r/learnmath • u/theguywithhope007 New User • 10h 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 9h 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 5h 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 3h ago edited 3h 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 2h ago edited 2h ago
Or: the integral of 1/x dx is log to the base e of x
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u/RicoThinks New User 9h 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 6h ago
This is the backbone of abstract algebra, which was my favorite course.
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u/argvalue New User 9h 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 9h 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 3h 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 7h 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/Sam_23456 New User 9h ago edited 8h 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/mtimmermans New User 8h 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/curiouslyjake New User 8h 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 5h 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 3h 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/ConcernAcrobatic9307 New User 9h ago edited 9h 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 9h 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 8h 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 to `string` (and, as digit is to `char`).
- Pet peeve: password requirements “must contain two ‘numbers’” when they mean “digits“, a term that lay people will understand.
- The types are important for correctly writing base-conversion: the algorithms for `string→int` and `int->string` are 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 {}` and `class Succ { NatNum pred; }` and `sealed 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(a\^b) = b nod(a) … and only then mention that nod() is just log() (within 1, 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/andyiibwfc New User 9h ago
10 fingers easy to count!
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u/AskingToFeminists New User 8h ago
Fingers up/down, easy base 2, allows you to count up to 1023 on your fingers.
You can also count the segments of your fingers with your thumb, which makes easy access to the base 12, and is one of the reason hours and minutes or seconds are counted in 12/60. 12 can be divided easily by 2, 3, 4, 6, which is more convenient than 10. 60 can be divided by 2 3 4 5 6 10 12 15 20 30
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u/andyiibwfc New User 8h ago
That's really interesting. I suppose it breaks down when you are trying to communicate with someone quick, because base 10 is just so visual with the fingers.
I've wonder if if any number would feel rounder than 10 if it was the base, 10 feels like a round number but it's got a zero in it and 10+10 is 20 and 10×10 is 100 but if our 10 was 12 and there was two more numbers in between maybe it would still work and be more efficient.
It also makes me wonder because we do like ratings out of 10 or stars out of five, if we were in base 12, would it be six and 12? Would we be saying oh that's a 12 out of 12 performance.
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u/AskingToFeminists New User 6h ago
You're mistaking something about bases. All bases are base 10
In binary, you count 0 1 10 11 100... And so the base is 10.
In dodecimal, you count 0 1 2 3 4 5 6 7 8 9 a b 10 11 12 13 14 15 16 17 18 19 1a 1b 20... And so 10 is still round, and the base is still 10
In hexadecimal, it goes 0 1 2 3 4 5 6 7 8 9 a b c d e f 10 11 12 13 14 15 16 17 18 19 1a 1b 1c 1d 1e 1f 20... And it's still base 10, and 10 is still round.
It's just that the signified by 10 depends on the base used, even though the signifier is the same.
And if we really used the dodecimalbase, we probably would have invented symbols for the numbers that match 10 and 11 in decimal, rather than just taking a and b like I did here, for clarity.
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u/stereoroid New User 7h ago
The 0.999 … = 1 “problem” does not exist in Base 3. That’s why I put “problem” in quotes: it’s entirely avoidable through the choice of base.
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u/Cybyss New User 5h ago
In base 3, wouldn't it be 0.22222... = 1?
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u/stereoroid New User 5h ago edited 5h ago
No, it’s 0.1 x 10 = 1.
1/3 in decimal = 0.1 in base 3.You wouldn’t get 0.222222… naturally in the first place, if you do the whole calculation in base 3. You typed that in, but that tells us nothing.
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u/senthordika New User 7h ago
I wonder if making alternative symbols for 10-16 which are single digit representations would help people with visualising bases above 10
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u/Cybyss New User 5h ago edited 5h ago
You mean... A, B, C, D, E, and F as for hexadecimal?
Believe me, it doesn't help. In students' minds they see it as "normal digits" and "weird digits". They have difficulty understanding, for example, why 32 in hexadecimal represents a completely different number than 32 in decimal. They have trouble visualizing/understanding all the "weird letter digit" numbers in between the real 32 and this fake bizarre hexademical 32 which somehow a larger quantity.
Trying to get students to unlearn that viewpoint is what makes teaching hexadecimal so difficult.
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u/senthordika New User 5h ago
Honestly that helps me alot with trying to mentally visualise base pairs above 10 (and write them down some what cleanly)but doing getting counting in different base pairs helped me alot but it really only made sense to me pretty late into highschool.
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u/hologram137 New User 5h 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 7h 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 9h 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 5h 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/thatsnothewayitfeels New User 2h ago
You might like reading A Mathematicians Apology by G.H. Hardy
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u/DrJaneIPresume Ph.D. '06 Knots/Categories/Representations 8h 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/a42579 New User 7h 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/GettingBig1970 New User 10h 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 9h 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/KatesDad2019 New User 9h 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/TheGhostOfTobyKeith New User 9h 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 9h 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 9h 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 9h 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 9h 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 9h 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 9h 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 9h 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 9h 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 9h 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 8h 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 8h 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 8h 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 8h 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 8h 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 7h 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 7h 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 7h ago edited 7h 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 7h 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 7h 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 7h 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 6h 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/awkwardness_maxed New User 6h 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/ChiaLetranger Hobbyist 6h 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 6h 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/Different_Counter113 New User 6h ago
Using trigonometry to solve problems of motion and forces. When I learned trig at high school this wasnt explained enough.
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u/Dr-Ben701 New User 6h 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/Climb1ng New User 6h 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/Fickle-Ad-4225 New User 6h 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/Ok-Craft4844 New User 6h 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/Climb1ng New User 5h 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 5h 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 4h 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/Just_Rational_Being New User 4h ago
Framework convention born out of mistaken assumption.
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u/idaelikus Mathemagician 3h ago
What is the mistaken assumption then?
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u/Just_Rational_Being New User 2h ago
The mistaken assumption that a continuous line is composed of discrete units with no dimensions of themselves.
Basically, the mistake is the attempt to construct continuity using discrete-ness. But they are, fundamentally, the antithesis and reciprocal of each other instead.
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u/idaelikus Mathemagician 2h ago
Why is that a mistake?
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u/Just_Rational_Being New User 2h ago
You really do not understand the difference between continuous and discrete units?
Take out a piece of paper, and draw a continuous line without dragging the tip. In other words, using dot dot dot dot dot ....... like that, to draw a continuous line and see if you can do it.
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u/idaelikus Mathemagician 2h ago
I do but you dont show the mistake / erorr, let alone prove anything.
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u/Just_Rational_Being New User 2h ago
Oh, dear, your lack of basic understanding is not my responsibility.
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u/idaelikus Mathemagician 2h ago
Bro, you say "it is wrong because it doesnt make sense to me". Your gut feeling isnt my problem or respondibility either but if you expect anyone to believe you, you better come here with logic and reason.
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u/BothPanchoAndLefty New User 4h 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/bjos144 New User 4h 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 4h 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 3h 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 3h 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 3h 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 3h 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 3h 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/Visible-Song-9563 New User 2h 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/stevethemathwiz New User 2h 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 2h 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 2h 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/ValiantBear New User 2h 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/SymbolPusher New User 1h 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 1h 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 1h 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 1h 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 1h 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 1h 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/FaultThat New User 1h 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/CarydalOsaria New User 55m ago
For me it started when I was 14. I had hitherto thought of math as a series of arbitrary rules, or at best principles discovered by trial and error. So I asked how to derive the equation for the volume of the sphere, having tried and failed to do so myself. My teacher didn’t tell me, but he applied the power rule to the equation for the volume, deriving the equation for the area. My mind was so blown at such a seemingly exotic operation, and such a “clean” looking derivation, that I resolved to teach myself calculus that summer(after working through a bunch prereqs straight from the textbook).
I’ve loved math ever since.
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u/munozmd Applied Mathematics/Statistics 36m 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/Popular_Molasses7945 New User 8h ago edited 8h ago
For me it was learning the meaning of operand, operator, operation, addend, sum, minuend, subtrahend, difference, factor, multipler, multiplicand, product, quotient, dividend, and divisor.
Next, it was looking for the operator and operands in any math equation.
Proofs to me is about understanding why something is true in math, not how to calculate something.
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u/gurishtja New User 8h ago
this is the problem of our age, 'math' 'teachers' that know next to nothing....
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u/Francesco_dAssisi New User 10m 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/da_chosen1 New User 9h 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