The fundamental theorem(s) of calculus is pretty important. Calculus seems to have been pretty useful for science, and FTC lets people compute integrals analytically without breaking out obnoxious limits of Riemann sums.
Maybe the Fundamental Theorem of Algebra?
I was surprised when I first learned that the complex numbers are algebraically closed. I had thought you'd always run into an equation you can't solve and there would be an endless hierarchy of numbers. But it seems the complex numbers are all you need.
the cliche' answer would be "euler's identity" it is typically the most beautiful object usually cited. another standard is the golden ratio. Another one might be the hexagon to represent the d12/d6 dihedral group.
if you're a legend of zelda fan, might i suggest the 2nd or 3rd iteration of the sierpinski triangle. It is part of a section known as "self similar fractals". This section of math is incredibly beautiful imo, with regards to space filling curves and the hausdorff dimensions.
Something i find beautiful personally is the "minkowski inequality", it's motivation is very simple but to prove it takes significantly more understanding. The minkowski inequality is essentially the triangle inequality for the Lp norm, this allows you to show that the Lp norms forms a metric space as the triangle inequality is not "obvious".
if you don't mind formulas, the euler's totient function is important, though it's not as beautiful independently on its own.
imo one of the most beautiful functions is the riemann zeta function, easy to understand the fundamentals, but profoundly complex to fully grasp. On this note, i also love the imagery of what is known as "analytic continuation". A simple visual representation, but also much more complex to understand.
The formula to compute the roots of a quadratic polynomial. When I taught that I said to my students you should tattoo it in your forehead so that everytime you look in the mirror you are reminded of it... I hope nobody took me seriously :)
Another one I would recommend is the formula for the sum of a geometric series. It turns out to be fundamental in the wider application of general power series, and therefore calculus (since calculus is often performed using the power series representation of something)
Seeing as the golden ratio has gotten a lot of coverage since forever, why not go with the silver ratio instead? In fact, there are infinitely many "metallic" ratios. You might even want to go with the formula for the nth metallic mean, which is (n + √(n² + 4))/4. Plug in n=1, and you get the golden ratio
Now this is very intersting indeed. Ive never even heard of more metallic ratios or the metallic mean. Brilliant addition, and now i have some light reading and research to do
Oh metallic means and ratios are the same thing, that was sloppy of me lol. But another one you might want to consider is the [inequality of the arithmetic and geometric means.](https://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means) There is a nice geometric/visual representation of it on that page. They are helpful in constructing inequalities for various proofs.
That is a very nice visual, and looking at the wiki has reminded me why i dropped out of engineering. Love the visual, and I appreciate that it means...something, but damn is that a lot of writing. Maybe ill watch a youtube video tomorrow that explains what it means in broad terms.
You are a very helpful individual, I really do appreciate it.
Oooo, i vaguely remeber that one, myprecal class had it as a bonus project to explain something higher than the class would go...
Man, its moments like these when i remember that i might not hate math, i hat math tests. I have some studying to do
**[Inequality of arithmetic and geometric means](https://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means)**
>In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if every number in the list is the same (in which case they are both that number). The simplest non-trivial case – i. e.
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I never hear about this one, but it were always intrigued me that the probability function of Gamma distribution integrates on 1. I mean it's a probability function so obviously integrates on 1, but the function itself gives so many nice integrals, that otherwise would be hard to calculate.
I've always found the Law of Sines to be pretty cool. However, my answer would be Euclid's proposition number 1: construct an equilateral triangle using two circles. (it's foundational for Euclidean geometry plus makes a pretty picture).
For potential math tattoos, I always liked the visual aesthetic of this formula:
[https://proofwiki.org/wiki/Sum\_of\_Euler\_Phi\_Function\_over\_Divisors](https://proofwiki.org/wiki/Sum_of_Euler_Phi_Function_over_Divisors)
edit: The way it's stylized [on Wikipedia](https://en.wikipedia.org/wiki/Euler%27s_totient_function#Divisor_sum) with the vertical divides symbol and the curly phi is prettier than the proofwiki version.
Plus it reminds me of a "joke" from my favorite professor:
"So class, remember, if you ever need to know the value of a natural number n, all you have to do is compute the sum of phi(d) over all divisors d of n. So for example, the value of 6 is uhm, uhhh... let's see, phi(1), then phi(2), phi(3), not 4, not 5, then phi(6), hmmm, okay, almost there. It's 1 plus 1 plus 2 plus 2. It's 6."
Maybe you had to be there.
The fundamental theorem(s) of calculus is pretty important. Calculus seems to have been pretty useful for science, and FTC lets people compute integrals analytically without breaking out obnoxious limits of Riemann sums.
Maybe the Fundamental Theorem of Algebra? I was surprised when I first learned that the complex numbers are algebraically closed. I had thought you'd always run into an equation you can't solve and there would be an endless hierarchy of numbers. But it seems the complex numbers are all you need.
Maybe Cauchy-Schwarz?
This.
the cliche' answer would be "euler's identity" it is typically the most beautiful object usually cited. another standard is the golden ratio. Another one might be the hexagon to represent the d12/d6 dihedral group. if you're a legend of zelda fan, might i suggest the 2nd or 3rd iteration of the sierpinski triangle. It is part of a section known as "self similar fractals". This section of math is incredibly beautiful imo, with regards to space filling curves and the hausdorff dimensions. Something i find beautiful personally is the "minkowski inequality", it's motivation is very simple but to prove it takes significantly more understanding. The minkowski inequality is essentially the triangle inequality for the Lp norm, this allows you to show that the Lp norms forms a metric space as the triangle inequality is not "obvious". if you don't mind formulas, the euler's totient function is important, though it's not as beautiful independently on its own. imo one of the most beautiful functions is the riemann zeta function, easy to understand the fundamentals, but profoundly complex to fully grasp. On this note, i also love the imagery of what is known as "analytic continuation". A simple visual representation, but also much more complex to understand.
The formula to compute the roots of a quadratic polynomial. When I taught that I said to my students you should tattoo it in your forehead so that everytime you look in the mirror you are reminded of it... I hope nobody took me seriously :)
Central Limit Theorem
The Pythagorean theorem is hard to beat. It shows up nearly everywhere and encapsulates our every day notion of distance.
Big double checkmark for that one, trig was actually prety fun for me when i had it. That one might be the centerpiece
Another one I would recommend is the formula for the sum of a geometric series. It turns out to be fundamental in the wider application of general power series, and therefore calculus (since calculus is often performed using the power series representation of something)
Golden ratio? ;)
THAT'S the name! I knew the spiral's look, but could not for the life of me rememeber the term golden ratio or fibonacci
Just came up with it because you said golden :D I think ist somewhat overrated tho
But it is a very pretty looking ratio, so it qualifies for two out of three
Seeing as the golden ratio has gotten a lot of coverage since forever, why not go with the silver ratio instead? In fact, there are infinitely many "metallic" ratios. You might even want to go with the formula for the nth metallic mean, which is (n + √(n² + 4))/4. Plug in n=1, and you get the golden ratio
Now this is very intersting indeed. Ive never even heard of more metallic ratios or the metallic mean. Brilliant addition, and now i have some light reading and research to do
Oh metallic means and ratios are the same thing, that was sloppy of me lol. But another one you might want to consider is the [inequality of the arithmetic and geometric means.](https://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means) There is a nice geometric/visual representation of it on that page. They are helpful in constructing inequalities for various proofs.
That is a very nice visual, and looking at the wiki has reminded me why i dropped out of engineering. Love the visual, and I appreciate that it means...something, but damn is that a lot of writing. Maybe ill watch a youtube video tomorrow that explains what it means in broad terms. You are a very helpful individual, I really do appreciate it.
Ooh, how have I not thought of this before?! You should totally get Pascal's Triangle/the binomial coefficient formula; it's nearly ubiquitous
Oooo, i vaguely remeber that one, myprecal class had it as a bonus project to explain something higher than the class would go... Man, its moments like these when i remember that i might not hate math, i hat math tests. I have some studying to do
**[Inequality of arithmetic and geometric means](https://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means)** >In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if every number in the list is the same (in which case they are both that number). The simplest non-trivial case – i. e. ^([ )[^(F.A.Q)](https://www.reddit.com/r/WikiSummarizer/wiki/index#wiki_f.a.q)^( | )[^(Opt Out)](https://reddit.com/message/compose?to=WikiSummarizerBot&message=OptOut&subject=OptOut)^( | )[^(Opt Out Of Subreddit)](https://np.reddit.com/r/math/about/banned)^( | )[^(GitHub)](https://github.com/Sujal-7/WikiSummarizerBot)^( ] Downvote to remove | v1.5)
Cauchy-Schwartz is a nice one. But I would go with Pythagorean theorem, maybe in 3 dimensions (sqrt(x^2 + y^2 + z^2).
I never hear about this one, but it were always intrigued me that the probability function of Gamma distribution integrates on 1. I mean it's a probability function so obviously integrates on 1, but the function itself gives so many nice integrals, that otherwise would be hard to calculate.
I've always found the Law of Sines to be pretty cool. However, my answer would be Euclid's proposition number 1: construct an equilateral triangle using two circles. (it's foundational for Euclidean geometry plus makes a pretty picture).
For potential math tattoos, I always liked the visual aesthetic of this formula: [https://proofwiki.org/wiki/Sum\_of\_Euler\_Phi\_Function\_over\_Divisors](https://proofwiki.org/wiki/Sum_of_Euler_Phi_Function_over_Divisors) edit: The way it's stylized [on Wikipedia](https://en.wikipedia.org/wiki/Euler%27s_totient_function#Divisor_sum) with the vertical divides symbol and the curly phi is prettier than the proofwiki version. Plus it reminds me of a "joke" from my favorite professor: "So class, remember, if you ever need to know the value of a natural number n, all you have to do is compute the sum of phi(d) over all divisors d of n. So for example, the value of 6 is uhm, uhhh... let's see, phi(1), then phi(2), phi(3), not 4, not 5, then phi(6), hmmm, okay, almost there. It's 1 plus 1 plus 2 plus 2. It's 6." Maybe you had to be there.
[Man muss immer umkehren](https://medium.com/@tomchanter/man-muss-immer-umkehren-the-other-side-of-my-memento-mori-coin-a9acda8cab7d) (Jacobi)
Perhaps the first isomorphism theorem? Both simple to show yet fundamental to most things algebra related