It is Carleton University's freshman/first year Discrete Structures/Math course (COMP1805). They cover a lot of the same material, but at a more leisure pace and assume a lot less background.

cdelahousse 12y ago
Looking over the second part of the MIT course, I see that a lot of the same material is covered in this free textbook. http://cglab.ca/~michiel/DiscreteStructures/ http://cglab.ca/~michiel/DiscreteStructures/

cdelahousse 12y ago

Oh man. This course brings back so many memories of suffering through this material. But I can't emphasize how important this stuff is to know if you want to know anything significant in Computer Science. What helped me ace the the exam was doing a hundreds of problems. I couldn't answer them most of the time, so I'd need an answer key nearby to get unstuck. Drill, drill, drill. Just like you'd expect to drill for calculus, you can drill for this class. What I found helpful was Rosen's "Discrete Mathematics and Its Applications". The material wasn't the best, but the problems are excellent. You can buy an older edition of the book and its solution manual for next to nothing on used book sites like abebooks.com

georgehm 12y ago
> I couldn't answer them most of the time, so I'd need an answer key nearby to get unstuck. This happens with a lot of books out there. It would have been great if the authors of MCS gave solutions for at least some select questions.

carlsednaoui 12y ago

I'm looking to gain the math foundation needed do inch my way towards data science. Could anyone on HN recommend a good path to take? For simplicity's sake, imagine my math level is 0. Also, curious to know if anyone has good book suggestions or places with practice problems. Edit: This is the best list I've been able to find so far http://datasciencemasters.org/#math http://datasciencemasters.org/#math

hevelvarik 12y ago
I am self-teaching myself mathematics for a few years now and will suggest the following: Algebra and Trig, a subset of which is known as pre-calc. Don't short yourself on this because it is basic and you don't want to be struggling with it at the same you are struggling with the higher level stuff. I cannot recommend Sheldon Axler's Algebra and Trigonmetry highly enough. After much searching I found that and have been through it 3 times. Next Calculus: I recommend Gilbert Strang's text book which will take you through what in uni is called Caculus 3, 1 being intro, 2 being differential and 3 being integral. I in the midst of this book and so can't speak to further but my plan is to move onto Strang's linear Algebra. After that, where to go depends on particular interest but leaving any of that out, in my opinion, will just handicap your further studies. Moder education does a lot of things wrong but the standard math sequence in use everywhere isn't one of them imo.
- carlsednaoui 12y ago
  Thanks so much for the recommendations, hevelvarik. Just to confirm, are these the right books? - http://www.amazon.com/Algebra-Trigonometry-Sheldon-Axler/dp/047047081X http://www.amazon.com/Algebra-Trigonometry-Sheldon-Axler/dp/... - https://open.umn.edu/opentextbooks/BookDetail.aspx?bookId=10 https://open.umn.edu/opentextbooks/BookDetail.aspx?bookId=10 / http://ocw.mit.edu/ans7870/resources/Strang/Edited/Calculus/Calculus.pdf http://ocw.mit.edu/ans7870/resources/Strang/Edited/Calculus/...
  - hevelvarik 12y ago
    Sure :) They are.

abhididdigi 12y ago

Being a CS student in my Bachelors I was able to spend sometime on Number Theory. But the lectures here blew mind.

daveed 12y ago

I just want to put a plug in for the video lecture portion of this. Tom Leighton is an excellent lecturer. There is plenty of electronic textbooks/courses, but I really do think he explains the course material in an engaging and lucid way.

deevus 12y ago

Having done an equivalent course in Australia I can say: despite being a struggle at the time to get my head around the content, in hindsight knowing this stuff is invaluable as you continue with your degree. Discrete Mathematics has been invaluable in Formal Languages (Turing machines), Algorithms and Data Security.

nimish 12y ago

Concrete Mathematics is also a good background.

qntty 12y ago
One of my favorite math books ever, but definitely not for the faint of heart. It's starts off slow but gets hard fast.

JAFTEM 12y ago

I believe this is a pre-req to CLRS too.

ilaksh 12y ago

Math is just manually running code in your head. Which is just an ignorant and wasteful obsoleted vanity, like remembering and performing a series of notes.

oskarkv 12y ago
Is thinking obsolete too?