# This is what a pure mathematics exam looks like at university

Today I’m going to take a look at a pure mathematics exam from University. Now pure mathematics deals with mathematics that is more abstract in concepts rather than applied. That doesn’t mean that pure mathematics doesn’t have applications, but it means you’re dealing with things like the theory behind numbers and functions and the abstract nature of pure mathematics makes it quite difficult. I will say that I took pure mathematics throughout my physics education, not only because I did a double major in math, but because it is useful for physics. Now the exam I have here is a real exam that would be taken I think your second or third year of university study, that’s undergrad. It would take you three hours to complete and in this booklet there are eight questions. Four of them deal with real analysis and four of them deal with complex analysis. The difference is that complex analysis deals with thinking about the theory behind functions in the complex pl ane. So we’ve got a bunch of i (the square root of negative one). We’ve got a bunch of is in the working. This particular exam paper I have is from the University of Manchester because they uploaded all of their exams for the public to see so shout out to them and let’s have a look at it. Even though there are eight questions in this exam you’re only going to have to answer five of them, so it wants you to pick two questions from real analysis, two questions from complex analysis and then the last question you choose to do is going to be up to you. Because of that this exam is really quite long, so it will take you much longer than three hours if you were going to try and attempt all of the questions. So let’s look at the first section, which is the real analysis section. So real analysis deals with the properties of real numbers and functions. You want to understand the theory behind mathematics and ideas like sequences, convergence, limits continuity and smoothness. So the first question we have is a question about limits. You’re asked to find essentially or show that the limit of a certain function is a certain number or a certain limit and you do that with various techniques. I won’t go into exactly the details of how you would do it, but using the epsilon-delta definition you want to show the limit. So essentially you just want to take x to be really really close to a certain value and see what the entire function essentially is bounded by or what the limit is. Next we have the intermediate value theorem, now the intermediate value theorem talks about if you have two points connected by a continuous curve. You can actually draw what we’re working with here… you’ve got an axes, you’ve got a point A and you’ve got a point B. If that’s a continuous curve and you’ve got say a certain line You’ve got one point below the line and one point above the line, then at least one place on the curve is going to cross that line. So essentially you need to cross a certain line to get from point A to point B. That’s what this theorem is talking about and then you’re asked to prove some things kind of related to it. Question part 2 down here is talking about what’s probably quite a familiar idea in calculus. Assume that a function f is differentiable on a certain range [a,b] with a local maximum at c. Prove that the derivative of the function at point c is 0. So really you’re proving that if a function has a local maximum, that the derivative at that point is 0 and that’s an idea that underpins a lot of calculus and optimization problems, but you’re not often asked to prove it. We’ve got another question here that kind of deals with calculus ideas. We’ve got being asked for a proof of the product rule for differentiation and a few other things about differentiation, functions and limits. Our last question in the real analysis section is this one here. I probably won’t go into this one very much. You can look at the kind of language that these questions use it’s quite abstract and when I took this for my own real analysis course it was definitely my least favorite math course and it’s because there’s so much new language involved that it’s just really hard to talk about or to think about, well that’s what I found. It’s really hard to describe to other people as well so when I was always complaining about real analysis I didn’t quite know how to tell people what it was about. it’s the theory of mathematics essentially. But yeah, it’s also very hard to Google questions like this because there are so many symbols, not a lot of language. Moving on to Section B. This is the complex analysis stuff. Now I actually really like complex analysis in comparison to real analysis because complex analysis seems to have more applications. Complex analysis I have actually seen used in physics and thermodynamics, and fluid mechanics and essentially like I said before, we’re extending the real functions like exponentials, logarithms and trigonometry into the complex domain and range. So we’ve now got functions that can be separated into real and complex parts. This first question deals with something called the cauchy-riemann theorem and essentially it’s a set of partial derivatives that can be used to test if a function is complex differentiable. That means that the derivative exists, it’s also called holomorphic if that is true. These questions deal with being asked to show that certain functions are holomorphic and why, proofs surrounding that really. Number two in the complex analysis part, this is dealing with mainly definitions of trig functions in the complex plane so you can see the definition of sine there using complex numbers. You can see it’s being used with a definition that includes hyperbolic sine and cosine so some of these definitions of trig functions you might not have seen before if you haven’t done much work in the complex plane. These questions are actually I think the most straightforward in this exam paper, but they still look pretty alien if you’re not used to definitions like this. Now we’re getting into my favorite part of complex analysis, and why I like it. It’s because there are some really cool applications to integrals using complex analysis and they don’t sort of strike you as being obvious, but once you start to learn about them, I think they’re pretty cool. So essentially here, we have this big crazy figure and what we’re doing is taking integrals around certain paths. Now usually or often I guess certain integrals are quite hard to do normally, but there’s this thing that you learn about in complex analysis where you can take an integral that’s usually quite difficult to do, you can describe it as a function, take that function and transfer it into the complex plane, do some crazy maths with things like the residue theorem, Cauchy theorem, work out the integral in that complex domain and then convert it back into the real space where you end up with the answer that you wanted all along. And the last one is a similar vein this is using residue theorem which again allows you to carry out integrations in the complex space that otherwise would have been quite difficult just in real space. I have included this entire exam in the description so you can have a good read of it there. I know it’s probably quite hard to see the details here, so have a look, I haven’t provided solutions, that’s because this is actually a real exam from the University of Manchester website, but hopefully that gave you a good insight into what some pure mathematics looks like at university. This by no means is the most advanced you’re going to do, like I said, this is probably second to third-year work. It might not necessarily be combined into one paper I personally did courses where real analysis and complex analysis were separated into two different courses. But yes second to third-year levels probably about right. There is definitely more advanced work than this. Hope it didn’t scare you off. Thanks for watching.

The only thing I understood was that this was a maths paper from the University of Manchester

As soon as you flipped from the first page I got some serious Vietnam flashbacks, I dunno how I came here but I am out of h

I have a PhD in Sociology. The closest I get to that is Statistics and statistical analysis, which is…… a beast of its own. Kudos to you and those alike

I am not joking but 5 of the questions there were directly from my intermediate(+1,+2) text book of our state(Andhra Pradesh, India)

My nightmare

When your professor says your test will take 3 hours, but you finish eating it in 2 minutes.

Im studying Computer Science at University of Crete (Greece) and we do these things the first year..

I just recently graduated high school and I’m going to work on earning a physics degree starting this fall in college! This video is both interesting and frightening. The math is going to beat me up lol

can Terrence Tao do this?

I know it seems really tough, but a lot of the difficulty in maths at University comes from the layers of jargon and terminology. If you deconstruct the vocabulary, and you're familiar with the processes necessary it's not really any more difficult than algebra or geometry in highschool.

Questions A1-A4 was quite easy in my high school. I’m Asian.

Is it ok if I can solve the rolle's theorum questions while being at the last year in high school?

They hardly give in class exams in higher levels because it is pretty silly and by no means it judges a mathematicians skill. We usually get take home that are like ten pages and it's pretty obvious if you copy or cheat.

I get to sit this in 2 weeks woohoo

Guys don't get discouraged watching this video. Of course, you can't compare this level of maths to what you do in A-levels. Doing mathematics in university means that you solely focus on maths only!

I mean what do you except? I actually expected worse than this.

Mathematics is a language of it's own after all. So expect to communicate more in it. If you decide to take your mathematics skills on a higher level that is.

I understood the first ones but after that😮😮

What am I even doing here

i have learnd them by myself no teacher only book

This is 12th grade maths

–

A JEE aspirant

Mom come pick me up I’m scared (*´ー｀*)

I was watching this video with my dog. For some reason when we finished we both were wearing monocles.

What book do you recommend for undergrad math real analysis ll? Could you please post exam questions for real analysis ll or abstract algebra if you have it.

Me 0:30 : She is cute.

Me: 1:00 : She is Sweet.

Me 1. 35 : ok. I m done with the vid

id like to use my brain to go places by thinking on my own, rather than thinking the way someone else tells me to think. Ive always been very anti school.

When are we going you use this in the real world. Except maby building a rocket or bomb

Which one is the hardest to learn?! calculus or algebra.

Right now I’m still learning algebra in college.

Is there like a rave going on in the room above you lmao

5 question 3 hours wtf here in India 30 question 3 hours and no calculators are allowed

and im here thinking GCSE maths is hard😭😭

LOL I'm 17 years old High school student in Greece and most of that was in my curriculum this year.

I completed these whilst watching this post .

Ok… so this is how exams look like in the uk 🤔

I guess even a postgraduated student won't even pass high school in my country 😕

This exam looks surprisingly doable(as an engineering student).

Fake and staged

Bruh I don’t understand how these are even questions

ok 🙂

The questions at the beginning are hard but for some reason I know the gist of is despite only taking high school mathematics. Although i wouldn’t get it right because I’m shit at it I would know how to approach it

Adderall

Triggered seeing that in france half of the exam is what we do in high school…

We should hand yall a Fluid Mechanics + Thermodynamics exam and see how u like it. Just when you think you're right, bang u fked up the whole thing…

Yep I'm 16 years old and I'm terrified..

I think you should do ASMR videos.🤣🤣

Check iit bombay exm.

I know its probably way easier but our midterms and finals in integral calculus are like that where the questions are very long and you only need to answer specific amount because it’ll take you longer to finish all

I can barely do times lol

I also study in UoM.😃😃 major in Atmospheric science

niceeeeee

love it

Aaaand that's why I'm stopping at Alevel

Reminds me of the intro to sections in Calc 3 and DiffEQ classes that they don't test you over.

Dmn. That's a very beautiful accent. I wish I have little bit of British accent. It's just so cool

There are times when I just chuckle and smile ear-to-ear after reading the first question in an exam at uni

For part 1 of that exam, I did all of that stuff in calculus (delta-epsilon proofs, IVT, etc). We had to do delta-epsilon proofs

beforewe got to differentiation.How manny marks do you get for writing just your name.

Its funny how i was having fun answering those seriously on class

After a year i feel like the second dumbest person in class

I’m a lit grad so literally no idea what’s going on here. But is it possible for some super smart person to look at these questions and know the answer instantly, or do all these questions have to be worked through?

Such trivial

Don’t know why I’m watching this, I didn’t even pass gcse maths

Teacher : pff MUHAHA its ez idiot

Me : :c

it happened to me:/well this is what i did in highschool… back here in africa we do up to calculus 4

Aaaahhh, yes.

The dark tongue of Mordor.

It's been a while, and I'm still feel stupid as shit.

Me( on looking video is about math): bbye,seeya next video.

Meanwhile there is me who got 4 at math and say to my parent i got 8 at math

It's a nightmare

Can a someone with a 100 IQ even do this

WTF I know how to solve some of those questions by youtube tutorials I've seen

I like her and that Schrodinger cat in the back

Estimated time: 3h

Me as a 13 year old maths nerd who's been literally asking his dad (a maths professor at an university) to give him the exams for the students who study complex math there: lmao so ez done

me:

doesn't understand anythingalso me:

continues watchingIn Greece we do these maths at the age of 16,17,18 the 3 last school years they are not even hard to be honest and at the university it gets a lot harder

Rolle's theorem … yaaayy!

This is like IB hL math 1. Not too bad…people with BC calc can do this too I think. Idk tho

What a 🤓

People don't watch Yt for exams

Is she the main character at Nymphomaniac??

after finishing intro to real analysis 1 and 2, i feel much better about how I did after seeing the real analysis portion of this exam. those questions were much more like homework level questions than test level questions. thanks for the confidence boost i guess haha

Math!

Oh no!!! not again

😞

I leave C.S for this kind of things

i'm definetely not going to take math class when i'll be in highschool

looks like 12th std maths…… CBSE india

The link in the description doesn't work

This is pretty much what we had to do at our first analysis course. At the time it was tough, but if I look at it now most of these questions are pretty easy. It's just a matter of getting used to the language of maths and then it's not so scary anymore. Also you've just gotta work hard and practice a lot of similar problems beforehand. If you did that then tests like these are very doable.

I felt Like section A was too easy for me .. …But section B was Too scary

I have PTSD from doing exams just by watching this video

Someone can suggest me calculus book for best teaching .

High School to College level.

Please please help me

Can you make videos on the higher mathematics , I have problems regarding concepts if u can make it will be of great help

Please share something about how to study pure mathematics ☹

I want to download the exam but the link in the description doesn't work. Does someone have the exam?

An aussie woman that is also not a basic white girl>? I am taken back wow

Can you make a video on laplas transform and from where laplas transform comes ?

Proof of laplas transform equestion😢??

Plzzz

Lmao Riemann sum got me thinking hard 😂

Good video ❤️

Your happy demeanor is a breath of fresh air from stem.

I'm a medical student who's watching this and thanking God for not studying math 🤣🤣

Coochie-reimann theorem

I like your golden record poster. I have a replica golden record in my home as well.

Anyway, I saw you on physics girl, but didn't realize you majored in math until I searched for "columbia univsersity math major". I suppose you didn't go to columbia, but you still came up.

I became nervous when you flipped over the first page.

you hsould do a collab with 3blue1brown

I went to the library and asked for a book on Pavlov’s dog and Schrodinger’s cat.

The librarian said it rang a bell but she didn’t know if it was there or not.

South Carolina was here. Math is beautiful and elegant..

my goodness!!!!!!!!!!!!!!

after my highschool i thought "How Hard can MATHS be ??"