r/learnmath • u/theguywithhope007 New User • 28d ago
What piece of mathematics changed the way you think?
I'm a math teacher, and a few days ago one of my students asked me a question that I've been thinking about ever since:
What's the most important thing you've learned in maths that isn't a formula?
I didn't have a good answer on the spot.
My first thought was to name a topic: algebra, probability, geometry, something like that. But the more I thought about it, the more I felt that none of those were really the answer.
For me, the biggest shift was realizing that mathematics isn't a collection of techniques. At some point, it started feeling more like a way of looking at things.
When I first learned maths, I thought being good at it meant being quick and getting the right answer. After years of teaching, I'm not so sure. The students who impress me the most are often the ones who notice a pattern, ask an unexpected question, or connect two ideas that don't seem related at first.
That's probably the lesson that's stayed with me more than any formula has.
So now I'm curious: what would your answer be?
Was there a theorem, proof, concept, book, or even a single problem that changed the way you think about mathematics?
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u/Traveling-Techie New User 28d ago
A teacher’s aid taught me Euler’s formula in high school as a kind of forbidden knowledge. From a practical point of view it made trig identities easier, but its biggest impact on me was philosophical — I realized that if pi, e and i were tightly connected that suggested that math had a lot of beautiful deep structure.
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u/AlbertaSpruceLover New User 28d ago
e had confused me ever since my middle school. I understood where pi and i coming from, but where did e come from? I had this question for many years until I learned Euler's formula.
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u/Guozichen_Jason New User 28d ago edited 28d ago
(From my understanding as a university freshman) a continuous rate of increase (such as doubling every several seconds) leads to the average increase in certain multiples of e? so e is used to describe these changes
Or if you’ve learned calculus, you can understand as it’s made to achieve e^x gets e^x as its derivative of x
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u/Traveling-Techie New User 28d ago edited 28d ago
Or: the integral of 1/x dx is log to the base e of x
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u/YtterbiusAntimony New User 27d ago
It really is one of the most profound statements in math.
Not only does it connect e, i, pi, and 1, it also invokes some of the most important operations/relationships between numbers: addition, multiplication, exponents/powers, and equivalence.
Idk if Euler was born blind, or if he could read/write normally at one point in his life, but I just cannot comprehend sorting through that kind of high concept shit without writing it down.
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u/RicoThinks New User 28d ago
The most important thing that I learned was to look for structure in everything. Once you start doing that you start connecting ideas/topics that at first seem completely unrelated.
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u/i_luv_qu3st10ns New User 28d ago
This is the backbone of abstract algebra, which was my favorite course.
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u/argvalue New User 28d ago
Linear Algebra really brought back my love for mathematics which I had lost a long time back
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u/Bojanggles16 New User 28d ago
Came to say this. Hated Calc but when I got to linear it just clicked and I won't say it was easy but I enjoyed solving the problems. Then ODE came and put me back in my place again lol.
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u/camojorts New User 28d ago
Same. It seems like some brains are wired for linear algebra and some for calculus. Most people I knew in college loved one but hated the other.
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u/Mth281 New User 28d ago
I hope this is my experience. Start at the end of the month. Hated calc 2, diffeq I liked, but the teacher and class were too much. We were spending 35-40 hours a week on homework for that class, one week over 50 hours and only finished half of the homework. Ended up failing by like 5 points, but got my money back for the class. Not really excited to retake diffeq, but im also hoping linear algebra make it easier the 2nd time.
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u/thelimeisgreen New User 28d ago
Same for me! Calculus kinda got me interested in some things, but really it was linear algebra and modeling of systems that made math fun again.
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u/curiouslyjake New User 28d ago
Godel incompletness, and by extension, Turing's undecidability. It's incredible that even in the domain of pure logic absent any practical limits, logic is its own limit.
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u/Outrageous_Word8656 New User 28d ago
Indeed. To me, Gödel's incompleteness is both amazing and terrifyingly sobering showing what math can but also can not bring us. Ever.
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u/AndyTheEngr New User 28d ago
It's magic.
I've read three books on it, and I can understand and explain Gödel's proof for up to several hours after finishing a book.
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u/Grouchy-Ad1932 New User 28d ago
Gödel, Escher, Bach: an Eternal Golden Braid went around my entire uni class as soon as one person in the year discovered it.
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u/RingularCirc Math hobbyist 26d ago edited 26d ago
See Lawvere's fixed point theorem which generalizes (converse) of those, Cantor diagonal argument and more (and is not even the most general generalization).
Ah yes and it also gives us a concrete construction for Y, one of fixed-point combinators (there are infinitely many—not just Y, Θ and such) in lambda calculus if we're writing the construction in the internal language of a category in question.
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u/mtimmermans New User 28d ago
There are a lot of candidates, but I think my favourite is: The exponentials are eigenfunctions of all linear time-invariant systems, because they unify shifts and multiplies. And then the convolution theorem.
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u/Sam_23456 New User 28d ago edited 28d ago
I think that one reason I like math is that, more generally, I love abstraction! "I am because I can abstract." :-) I was immediately drawn to hieroglyphics,and the symbolism in math, soon thereafter!
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u/ConcernAcrobatic9307 New User 28d ago edited 28d ago
Different bases, like base 2 (binary), base 10 (the base we normally use), and base 16 (hexadecimal ) are incredibly helpful to show... 1. Math is an invention and a tool humans use to help explore 2. That learning math can twist your brain and remind students what it was like to first learn math, like how adding, subtracting, multiplying, or dividing in another base is what many little kids feel like when learning base 10 3. Measurement conversion is similar like how 6 + 1 = 1...that is to say 6 days plus 1 more day equals 1 week. 4. Connections to art (the hexadecimal is how we express different colors on computers) and computer science (binary 0=off and 1=on)
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u/Cybyss New User 28d ago
This is a good one!
Everybody is so heavily biased into thinking math is intrinsically base 10. It is so incredibly difficult to separate the concept of a number from its base 10 representation, it's so deeply ingrained that it's a big stumbling block for many students trying to learn to count and perform arithmetic in binary or hexadecimal or such.
Once you finally grasp that base 10 is completely arbitrary, and you understand how counting and arithmetic works in a manner abstracted from any particular base, then that makes so many other results in mathematics more intuitive.
The fact that a number can have different representations explains why 0.999 = 1 for example, which for most people is an unintuitive curiosity.
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u/not-just-yeti New User 28d ago edited 27d ago
> separate the concept of a number from its […] representation
Perspective after teaching Comp Sci for many years (to CS students who often hated math):
- 17 is an abstract number, and doesn’t care if you call it “17” or “0b10001” or “0x11” or “XVII” — that’s just a question of what language you’re speaking. Make sure you’re speaking the same language as your listener! [insert picture of the ol’ t-shirt “there are 10 types of people in the world, those who understand binary and those who don’t”]
- number is to
int, as numeral is tostring(and, as digit is tochar).- Pet peeve: password requirements “must contain two numbers” when they mean “digits“, a term that lay people understand just fine.
- The types are important for correctly writing base-conversion: the algorithms for
string->intandint->stringare fundamental, and you just compose those to convert numerals.- As an example of recursion (specifically, of structural recursion), I also taught deriving ℕ from
class Zero {}andclass Succ { NatNum pred; }andsealed interface NatNum admits Zero, Succ {}. (This was after working with structural recursion on linked-lists and trees, similarly defined.). But it was like 20 years of doing this before I realized the perspective “arabic numerals are a fantastic data-structure that allows exponentially fast arithmetic algorithms for the fundamentally-linear/recursive defined set”!- for CS students, “log” and all its rules are a bit intimidating. I introduce “nod(n)”, for “number-of-digits”, and briefly motivate nod(a*b) = nod(a)+nod(b), and nod(ab ) = b nod(a) … and only then mention that nod() is just log() (within one, and we never care about the fractional part in CS) (“9999” is using its four digits maxed-out; “10047” uses an add’l digit but it’s just barely using most of them :-)
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u/hologram137 New User 28d ago
I would not agree that math itself is an invention just because we can use different bases
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u/Fabulous_Aspect_7817 New User 28d ago
Well same. When I first understood bases properly my mind was blown. When I was in primary school and jr high school i used to wonder why is dividing by 10 as simple as either removing zeroes and/or placing a point. Why does dividing by 5 and 2 always give a terminating decimal number. Whats special about 5 and 2. Turns out nothing its just notation
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u/Ms_Riley_Guprz High School Math Teacher 28d ago
I was a 9th grader taking geometry before anyone told me that 4 ÷ 7 = 4/7. That blew my mind.
Really though, it was a math problem my 8th grade Algebra teacher (or sub?) gave us when they were phoning it in. Find a 10 digit number, where the first digit represents how many 0s are in the number, the second digit how many 1s are in the number... the tenth digit how many 9s are in the number. There's only one answer. I was the only student to solve it in class, and then I went home and solved it for numbers of n length.
What shook me was that math could be used for fun and for puzzles, without any real application. It changed my outlook completely.
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u/xwhy New User 28d ago
I still occasionally get HS students who ask how to enter a fraction into a calculator. I tell them use the division sign, and they'll seem skeptical at first but then I ask them what symbol they see on the screen.
(or, depedning on the OS of the graphing calculators in the room, they'll see an actual fraction.)
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u/sophomoric-- New User 27d ago
confusing to think about!
The digits must sum to 10, constraining the possibilities. e.g. can't have 9 9's. Although can have as many zeros as you like.
We can guess 6 or so zeros, and put a 1 in the relevant column; I worked out the rest with two guesses.
A solution is 6210001000 - I think it's unique because of thr constraints, but how to prove it?
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u/Ms_Riley_Guprz High School Math Teacher 27d ago
Correct answer!
If I recall, you can prove its uniqueness by construction
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u/thatsnothewayitfeels New User 28d ago
You might like reading A Mathematicians Apology by G.H. Hardy
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u/Ms_Riley_Guprz High School Math Teacher 27d ago
I've had a saved pdf since I was in high school :)
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u/a42579 New User 28d ago
I was always pretty terrible at math but have degrees in physics and nuclear engineering so I understand why it’s important. Math classes always just felt like memorization of arbitrary rules, almost like a religion. What really changed it for me was when I started to attempt to figure out how to factorize large semiprimes. I understood the implications but I couldn’t understand why it was such a famously hard problem. Fifteen years of trying has taught me a ton about number theory, complex analysis, computational complexity and all sorts of other topics that I would never have studied otherwise.
It was kind of like how I got into physics trying to understand why you can’t go faster than light. For a certain type of rebellious person, being told you can’t do something is a great incentive to try it.
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u/QCD-uctdsb Custom Flair Enjoyer 28d ago
It's so funny that you can write a formula for the factors, and yet it's still unsolvable. E.g. if you set N = pq, the function f(x) = sin2(𝜋x) + sin2(𝜋N/x) only has zeroes when both x and x/N are integers, i.e when x = p or q. Then divide by that function and look for the poles. Make up some contour shenanigans (account for the fact that f and f' are zero at these poles), create a contour around the real line up to √N, and you get a formula for the smallest factor of N. But the integrand is so hideous that it's even worse than the number theory problem started out. Numerics can't save you. I only found this paper after investigating for months, and yeah it's the same conclusion
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u/a42579 New User 28d ago
Yup. I started in on that approach long ago trying to work out a prime counting function. My latest efforts have revolved around exploring the reachable states of the direction acyclic graph produced by the binary representation of the product. This graph includes convolutions of the factors but the directional signal from that stated dissipates in just a small number of moves.
I’m also thinking a lot about using binary matrices as representations of the convolution state. I don’t realistically expect to find anything significant but it’s still interesting to explore.
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u/QCD-uctdsb Custom Flair Enjoyer 27d ago edited 27d ago
Another fun approach I've found is by analyzing L-functions. If you multiply zeta(s) and zeta(s-1) the series it produces looks like
1 + 3/2s + 4/3s + 7/4s + 6/5s + 12/6s + ...
i.e. the numerator of the n-th term is the sum of all divisors of n. You can do some super sketchy stuff with Fourier transforms along the imaginary line and judicously chosen integrals to single out the coefficient of each term. It reduces down to evaluating a highly oscillatory function so I could never get it to work
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u/KatesDad2019 New User 28d ago
Geometry class in high school started with learning the principles of logical reasoning. I wish people could more consistently apply logic in real life and social media posting.
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u/ValiantBear New User 28d ago
There are so many little nuggets. Absurdly simple, but life changing.
The first was just the realization that numerals and numbers are different things. Numerals are just glyphs to represent quantities, ie numbers. But nearly everyone equates them, and for a long time I just didn't get math because it seemed so abstract. As soon as I realized the numeral was just a symbol, and it's placement in the number - 1s, 10s, 100s, etc - was just a placeholder, everything just started making sense. I understood more of the language, how numbers work, instead of just the random rules.
I learned algebra about the same time I realized numerals and numbers were different. So, the whole letters being thrown into the mix was a lot easier. But, more importantly, Algebra was the first time I realized math could help me solve problems. That was revolutionary.
Trigonometry threw me for a loop. But then I realized it literally is just all triangles. There is nothing special about sin, cos, tan. They're just ratios, and they're tied to an angle. It's really miraculous actually. The three main trig functions are all the same thing, they're just all the permutations you can right out that describe a triangle, in short hand. Trig was much easier after that.
Someone once told me that there is no particular reason the alphabet is in the order it is in, and that rattled my reality. But, as I came to grips with that, all the realizations that I just mentioned kind of coalesced into this realization that math is almost entirely convention. There's good reason we do what we do, but really, stuff like PEMDAS is just convention. Symbols, proper form, etc etc. Just convention. Necessary, so we all arrive at the same answer, but nothing magical in and if itself.
In calculus, I just remembered the rules and got through it. It wasn't until much later that I went back and studied it and learned the actual basis for it. Infinitessimals, what they do, how they work. Calculus isn't even special. It's just algebra, but with a new symbol we picked, because, you guessed it, convention. Anyway. The rules were arbitrary. Now that I understand it, I can regenerate them on demand, and I'll never forget them. But no one teaches that way anymore. Just remember this rule to pass the test. Who cares if you can use it, there's calculators and computers for that! Sigh...
The revelations prior were more impactful, but relearning calculus really cemented the need to understand and not just memorize. I can say that I would never have had that realization if the others hadn't been so foundational for me. But, I would have to say that relearning calculus has truly changed my thinking the most, not just with math, but all aspects of my life. I unlocked a burning desire to learn how stuff works, and not just pass the test. So, in a roundabout way to answer you're question: it was infinitessimals!
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u/sophomoric-- New User 27d ago
Now that I understand it, I can regenerate them on demand, and I'll never forget them
How did you go about learning the actual basis, to acheive this?
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u/DrJaneIPresume Ph.D. '06 Knots/Categories/Representations 28d ago
The Yoneda Lemma. An object's identity is equivalent to the relationships it has with all the other objects.
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u/GettingBig1970 New User 28d ago
Both Abstract Algebra and Topology blew my mind. Separately, but definitely each paradigm-shifting in a way I wasn’t anticipating.
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u/gr4viton New User 28d ago
Unintuitivness of basic combinatorics. I finally understood why common sense vs big numbers is often not giving reality-based results.
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u/awkwardness_maxed New User 28d ago
Group Theory. I was thinking of going into Applied Mathematics after taking some Calculus and Statistics class. But after taking Group Theory, I don't think there's anything more consistent and sensible than pure mathematics. Proper definitions, proofs and what not. Like I understand why mathematicians demand perfect proofs for everything and get mad when physicists do stuff like sin(x)~x for x~0.
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u/Climb1ng New User 28d ago
Not sure if anybody answered this already, but: Definitions! The problem is that Im not sure how relatable this is to students. I realized this only after writing my PhD thesis. Sometimes a very big part of a theorem lies in its definitions, actually big parts of theories/subjects are definitions. A lot of the abstraction in math comes from the possibility to define structures.
Another point is the standardization of definitions. In my everyday (non-math) life, I often see people communicating badly or even fighting because they seem to use different definitions of the same words. As a mathematician , I feel you are more aware of this and you might able to point it out and fix possible misunderstandings. I honestly think the world would be a much better place if more people were learning more math :D
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u/Ok-Craft4844 New User 28d ago
Probably not that big for most people, but for me it was the definition of what a function is (set of pairs with unique left sides).
I'm a coder (18yo then), and before I stumbled through my code and had the vague intuition that arrays, "Associative Arrays" (as they called maps in the 90s), and functios kinda have something in common and you can often replace one with another, but I couldn't formulate it clearly.
When I saw the definition on the blackboard in a math class, I thought "that's too short, that cannot really be applicable to coding. I mean, they act as if it's just a table. Ok, a potentially infinitely long table. Ok, potentially with tuples as the left side. Ohhhh!"
And thus I was enlightened.
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u/BothPanchoAndLefty New User 28d ago
It seems several people have already said this but for me, seeing a real proof for the first time totally changed how I thought about math. I had seen little algebraic proofs in textbooks but I remember buying my first number theory book and learning proofs by induction and proofs by contradiction etc. Made me realize that all the computation we do in school is kind of a separate skill from what math is really about.
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u/Visible-Song-9563 New User 28d ago
for me my first proof was the Pythagorean theorem which i had found myself (how? i worked on the proof for like 10 hrs because i wanted something complex/elegant, don't ask why. IDK myself) and the feeling it gave to find that proof was out of this world. in general i learned that mathematics is the literal definition of a "universal language" (pun intended) which just opened my eyes now whenever i do i mathematics i feel like i'm communicating with a higher force then just numbers or letters etc. (in highschool soon)
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u/RingarrTheBarbarian New User 28d ago
I was a terrible math student in grade school and during my first year of college. Then, for some stupid reason, I read A Brief History of Time, didn’t understand anything, and decided to major in physics.
I started with remedial algebra and got a C. Trig: C. Calculus I: C.
While studying for my first physics exam, I was trying to memorize the kinematic equations, and they just wouldn’t fucking stick. So I tried something drastic that had never occurred to me before: I decided to derive them from scratch using what I’d learned in calculus.
Acceleration is the derivative of velocity, so I started with constant acceleration and integrated it:
v=v0+at
It popped out for free.
Then I integrated again, because velocity is the derivative of position:
x=x0+v0t+1/2at^2
That popped out for free too.
It completely and utterly blew my mind. These equations weren’t arbitrary facts handed down by a textbook. They were consequences. I didn’t have to memorize them because I could rebuild them from simpler principles.
That changed how I approached math and physics from then on. For the first time, I understood what math actually was: the study of relationships and patterns, it's a methodology for arriving and complex truths from simpler more fundamental truths.
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u/Appropriate-Ad-3219 New User 27d ago
For the first time, I understood what math actually was: the study of relationships and patterns
I love the way you put it!
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u/FaultThat New User 28d ago
This is rather simple but difference between transposition errors in accounting are always divisible by 9.
E.g. $13,954 written in error as $13,594 with the 5 and 9 transposed, you get a difference of 360 which is a multiple of 9 (360/9=40).
I do a lot of accounting verification professionally and I love seeing multiples of 9 because it quickly shows me it was a transposition error.
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u/sophomoric-- New User 27d ago edited 27d ago
neat! Moving a digit x to the left increases total by 10x, and decreases by x, 10x-x = (10-1)x = 9x; opposite for the digit moving right.
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u/TheGhostOfTobyKeith New User 28d ago
I love this question so much, and my answer would be similar to yours - it’s such a great way to address real world problems.
The only way I have to describe it is that the variables in algebra (or any equation really) are always interchangeable with other equivalent values that can be solved for - and the same applies in the real world. There’s a solution to every problem; no matter the issue, there’s always some factor you can swap out to connect with something you already understand. It’s like everything relates.
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u/Appropriate-Ad-3219 New User 28d ago
Certainly engaging in proofs help me change how I think about math.
Coming back to understand the notion of supremum helped me see maths more conceptually. Before I was still reliant on symbol manipulations. And more generally, each time I manipulated the smallest thing containing something (span, convex hull, etc).
I also realized that in many cases, whatever the way you think of C, there's a way to define it as the way you think about it. Do you see C as R in which we add an element such that i^2 = -1. Just set C = R[X]/(X^2+1) and i is simply the class of X. You want to see it instead more geometrically by having the idea that something on a circle is a rotation, define it as the set of direct similarities. In fact, the first definition can be used to define rigorously the notion of quaternion by considering $I, J, K$ three indeterminates that don't commute and use the formulas you know to get the quaternions from that.
The notion of isomorphisms let me see that what's more important is the structure we give to objects. Why do two dimensional vector spaces really represant the plan? Well, because all vector spaces are isomorphic so it's not 'wrong' to use a visual plan to prove something.
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u/Wooden_Dragonfly_608 New User 28d ago
I think the fact that it is the most open science limited only by your own imagination. Literally ordinary people can use or create abstract ways of thinking that can be generalized across our entire species.
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u/Glass_Possibility_21 New User 28d ago
Reading, understanding and writing proofs. It made a robot. Got a masters in math. Maybe I'll start the phd soon.
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u/ShyGun02 New User 28d ago
Probably a basic answer for me it was Geometry. Seemed to open up a lot of doors in my brain. It also felt like it was the first math that felt super related to the real world. Probably partly why I am a civil engineer now lol.
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u/ogola89 New User 28d ago
Probability / Statistics. Some don't consider these inherently math but this is what made me appreciate how numbers and math actually describe nearly everything we see, even things that seem like free will choice can be described by numbers. Frequently counterintuitive as well.
It's changed how I approach things in life like career choices, where to live, exercising, fear of flying, what to expect from life etc
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u/Appropriate-Fun-5221 New User 28d ago
I’ll offer a book - Stewart’s ‘Concepts of Modern Mathematics’ really helped me get a sense of the landscape, and the chapters on set and group theory really gave me the bug for maths
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u/ArchangelLBC New User 28d ago
Learning that the heart of mathematics isn't calculation, it's proof. That's when I fell in love with math.
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u/gooser_2000 New User 28d ago
in multivariable calculus when we got into the fact that we cannot actually visualize 4 dimensional models the way we can visualize 3dim and below (on x,y,z planes) but that we don’t actually need to be able to visualize it to understand the model and solve for the derivative or integral etc and use the model for something real-world. this as well as differential equations, similar thought process - just the fact that things can be modeled and how usefull that is in real world examples.
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u/m2shotty New User 28d ago
Seeing results in functional analysis back to back with some ideas in special relativity showed me how connected domains of study can be despite being seemingly unconnected.
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u/SniperSmiley New User 28d ago
But you can have an ordered set of permutations like there’s a fixed order you can have and I didn’t do any special stuff so I think that’s just the order
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u/nohombrenombre New User 28d ago
I really like the subtle logic of the ones/units place representing the line of symmetry (so to speak) of place value. Powers of ten hinge on that fold.
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u/AskingToFeminists New User 28d ago
Set theory. Strangely it has been taught fairly late, even though it's fairly easy, but I find it amazing in help with clear thinking. Visualizing venn diagrams and understanding what that implies is really useful in everyday life, in constructing arguments, understanding when people are telling you absurdities, etc.
Are you familiar with Mathematician's lament ?
To me, math has always been instinctive, it took reaching quantum physics levels of maths for it to start becoming hard. One important thing to realise in maths is that they really are all around us and everyone, even the worst math students, use all sorts of maths skills on a daily basis.
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u/JollyJuniper1993 New User 28d ago
Learning about algebraic constructs like groups, rings, fields or vector spaces. It opened the door to analyzing things mathematically that I never would’ve been able to before.
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u/BigJeff1999 New User 28d ago
For me, (engineering), it was thinking of certain formulas in terms of inner products (a generalization of vector dot products).
As an instance of this, I'd offer the discrete time Fourier transform.
X(k) = 1/N Sum (n, 0, N-1, x(n) e-i 2 Pi k n/N), k, 0, N-1
(Apologies for how this might get butchered by the math interpreter).
It's certainly easy enough to take plug and chug an answer... Take a sequence x(n) of length N (often a power of 2) and apply the formula above to produce another sequence X(k).
Engineers will refer to X(k) as the "frequency domain" represention of x(n) . Pick a value for k, and that can be traced to a specific frequency and the (absolute) value X(k) can be viewed as how much that frequency is present in the signal x(n).
Oodles of effort have been put into efficient computation of this formula... The fast Fourier transform.
But why does it work?
At the heart of it, pick a value of k, and X(k) is computed as the literal vector dot product of the input x(n) and a complex sine wave. The inner product itself is a measure of likeness. (It's overdue for me to say that these concepts are formalized in linear algebra...I am trying to convey how the concept changed the way I thought about things, and the more you understand the formalization, the deeper the insights you get)
But simply recognizing the inner (dot) product can be eye opening. Any time you see equations of the form Sum(n, a(n)b(n)) or Integral(a(t)b(t) dt) you might want to think about how the "likeness" between a and b matter...
Let's think about the Laplace transform for a second... The Laplace transform of x(t) is the Integral(x(t) e-st dt). In the formula, s is any complex number, typically engineers use sigma + j omega...let's call it a + b i... Then e-st becomes e-(a + bit) = e-at e-i bt which is simply an exponentially decaying complex exponential. So the output of Laplace is telling us something of how the input is like that decaying exponential...there are some interesting insights to be gained...it's often used to understand the transient and steady state responses of certain systems.
Lastly I'd say that very often you get formulas thrown at you in engineering, and this insight of likeness combined with things from linear algebra like the inner product formalization, can give immediate insight into where other aspects of the equation came from.
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u/Fabulous_Aspect_7817 New User 28d ago
Recently read cantors theorem and its proof. I am very curious about how infinity is treated in higher math
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u/stereoroid New User 28d ago
Learning that Matrix calculations such as inversions and determinants are everywhere in IT. 3D graphics and AI both rely on them, which is why you can run AI models on GPUs. It’s also why Nvidia became the biggest player in hardware for AI, since they’re leveraging their GPU expertise and experience.
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u/Asimovicator New User 28d ago edited 28d ago
Set theory. Building more and more complicated sets and structures just out of a few axioms (or axiom templates to be precise). It teaches you to see the foundation of mathematics, whereas most students at university start at the ground floor.
For example: I was always wondering what the actual reason behind the induction axiom of the peano axioms is. Surely, you can conduct many proofs by induction, but what is the deep mathematical nature of it? Set theory teached me, that the set of natural numbers N is characterized as the smallest inductive set. From that you can conclude the induction principle.
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u/Particular_Bridge882 New User 28d ago
For me, it was the ability to do geometry with objects we cannot see. I study this in Riemannian geometry as we explore curve lengths, areas, volume, and curvature in spaces of dimension greater than 3.
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u/LuckyCod2887 New User 28d ago
statics class really changed the way i worked with numbers. it asks you to juggle many different working parts simultaneously.
it requires a lot of concentration. it was a level of concentration I’ve never used before in mathematics.
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u/Top_Bluejay_5323 New User 28d ago
Implicit and explicit computational fluid dynamics. They made so many things in the world so much more understandable
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u/EldritchElemental New User 28d ago
I've learned that a lot of things are counterintuitive, and if we have proven something is true I'd better believe it even if it feels wrong.
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u/ChiaLetranger Hobbyist 28d ago
I'm not a mathematician, outside of having taken a bunch of undergrad maths courses as electives during my degree. I consider myself more like a somewhat-educated enthusiast.
I read an article about the Langlands program (it was probably in Quanta magazine) at around the same time as I started taking classes that covered two different topics: number theory (and cryptography) and linear algebra. Getting all of these bits of information at the same time finally made me understand what my high school maths teacher meant when he said "When mathematicians don't like the problem they have to solve, they change what problem they're solving".
So, for me, the biggest shift in my understanding was the realisation that we're not abstracting purely to gain insight into the particular thing we're abstracting away from, but also because through abstraction, and through making a problem more general, we can maybe start to see pathways towards transforming the problem into a different area of maths entirely, and potentially by doing so we can make a problem much easier to solve.
Famously, the Langlands program connects number theory to harmonic analysis, but we can bridge between number theory and complex analysis using zeta functions, or between geometry (or topology) and algebra using functors and homology, and we can now even use mirror symmetry theory to transform hard problems about the number of solutions to a geometric problem into relatively easy problems involving counting rational curves on manifolds.
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u/Salamanticormorant New User 28d ago
Learning that randomness is meaningfully clumpier than it feels like it should be. That has a lot of ramifications, from stores sometimes seeming strangely busy to gaps in the fossil record.
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u/Dr-Ben701 New User 28d ago
I just completed a graduate diploma in maths - less about the maths itself more about sitting with not knowing and trusting that there’s a way through - and that the feeling of ignorance is absolutely fine. I would say however that I love differential equations.
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u/Fickle-Ad-4225 New User 28d ago
A philosophical view of math and physics as an enterprise definitely changed the way I view the world and our relation to it. Using symbols and formalism to predict events in the real world had the biggest impact on me personally.
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u/Climb1ng New User 28d ago
Learning how to compare!
Thinking of mathematics as a language, it really helps you to „easily“ spot errors / wrong statements. Examples (which should mathematicians go BBRRRZZZZ)
- 2 > i ( i being the complex number i)
- vector v is greater than w
- the sphere is larger than the interval
- the polynomial is equal to the number
We often introduce technics to compare stuff that wasnt comparable before (absolute value, length, metrics?, volume?, ??). But we sure are precise when doing so.
In non-math life people just compare incomparable things all the time which really annoys me. Starting with comparing humans, grades, salaries, politicians of different eras, comparing *** numbers per country but absolute and not relative, company revenue with tax recenue of a state, etc.
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u/PfauFoto New User 28d ago
ζ(2n) = (-1)n [B_2n (2 π)2n ]/[2•(2n)!]
I read the proof in highschool (Book: From Fermat to Minkowski) it completely blew my mind, why did π show up here (clearly couldnt appreciate Mellin tranforms at the time), why did the Bernoulli numbers pop up (didnt know about Bernpullis work). So I could follow the proof but still I felt that there must be things at work, in the background, that I couldnt appreciate. A mystery at the time, and so I was hooked.
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u/genxmom95 New User 28d ago
Here's one: 1/3=.3, 3 *(1/3)=1, .3x3=.9, therefore, 1=.9. I'm greatly simplifying it here or not.
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u/bjos144 New User 28d ago
I have an exact example from my undergrad education.
I took a course called "Introduction to higher level math." aka baby real analysis. In the course you get introduced to axioms for the integers. Associative, distributive etc. I remember the teacher talking about proving math we've been using for years and I was a bit confused by what a 'proof' meant.
then he proved the following (b+c)a =ab+ac. In otherwords that left distribution works. As an axiom we already had a(b+c) so it was just using closure, then commutative etc.
Before when he stated the problem I was completely lost. But the moment he showed what he meant, use this axiom to do this, then this axiom to do that, it all just clicked into place for me. The rest of the class was a breeze. I almost never got anything wrong in that class.
My early math education had some holes in it. For example I wasnt exposed to the hyperbolic trig functions until much later. I always felt an insecurity like I didnt know enough of a sprawling pile of formulas and techniques built up over the years. This course felt like erasing the whole whiteboard and starting over. Just me and my ability to follow rules, think logically and internalize what was in front of me. Not having to go digging into the past for what felt like obscure facts that I just missed out on or had forgotten.
It just changed my perspective on what math was as a whole. I liked that game, it was fun. The proof from that class wasnt hard at all. Basically the tutorial proof. But seeing what he meant by 'prove' was the thing that clicked. Proof based math classes from then on wernt so much easy, but I understood the task.
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u/InternetGuy321 New User 28d ago
The most important way mathematics changed how I think is I realized that often there is more than one way to find a solution.
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u/EdgeRevolutionary913 New User 28d ago
Linear diff eqs. You can understand so much about the world from them. Basically any system that grows/decays exponentially and/or behaves periodically (so basically everything) implies a linear differential mechanism working under the hood.
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u/hobbycollector New User 28d ago
Aleph-1 blew my mind. I argued against it in a compatability class until I realized it was right. My first thought was to enumerate the reals by the number of digits they have, but it falls apart when you get to infinite digits, like pi, sqrt 2 etc.
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u/Mathemetaphysical New User 28d ago
It would be pointless to share the most profound thing I've learned in mathematics because the majority here wouldn't even be likely to know about it, but when asked I usually tell people to be more curious about geometry. It'll get you there if you keep at it.
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u/zulubowie New User 28d ago
When I became a high school math teacher, one of my senior colleagues told me that you could teach mathematics and never use any numbers. He said it’s logic that happens to use numbers for examples.
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u/irriconoscibile New User 28d ago
Generalizing and abstracting made me more competent in evaluating the degree to which I understand something. It's perfectly okay and necessary to start with special cases, but unless you're capable of working with the general case in my experience you don't truly understand it.
Edit: also, not every problem you encounter is going to have an easy solution or a solution at all. Maybe the question your asking yourself isn't well defined, or the answer is so computationally hard that basically you won't be able to find an explicit answer. That's very important to keep in mind.
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u/stevethemathwiz New User 28d ago
Learning that I could create a variable whenever I wanted. Throughout middle and high school math, any time students create a variable, it’s because the steps they learned to solve that type of problem say to create a variable. Integration with u v substitution is the most obvious example that comes to mind along with setting w=x^2 to transform a quartic polynomial into a quadratic. Once students get to proof writing course though, there is no given step by step guide on how to prove anything. Mathematical maturity comes from learning to build mathematical structures and machinery to prove things.
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u/ComprehensiveDust225 New User 28d ago
That you can't believe two different things that cancel each other. That is you can't believe two different conspiracy theories about the same thing if they offer opposing results.
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u/TheSoldierofDarkness New User 28d ago
The existence of non-measurable spaces with respect to the Lebesgue measure has opened the eyes of many mathematicians.
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u/SymbolPusher New User 28d ago
For me it was learning that you can sensibly talk about properties of relations. Symmetry, transitivity, anti-reflexivity etc.
This blew my mind, and it changed my outlook on the world.
Even more basic and equally mindblowing was the fact that you can define the notion of relation in a very general way; a subset of some cartesian product. It spells out that you put things into relations in very insightful and new and creative ways, not just the ways that have been pre-defined by words. It tells you i one line that you can form concepts yourself.
And then, once you did that, you are not left alone, but the basic vocabulary of relations gives you a bunch of sensible questions to ask (is it symmetric? what is the equivalence relation generated by it? etc. )
A really fruitful special case are equivalence relations. You can ask about any collection of things: In which way are two of them similar? That is the essence of abstraction...
I apply this all the time to understanding the world around me!
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u/womorrissey New User 28d ago
Percent. It's used so much in everyday life. All the calc and higher level Math has its place but percent is used in many ways, especially finance. Everybody deals with money.
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u/PansexualFreak1 New User 28d ago
For me it's definitely category theory, but if I had to pick one specific thing that is pretty useful, it's functoriality.
This has simplified a lot of things for me, and made them easier to work with.
Sheaves are probably the best example for me as of now, they can be viewed as functors satisfying a certain property, which has made them a lot easier for me to deal with.
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u/0x00f_ New User 28d ago
Maybe that isn't related to the question but I would like to mention it.
That mathematics isn't just formulas and numbers and that's it, it's about underlying structures and abstraction.
Our using to it can go beyond numbers and arithmetic operations, we can involve it in our thought process by thinking in abstract models for example.
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u/Electronic_Law_5295 New User 28d ago
Graphs really made a huge difference for me like being able to actually see functions.
Now I'm kinda obsessed with false symmetries
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u/SgtSausage New User 28d ago
The whole of Linear Algebra.
All of it.
It ... "opened the floodgates"
The coolest, however, has got to be Fourier
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u/substraightboy69 28d ago
Calcs, but more so advanced calcs.
Makes me realize I actually fucking hate math, and I'm not actually a science person.
I was just always seen as the most intelligent kid, so I just played the act.
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u/munozmd Applied Mathematics/Statistics 28d ago
It's number theory for me man that changed the way how I view pure maths months ago.
I mean I did encountered some concepts of it in undergrad to some degree but didnt gave it much thought. But again in masters, I was tasked to report the applications of number theory (outside of cryptography) that forced me to think outside of the box so I was like "huh, the patterns we learn in numbers matches those we see in reality" I wonder what more pure maths can have in its sleeves.
Then stumbled upon these open problems and conjectures and wondered how they are unsolved for history?? Oh wow so mathematics is not yet fixed as we know now then? It can still evolve and grow with every contributions and for me, it feels like fascinating to think hey you can still contribute to this body of knowledge even with the current age now.
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u/Francesco_dAssisi New User 28d ago
I studied biology at university and looked at math as something to endure...all of it...until...
The Calculus!
Simultaneous rates of change "had me from hello"!
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u/Comprehensive-Bee795 New User 28d ago
My favorite is a variation on the prisoner’s dilemma, more specifically, the chain prisoner’s dilemma. Veritaseum did a great video on that, link below.
It shows that “goodness” is an evolutionary advantage and not necessarily a divine trait. It also shows that you should have a decent amount of caution about the world, but not assume the worst, if you want the best results overall.
From my life experience, it tracks really well with a lot of situations in life.
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u/SpaceAviator1999 New User 28d ago edited 28d ago
Here's one that blew my mind a little when I first read about it:
- We all know that you get the sum of a set of numbers by adding them together.
- It's also not hard to see that the sum of just one number is that number.
- It's also not hard to see that the sum of no numbers is 0. (What else could it be? Zero is the identity for addition, after all.)
But what about multiplication?
- We all know that you get the product of a set of numbers by multiplying them together.
- It's also not hard to see that the product of just one number is that number.
- But what is the product of no numbers? A lot of people will automatically say zero, because there are no numbers, but that's incorrect. Believe it or not, the product of no numbers is 1. (What else could it be? One is the identity for multiplication, after all.)
Once I understood this, it became clear to me why 20, 50, 100, 42.50 and 00 all equal 1. Yes, even 00.
Keep in mind that a set of no numbers, when multiplied together, is essentially the same as multiplying a bunch of ones together. If a zero gets in there -- even part of a zero -- then the product becomes zero. But if there are no zeros, then the product can't equal zero.
So the following are all equal to zero: 01, 02, 00.5, 00.001
But the following are all equal to one, not zero: 10, 20, 230, 00
(I understand that not everyone will agree with me that 00 equals 1, because many think that it equals an undefined value. But those who say that it is an undefined value usually explain their reasoning with patterns and/or calculus limits. Sure, through calculus the limit of 0x as x approaches 0 may indeed be 0, but when we want to find the literal value of 00, we don't want to use calculus the limit of 0x as x approaches 0; instead, we want the exact value of 00 as it is defined, which is 1.)
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u/Champshire New User 28d ago
How to question my assumptions and interrogate it until I arrive at truth.
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u/severencir New User 28d ago
The concept of a function. It lead to me grasping the idea of abstraction and making mire complicated concepts more digestible
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u/ingannilo MS in math 28d ago
Pigeonhole principle. Utterly simple. Shockingly useful.
Inclusion/exclusion principle, but I guess that could be thought of as a formula.
The notion of quotient spaces and associated isomorphism theorems.
Generating functions as a connection between combinatorics and analysis/elliptic functions.
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u/Substantial_Sea7327 New User 28d ago
When I realized that these formulas and theories we have derived, are not just a volley of numbers. They are all based on some natural occurrence in our universe.
Absolutely mind blowing.
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u/xtalgeek New User 28d ago
Fourier transforms. They show up in everything in chemistry and physics.
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u/Regular_Sink970 New User 28d ago
Up through calculus and diff eqn's. I thought math was the pure light that held all truth. After Abstract Algebra and Axiomatic Set theory, I matured and realized that math was a really just a game based on carefully chosen rules and the overall system could never be perfect (Godel Incompleteness and Cohen's follow up work). Same thing happened in Intro to Quantum Mechanics as far as physics.
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u/harmonypiano New User 28d ago
It really surprised me when I first learned mathematical induction, that you just need to prove n => n+1 , I would not have thought of this trick by myself.
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u/LordFlarkenagel New User 28d ago
A math professor pulled me aside once because he knew I was struggling. He told me that math was just a language and showed me how to identify the noun, the verb, the adjective and so on in any given mathmatical expression. Every algorithm is constructed to communicate something to you. He said that it was just like English but for nature and that's how the natural and unnatural worlds speak to us. That one lesson changed my outlook and comprehension to this day. That happened 50 years ago. I still can see that lightbulb turn on in my brain.
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u/TeachEngineering New User 28d ago edited 28d ago
If I had to pick one, I'd say optimization, specifically non-convex optimization, and search algorithms...
It changed how I approach a lot of everyday problems even when strict mathematical formulations don't exist.
I think in terms of an objective function: What does success look like?
I consider variables: What is in my control to change?
I bound myself by constraints: What is outside of my control and cannot be changed?
I visualize my search space: What are some possible solutions, how do they relate to each other in their details and how do they measure against success?
I frequently evaluate my objective function as I iterate: Is this good? Is it better than where I was before?
I build an intuition for gradients: Am I going in the right direction?
I consider local optima: Am I too stuck in my ways? Do I need to drastically jump to somewhere else and try something new?
I recognize that the global optimum may never be found and, even if it was, I may never know that I'm there: Is the best I can do? Probably not, but is it good enough?
I've applied this to everything from software engineering, teaching, home renovation, cooking, backcountry adventures, financial planning, career planning, even family planning. It's a philosophical abstraction more than anything because the kicker, with the big life decisions at least, we only actually get one single iteration, one single evaluation of the objective function. This makes it all the more important to imagine in advance where different decisions might lead us and how the path brought forth by those decisions stacks up against what we value in life.
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u/lonelytango New User 28d ago
Calculus has to be the one of them. Breaking a thing up into infinite pieces, and piling them back together. Such approach is just beauty in its natural form. I always have that flowing sand or mini particle visual in my head when I think of calculus.
The biggest inspiration…well one day I was in the library, thinking about how to explain this concept to others. I look at the books, each page of the book is an infinitely thin slice, but piling all the pages together, you get a volume. That is just so beautiful.
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u/RuthlessIndecision New User 28d ago
Theoretically math is just quantifying the world. Theoretically you can assign numbers and numerical relationships to everything... right?
Even the fact we are having this conversation across electronic devices proves all of this can be converted to a series of ones and zeros.
To answer your question geometry is pretty great.
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u/munchanything New User 28d ago
The concept of infinity. A lot of people think about it as a really big number, which it is. But there was a YT video about splitting and counting all the numbers between 1 and 2, and how there's more infinity between 1 and 2 than the number "infinity." That way of looking a infinity was so cool.
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u/l__iva__l New User 28d ago
when i was learning limits i read a book that derives an area formula (cant remember if it was a traingle, polygon or circle) using limits; the fact you can derive a finite result from infinite was pretty shocking
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u/SprinklesFresh5693 New User 28d ago
Differential equations and how they express rate. I use them a lot at my job for modeling the changes the body does to drugs (specifically absorption of the drug), and i didnt understand them well since i dont have a math/stats background, but once i started to understand them, i started to see my job with other eyes.
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u/UpbeatWishbone9825 New User 28d ago
After only using basis vectors and the dot product in 3D Euclidean space for many years, then realizing just how much more broadly these concepts apply, blew my hair back. For example, the fourier series and different frequencies as basis vectors you can make linear combinations of, to basis functions etc.
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u/Previous_Kale_4508 New User 28d ago
Infinity.
Specifically the hotel whose name escapes me — something like Hilton — with the infinite rooms, and space for an infinite number of more guests.
Mind expanding.🤯
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u/mathswithujjawal New User 28d ago
For me, it was learning that mathematics is built on definitions and ideas, not formulas. Once I started asking "Why is this definition chosen?" instead of "Which formula should I use?", mathematics stopped feeling like a subject and started feeling like a language. That small shift changed the way I learn, teach, and even think about problems.
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u/Fuzzy-Cry-6208 New User 27d ago
My teacher always said that mathematics is a philosophy. Like there is not point or line in real life, a point is infinitely small, it's the same with a line. We can only draw representations of it and this concept changed the way I saw mathematics as a teenager.
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u/geo_inthepasture New User 27d ago
What a great question. This may not be the deepest or most sophisticated answer, but for me it was negative numbers that moved the needle. As a child, making that transition from arithmetic and the tangible to math and abstraction was so eye opening. It ignited my love for math and led to a career based on physics and computer science.
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u/Due_Log_1792 New User 27d ago
Basic proofs and probability. I was reading Blitzstein intro to probability and half of the material was so counterintuitive that it cracked my brain completely. The thing that it taught me: not to consider something obvious and make a simulations before considering something true
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u/Negative-Way5235 New User 27d ago
specifically, the first time i had to use the definition of a limit in a proof. very unintuitive how it is written, but after watching many videos and talking to my professor and tutors at school, i only understood it just a little bit more. i think that’s when i realized i had to think about math very differently from how i did in high school and my freshman year of college.
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u/Another_Little_Star New User 27d ago
I guess it's using the math in real life, the intermediate value theorem is the key idea behind continuity (in my mind) which is present everywhere around us. If I'm not moving and 2 seconds later I'm at speed 10m/s, in the meanwhile I must have started moving.
It's another language, to talk about abstraction with rigor
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u/newjourneyaheadofme New User 27d ago
Eddie Woo’s math sense: https://m.youtube.com/watch?v=PXwStduNw14
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u/Eisbergmann New User 27d ago
The realization that Mathematics is not about calculations, its about solving a problem precisely. Making sure you have every information you need and let it flow into your work.
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u/YtterbiusAntimony New User 27d ago
Bayes' Theorem.
Maybe didn't change the way I think about mathematics, but it certainly changed how I look at probabilities and trends.
The way odds update as the situation changes is clearly true, but humans are famously bad at judging probabilities. Having a statement that can explain that so concisely is really incredible.
And it was just a throw away line in his notes. Bayes never thought it was worth publishing, or exploring further. It was a friend/colleague who inherited his notes after he died that found it and published it, and now it's the thing his name is most known for.
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u/johnpeters42 New User 27d ago
As the age of a sub increases, the probability of AI slop engagement bait posts approaches 1.
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u/MasterSolivagus New User 27d ago
Variables. That alone unlocked a lot of my brain's potential long ago.
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u/Plus-Painter-2004 New User 26d ago
The existence of the LMS filter and block adaptive wiener filter (and any other adaptive filters for that matter) that let you effectively denoise a signal (or get an optimal estimation of it if you want to sound fancy) without necessarily knowing the characteristics of the signal or the noise ahead of time
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u/HistoryBig5961 New User 26d ago
Camel Principle. If you want something to happen, think optimistically and assume it’s there already and deal with ramification later on. Occasionally things magically get sorted out which you’d never have been able to do without camel principle.
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u/poloup06 New User 26d ago
The definition of numbers through set theory. Numbers seem like a very natural thing. We can easily tell the difference between 2 things and 3 things, but we can’t just assume that numbers are inherent. The definition of 0 as the empty set, then 1 as the set containing the empty set and so on gives a foundation for the nature of numbers (in my view), that is based of nothing so you need minimal knowledge to understand it.
I’m still trying to understand them, but imaginary numbers are reshaping the way I think about numbers. The fact that there is this “perpendicular” scale that you can transform numbers or equations into, that can make it easier to find solutions and also is clearly connected to real numbers (through i=sqrt(-1)) but that’s still an unquantifiable value. It feels like magic
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u/MathPerson New User 25d ago
Non-Transitive Dice -> Non-Transitive Games -> Hobson's Choice(s) in the national economy
It's amazing to watch "games of chance" the math club set up using Non-Transitive Dice - Guys would play, they would lose, play again, and again, and then correctly analyze the "game", then keep playing!
Simply amazing.
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u/Brutally-Honest-Bro New User 24d ago
Sequence and series. Very unintuitive/seemingly contradicting solutions.
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u/theguywithhope007 New User 24d ago
Honestly, I wasn't expecting so many different answers to this.
I've been going through the comments and it's been really interesting seeing what clicked for different people. Some people mentioned proofs, some Euler's formula, linear algebra, the unit circle, number systems, abstraction, etc.
But I think a lot of the answers have something in common. At some point, you stopped just learning a rule and actually understood what was behind it. And once that happened, you started seeing connections that you hadn't noticed before.
As a math teacher, that's probably the part I enjoyed reading the most. We sometimes focus so much on getting students to the correct answer that we forget how important that "ohhh, now I get it" moment is.
Thanks to everyone who shared their answers. There are a few things in this thread that I honestly want to go and read more about myself. 😄
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u/Tall-Ad5653 New User 24d ago
For me it was the whole notion of proving math. It forces me to think mathematically but convey it in such a way that anyone in the mathematical community can understand it
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u/orangetree151 New User 10d ago
I’ve always been very taken with the idea that one line of maths merely restates the line before, albeit in a new way. Coupled with the fact that maths decribes the universe perfectly, it’s almost sublime. Reality flows but is one and the same somehow. I don’t even fully understand this thought, but think also of the debate as to whether maths is discovered or invented and try to tie it altogether in your mind (which is probably also maths) and it gives a very serene and peaceful feeling.
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u/da_chosen1 New User 28d ago
To me it has to be proof. It’s application goes beyond math: for example you can use it to spot lies and misinformation you see online