Mentioned statistics are findable if you search hard enough. Opinions presented come from personal experience and discussions with hundreds of undergrad math majors.
I unfortunately completed my math major about 8 years before you created this video :P But I stuck through it even though I ran into the same wall a lot of these kids went through because I knew I had enough progress to just barely finish it, and I wanted to challenge my perseverance on a real life goal - by bringing it back from the brink of failure - for the first time in my life. Also, real analysis was easier than complex analysis for me :)
Wow... As a person taking an undergraduate degree in Mathematics I can relate. When I first started Real Analysis, it was my weakest topic. I couldn't understand what half of the lecturers were stating. Besides, the topic is almost completely unintuitive for me. Meanwhile, I can't say I'm good at discrete mathematics or linear algebra, though I am definitely more confident in those topics than analysis (real OR complex). This video deserves more views.
Really awesome video! I’m a senior in high school right now wanting to major in math so I’m probably going to go to the library and borrow a book on basic proofs now 😂
Makes me so glad I didn't major in math or engineering. But seriously, I'm very pleased to see how many non-US students are watching your new vlog. I'll bet they got taught bridge in their schools.
I am contemplating the idea of studying mathematics because I have an interest in the theme, but I don't know if it's best to be self-taught for now. I started to really like how it can solve real life problems and improve my knowledge. I studied some logic and statistics in an English teaching degree and afterwards started a computer science related degree.
Weird , i recently started to study math in collegue (well , the equivalent of collegue here , in Spain) and we are already doing some kinds of proofs , and even we have a subject called "sets and numbers" which give us basic notions in proofs and mathematical logic.
Wait, what? I'm not from the US so I don't know the school system. Do you really not get to do proofs for 2 years even at university? At my uni in Germany you have to work on some kind of proof in the first week. What is taught in Calculus and what's Real Analysis? I think at my uni those are grouped into one series of lectures. But maybe I still don't get what Real Analysis is actually about.
I don't have a wholistic perspective on what happens outside the US, my only understanding of other school systems comes from my interaction with graduate students from other countries. What is taught in Calc vs. Real Analysis? Calculus focuses on problem solving techniques like the power rule for derivatives or formulas for arclength or surface area integration. Real Analysis focuses on proving formulas from Calculus, like using the delta-epsilon definition for a limit. I think the school system's rationale behind not teaching the proof techniques earlier than the 3rd year is that many engineering students need calculus for their jobs, but they don't need to know what a proof is. It's expected that only students who study pure math will take real analysis and therefore proofs are delayed to weed out the rest of the population.
When I was at school, in France, we started to do proofs (on parallelograms, I think) at what is the French equivalent of US eighth Grade (i.e "4ème") . It may unfortunately have changed since these happy days (1979), though.
Oh, interesting. Here we have different courses for math, physics, computer science students. I don't think one can get an engineering degree at my uni. The focus on proofs is different among those courses. The math students definitely learn Calculus tools while proving why they work (or more commonly the professor proving why they work). This way most students drop out in the first or second semester if they don't wanna do that kind of math. Even for comuter science students the most common dropout reason are math courses unfortunately. They might still be too proof focused.
@@eagelwizard290in america u learn way more math before U start with our Proof based uni system, but because of that u ll do a speedrun of what is tought in analysis 1,2,3 in real analysis
@@eagelwizard290 That lack of proofs until real analysis would be atypical for my experience too. At my math major program in Canada, we did proofs from semester one, because that's what made it a math program and not an engineering program.
6:10 what might also be helpful here is recognizing how the statement relies on the fact that the real numbers form a field and how this property fails in more general rings like Z/12Z. If you never saw those examples it's harder to guess what properties you need to use to prove the statement.
This is the whole point. A proof like this one will be shown on the first day of an introductory Analysis course. None of the students know what a group or ring or field is since Abstract Algebra comes later. It's possible to simplify the proof further by saying, "A field has no zero-divisors," but this is circular. You have to start learning about proofs somewhere and that's a big part of the challenge.
I was struggling with very basic proofs in high school, and that made me confused because im usually very good at math, i asked my teacher about it and then he told me about how in his analysis class only 4 of the 30 students finished the class, and they had to reduce the minimal grade so theyd pass, else only 2 had enough grade to pass the class
What you said about real analysis being the first course with proofs isn’t accurate in my experience. We did them in high school geometry and then I had to take an intro course for math majors which was linear algebra but proofs based and it was a good bridge between calculus and pure math. Also discrete math for my cs major. If you’re school does this they have a shit math department
Any proof-based course will have many of the same ingredients, so Linear Algebra with proofs will be a good introduction to the concepts. But Real Analysis is very focused on producing rigorous proofs from day 1 while Linear Algebra will still spend at least half of the time covering problem solving techniques like the Gram-Schmidt process. My guess is that Real Analysis is still a substantial bit more difficult than other introductory proof courses.
In Morocco we take real analysis 1 and 2 in the first year as well as a 2 proof based algebra courses and I'm not even a math major I'm finding it very diffuclt why the hell do we need to learn this as informatics students
Proof by contradiction If x&y are non-zero real numbers and their product is zero, then, since multiplication is repeated addition of a value, there must exist at least one non-zero real number which when added to itself a non-zero real amount of times must equal zero. However, we know that if you add any real number not equal to zero to itself any amount of times, it will never equal zero. Therefore their products must be zero, unless you’d like to say that anything other than zero added to itself n amount of times will equal any value not a multiple of themselves, which is ridiculous.
It's interesting how the proof presented here and in the video are kind of distinct. To use mathematical language, you are using the fact that the real numbers have characteristic zero while the video used the fact that the real numbers form a field. What makes them distinct is that there are rings with characteristic zero that are not fields and fields with positive characteristic (both cases fulfill the statement in the video). There are also rings like Z/pZ[x] that fulfill the statement but where both arguments fail which is pretty interesting.
Since x & y are not zero let x = m+k and y = n+k ; k ≠ 0 and m, n belong to R xy = (m+k)(n+k) = mn + km + kn + k => xy is non zero because k exists independently
I found this video 7 years late, now I’m in a math phd
I major in Yu-Gi-Oh and I use probably and statistics to win! More math majors should play TCGs, especially Yu-Gi-Oh!
That's a real major?
Is there also pokemon major
*probability
@@ZDTF 😅Only on RUclips, I have acquired a PHD in Dueling due to my studies in AI & The Hypergeometric Distribution!!!
As someone considering a math major this actually encouraged me, not that I think I'm any good at this, but rather it looks like fun!
I unfortunately completed my math major about 8 years before you created this video :P
But I stuck through it even though I ran into the same wall a lot of these kids went through because I knew I had enough progress to just barely finish it, and I wanted to challenge my perseverance on a real life goal - by bringing it back from the brink of failure - for the first time in my life.
Also, real analysis was easier than complex analysis for me :)
Wow...
As a person taking an undergraduate degree in Mathematics I can relate. When I first started Real Analysis, it was my weakest topic. I couldn't understand what half of the lecturers were stating. Besides, the topic is almost completely unintuitive for me. Meanwhile, I can't say I'm good at discrete mathematics or linear algebra, though I am definitely more confident in those topics than analysis (real OR complex).
This video deserves more views.
I was gonna scroll past this vid but the thumbnail made me laugh, subbed.
Really awesome video! I’m a senior in high school right now wanting to major in math so I’m probably going to go to the library and borrow a book on basic proofs now 😂
Makes me so glad I didn't major in math or engineering. But seriously, I'm very pleased to see how many non-US students are watching your new vlog. I'll bet they got taught bridge in their schools.
Leaving a comment to boost the algorithm
Great video btw!
Looks like learning to prove things like how opposite angles of a cyclic quadrilateral are supplementary wasn't a complete waste.
I am contemplating the idea of studying mathematics because I have an interest in the theme, but I don't know if it's best to be self-taught for now. I started to really like how it can solve real life problems and improve my knowledge. I studied some logic and statistics in an English teaching degree and afterwards started a computer science related degree.
I hope the math major you resurrected from quitting will weigh in :)
Weird , i recently started to study math in collegue (well , the equivalent of collegue here , in Spain) and we are already doing some kinds of proofs , and even we have a subject called "sets and numbers" which give us basic notions in proofs and mathematical logic.
I’m glad Spain is teaching math better than we are!
ive just started studying maths in spain this year too and we also have that class (along with intro to analysis, which also dives into proofs)
Wait, what?
I'm not from the US so I don't know the school system. Do you really not get to do proofs for 2 years even at university?
At my uni in Germany you have to work on some kind of proof in the first week.
What is taught in Calculus and what's Real Analysis? I think at my uni those are grouped into one series of lectures. But maybe I still don't get what Real Analysis is actually about.
I don't have a wholistic perspective on what happens outside the US, my only understanding of other school systems comes from my interaction with graduate students from other countries.
What is taught in Calc vs. Real Analysis? Calculus focuses on problem solving techniques like the power rule for derivatives or formulas for arclength or surface area integration. Real Analysis focuses on proving formulas from Calculus, like using the delta-epsilon definition for a limit. I think the school system's rationale behind not teaching the proof techniques earlier than the 3rd year is that many engineering students need calculus for their jobs, but they don't need to know what a proof is. It's expected that only students who study pure math will take real analysis and therefore proofs are delayed to weed out the rest of the population.
When I was at school, in France, we started to do proofs (on parallelograms, I think) at what is the French equivalent of US eighth Grade (i.e "4ème") .
It may unfortunately have changed since these happy days (1979), though.
Oh, interesting. Here we have different courses for math, physics, computer science students. I don't think one can get an engineering degree at my uni. The focus on proofs is different among those courses. The math students definitely learn Calculus tools while proving why they work (or more commonly the professor proving why they work).
This way most students drop out in the first or second semester if they don't wanna do that kind of math. Even for comuter science students the most common dropout reason are math courses unfortunately. They might still be too proof focused.
@@eagelwizard290in america u learn way more math before U start with our Proof based uni system, but because of that u ll do a speedrun of what is tought in analysis 1,2,3 in real analysis
@@eagelwizard290 That lack of proofs until real analysis would be atypical for my experience too. At my math major program in Canada, we did proofs from semester one, because that's what made it a math program and not an engineering program.
6:10 what might also be helpful here is recognizing how the statement relies on the fact that the real numbers form a field and how this property fails in more general rings like Z/12Z. If you never saw those examples it's harder to guess what properties you need to use to prove the statement.
This is the whole point. A proof like this one will be shown on the first day of an introductory Analysis course. None of the students know what a group or ring or field is since Abstract Algebra comes later. It's possible to simplify the proof further by saying, "A field has no zero-divisors," but this is circular.
You have to start learning about proofs somewhere and that's a big part of the challenge.
One might also see this fact as something encouraging, especially the less computationally inclined among us (that's including me haha).
Not me finishing my major watching this vid
I was struggling with very basic proofs in high school, and that made me confused because im usually very good at math, i asked my teacher about it and then he told me about how in his analysis class only 4 of the 30 students finished the class, and they had to reduce the minimal grade so theyd pass, else only 2 had enough grade to pass the class
1st semester math major here can confirm my ass is cooked
I will take real analysis next semester
What you said about real analysis being the first course with proofs isn’t accurate in my experience. We did them in high school geometry and then I had to take an intro course for math majors which was linear algebra but proofs based and it was a good bridge between calculus and pure math. Also discrete math for my cs major. If you’re school does this they have a shit math department
Very interesting video! How does proof-based linear algebra stack up to real analysis in terms of difficulty?
Any proof-based course will have many of the same ingredients, so Linear Algebra with proofs will be a good introduction to the concepts. But Real Analysis is very focused on producing rigorous proofs from day 1 while Linear Algebra will still spend at least half of the time covering problem solving techniques like the Gram-Schmidt process. My guess is that Real Analysis is still a substantial bit more difficult than other introductory proof courses.
@@LilBiteOEverything i appreciate the feedback👍
In Morocco we take real analysis 1 and 2 in the first year as well as a 2 proof based algebra courses and I'm not even a math major I'm finding it very diffuclt why the hell do we need to learn this as informatics students
Proof by contradiction
If x&y are non-zero real numbers and their product is zero, then, since multiplication is repeated addition of a value, there must exist at least one non-zero real number which when added to itself a non-zero real amount of times must equal zero. However, we know that if you add any real number not equal to zero to itself any amount of times, it will never equal zero. Therefore their products must be zero, unless you’d like to say that anything other than zero added to itself n amount of times will equal any value not a multiple of themselves, which is ridiculous.
It's interesting how the proof presented here and in the video are kind of distinct. To use mathematical language, you are using the fact that the real numbers have characteristic zero while the video used the fact that the real numbers form a field. What makes them distinct is that there are rings with characteristic zero that are not fields and fields with positive characteristic (both cases fulfill the statement in the video). There are also rings like Z/pZ[x] that fulfill the statement but where both arguments fail which is pretty interesting.
It's real analysis.
Since x & y are not zero
let x = m+k and y = n+k ; k ≠ 0 and m, n belong to R
xy = (m+k)(n+k) = mn + km + kn + k
=> xy is non zero because k exists independently
Wouldn’t you also have to prove that mn + km + kn =/= -k?
@@happypig8690 we can see if, k(m+n) + mn = -k. At (m,n) = 0 => k = 0. This is a contradiction bc by def. k≠0
I am not a math major but am taking Discrete Math which covers many concepts in writing formal proofs, is that not normal for math majors?
🙏🙏🙏
Is this bradybot?
Yes
BRADYBOT IM A MATH MAJOR BUT WTF
bump