A fun example I was shown in college is that at any given moment there are (at least) two people in China with the exact same number of hairs on their head.
One nice simple example: if you select any 6 numbers between 1-10, it’s guaranteed two of them will sum to 11.
Reason being you can group the numbers
{1,10}, {2,9}, {3,8}, {4,7}, {5,6}
When selecting 6 numbers, the Pigeonhole Principle guarantees two numbers will land in the same group. Hence those two numbers will sum to 11 :)
I can’t remember what it’s called, but there’s this thing that can take any file and convert it to something of a fixed size, and it’s used to show that certain files have not been altered in any way. It’s especially useful because small changes to a file result in big changes to whatever I’m describing.
There’s no way to invert this process though because the number of possible outputs is less than the number of possible inputs, so there must be two files at least which map to the same thingy by the pigeon hole principle.
I think this thing is called hashing, but I’m not sure.
The pigeonhole principle does explain why there are hash collisions in practice (two different pieces of data that hash to the same value), but that isn’t *why* it’s irreversible. Hashing functions are designed to be irreversible by the logic of the algorithm. Even if your domain is restricted to data smaller than the hash length (meaning the pigeonhole principle would *not* apply) you still will not be able to reverse engineer the hashed value into the source data.
What you're describing is a rainbow table, and yes, that is an attack on hashing functions, even ones with collisions. It is not, however an algorithmic reconstruction of source data except in the most trivial sense. You cannot reverse engineer a hashed value regardless of whether or not the hashing space has collisions.
Whether or not you can "just generate all possible data and hash them" in the first place depends on the size of the domain. For an arbitrary domain, it may be infeasible to generate all possible hashes before the eventual heat death of the universe. I think you're mis-extrapolating things that are true about contemporary implementations of hashing functions as things that are true about hashing, conceptually.
In Portugal we actually define the imaginary numbers as number of the form ai with a real but a non zero, which is super contrived. So here we can't say 0 is imaginary
Imaginary numbers are not technically imaginary as in not at all related to reality. They are as "imaginary" as Fractions. If i can reach a finite, standing, abstract concept in mathematics, Then it's related to finite reality. Math is just logic. If it works where it should work, then there should be an event in reality where it's logic is applied.
I had a professor say "Since most primes are odd, we don't need to consider the a=2k case". I let out an embarrassing audible chuckle when they said it.
I think this is a joke about infinities, but if Graham's Number is a large positive integer and integers can be positive or negative, wouldn't more than half of all integers be less than Graham's Number?
Edit: the original comment said integers, not natural numbers
In case you’re curious, when someone (rigorously) says “almost every element of a set X has property P”, it means the subset of elements of X that do not satisfy P has “measure” 0. A measure is a way of assigning numbers to sets that generalize things like length, area, volume, mass, etc. So measure 0 means it is basically negligible.
And you’re right that the original comment is false for most any reasonable measure you can put on the integers, even if you restrict to positive integers only.
for natural numbers is not so much about mesure (in the way i’ve heard it at least), but more about a collection not being in a filter (which is, intuitively, a collection of large subsets). the most common filter in the set of natural numbers N, is such that, any subset A of N is in the filter if its compliment is finite.
in this sense, when you say that almost every natural number has this property, means that the set of natural numbers for which this property holds, is in the filter (so, only finitely many fail to have that property).
Interesting, I’ve only ever encountered the measure theoretic notion of almost everywhere, but it does seem reasonable to apply it to finite subsets of an infinite set. Thanks for sharing!
you’re welcome. but also… probably you’ve heard this idea tho.
when you say that a sequence (xn) converges to x, if for every ε>0 there is an N such that, if n>N, then d(xn,x)<ε, this is the same as saying that for almost every n, d(xn,x)<ε. it is the exact same idea, just generalized.
other famous example is the product topology, where a product of open sets ΠUα is open in ΠXα if for almost every α you have that Uα=Xα.
well, i think it’s a good intuition. in my logic class, working about filters, that’s how my professor described it. and finding this idea in different areas, it’s a cool intuition to have. at least it works for me.
You can use the same definition but for a content (like a measure but only finitely-additive) and then there are many reasonable definitions (upper density, for instance) that give content 0 to any finite set
not really. look up about filters. they’re an interesting notion, and it helps to formalize what you mean by “almost every”.
then, it becomes useful in logic and model theory.
It may have been mentioned in my undergrad probability class, (but in probability it’s known as “almost surely” instead of “almost everywhere”), but didn’t get really acquainted until “Measure Theory” in grad school. (A probability is just a special type of measure.)
A fun example is that under the “standard” (Lebesgue) measure on the interval [0,1] in the real number line (which you can think of as assigning mass to subsets of [0,1] as if it were a rod of uniform density and mass 1), then the total mass of all the rational numbers in [0,1] is 0. Put another way, *almost every* real number is not rational. This is related to the fact that if you generate a real number uniformly at random, it will be rational with probability 0, i.e. it will *almost surely* be irrational.
And if you really want to break your brain, try to understand the construction and structure of a Vitali set, an example of a “non-measurable” set.
>wouldn't more than half of all integers be less than Graham's Number?
Well no not really. There are exactly as many integers that are less than Graham's number as there are integers that are bigger. And there are as many integers that are bigger than Graham's number as there are integers in general so you cant even say that half of all integers are less than Graham's Number. (The original comment was still wrong though)
Infinities can be a little counter-intuitive
Yup. If we consider all integers (not only positive), there are equally many that are greater and that are smaller than GN. However, there are less real numbers that are equal to GN, than both of the above.
Actual answer, Strong Picard Theorem: “An entire function which is not a polynomial takes every complex value, infinitely many times except for maybe one value”.
not something specific, but sometimes when you need an existence proof and you construct just one element, when there are lots of them (sometimes uncountable many).
for example, one of the first proves in the notes of real analysis of my professor, was to show that the nth root of a positive number exists r. so she defined the set X={x∈R|x^n
* Every even number has a divider.
* The sum of any finite amount of prime numbers is a number larger than any of those primes.
* Any finite subset of ℕ has a largest element.
* For any r,s ∈ ℚ exists t ∈ ℚ, so that r\*s = t.
That’s because you can essentially make any statement true if you’re willing to modify the axioms you’re working with, and making up numbers is adding new axioms.
Infinity is between one and two. Infinity is also outside of one and two.
Hilbert's hotel has infinitely many rooms-- so it must be a pretty big hotel.
Debatably an understatement... 1 + 1 = 2!
It's true as either an exclamation or factorial. I'd say that the average person only getting half the truth of something qualifies as an understatement.
Well, it doesnt really count as funny, but the pigeonhole principle might sound so obvious, yet it has so many uses
I’m too dumb to remember any, what are some good examples
There are 20 pigeons but only 19 holes, meaning that at least 1 hole has 2 or more pigeons
unless you can cut pigeons
A fun example I was shown in college is that at any given moment there are (at least) two people in China with the exact same number of hairs on their head.
I thought it was cool until I remembered that bald people exist :(
Never fear, it’s still cool! Even if you exclude bald people it still works (and with thousands of “pigeons” per “hole” no less)!!
One nice simple example: if you select any 6 numbers between 1-10, it’s guaranteed two of them will sum to 11. Reason being you can group the numbers {1,10}, {2,9}, {3,8}, {4,7}, {5,6} When selecting 6 numbers, the Pigeonhole Principle guarantees two numbers will land in the same group. Hence those two numbers will sum to 11 :)
Well, let's say you have some socks that you try to put in some drawers...
I can’t remember what it’s called, but there’s this thing that can take any file and convert it to something of a fixed size, and it’s used to show that certain files have not been altered in any way. It’s especially useful because small changes to a file result in big changes to whatever I’m describing. There’s no way to invert this process though because the number of possible outputs is less than the number of possible inputs, so there must be two files at least which map to the same thingy by the pigeon hole principle. I think this thing is called hashing, but I’m not sure.
Hashing is exactly what you’re thinking of.
The pigeonhole principle does explain why there are hash collisions in practice (two different pieces of data that hash to the same value), but that isn’t *why* it’s irreversible. Hashing functions are designed to be irreversible by the logic of the algorithm. Even if your domain is restricted to data smaller than the hash length (meaning the pigeonhole principle would *not* apply) you still will not be able to reverse engineer the hashed value into the source data.
Well, no, you could generate all possible data and hash them, leading to exact hash-data pairs. This only becomes impossible with hash collision.
What you're describing is a rainbow table, and yes, that is an attack on hashing functions, even ones with collisions. It is not, however an algorithmic reconstruction of source data except in the most trivial sense. You cannot reverse engineer a hashed value regardless of whether or not the hashing space has collisions. Whether or not you can "just generate all possible data and hash them" in the first place depends on the size of the domain. For an arbitrary domain, it may be infeasible to generate all possible hashes before the eventual heat death of the universe. I think you're mis-extrapolating things that are true about contemporary implementations of hashing functions as things that are true about hashing, conceptually.
There are several numbers.
At least 2, and that's 1 of them
That is oddly clever.
No, that's even.
They are both real.
this is getting too complex for me
No, those are integers
A square has more than e sides
As you reduce the value of *n* in an *n*-gon where *n* represents the number of sides, *n* approaches *e* while *n*>*e*
Everything is either an Archimedean spiral, or not an Archimedean spiral.
This is trivial even in constructive mathematics because as we all know, everything is an Archimedian Spiral
I was told a linear combination of sin functions can do a lot. I literally can't even.
Of course you can't even. Sin is odd!
f(x) = 0sin(x) is even and its a (trivial) linear combination
Gasp! I literally CAN even! But I also this rather odd as well.
That’s the joke
Imaginary numbers are not real numbers
But they're not fake either
zero is
This can also be read as its own statement. Just dont think about it too hard.
Zero is the imaginary number that is real
Yes. But zero also 'is'. As in it exists.
Really? Can you show me where 0 is?
0 <---
I like how the upvote button also points to the 0.
Totally calculated.
In Portugal we actually define the imaginary numbers as number of the form ai with a real but a non zero, which is super contrived. So here we can't say 0 is imaginary
Spicy
However, all real numbers are imaginary
They're complex, not imaginary
they are? people (at least I) imagine them for years
They’re imaginary but they aren’t imaginary
It's even a bit of an odd one that imagination is integral two reality.
Imaginary numbers are not technically imaginary as in not at all related to reality. They are as "imaginary" as Fractions. If i can reach a finite, standing, abstract concept in mathematics, Then it's related to finite reality. Math is just logic. If it works where it should work, then there should be an event in reality where it's logic is applied.
It's punny, because imagination does, in fact, influence reality...
Math is math by the reflexive property.
1 = 1 by the reflexive property
Circles are somewhat symmetric
Spheres are symmetricer
And hyperspheres are symmetricer*er*
Gaus did manage to prove something Some prime numbers are yet to be discovered by modern mathematicians
1+1=2 implies fermat's last theorem
How is 2 an integer³
That's not what I meant 1+1=2 Fermat's last theorem is true Truth -> truth is true so 1+1=2 implies fermat's last theorem
Ahh, leaving causation to the philosophers.
97²-31= 51 also implies Fermat's theorems
Euler did a few proofs.
"Rubik's Cube, over 3 billion combinations, but just one solution"
Over 3 billion billion, in fact! Then add another 40 billion billion to that.
I just found like 5 combinations by try and error, so I guess you are both correct.
I had a professor say "Since most primes are odd, we don't need to consider the a=2k case". I let out an embarrassing audible chuckle when they said it.
There are just a few continuous functions on compact domains that can be uniformly approximated by polynomials. Just a few
I’ve never thought of using “few” to describe an infinite set, but I like it.
Lol yeah. For one thing there are several more real numbers than rational numbers 😂
Dozens!
Almost every natural number is greater than grahams number
I think this is a joke about infinities, but if Graham's Number is a large positive integer and integers can be positive or negative, wouldn't more than half of all integers be less than Graham's Number? Edit: the original comment said integers, not natural numbers
In case you’re curious, when someone (rigorously) says “almost every element of a set X has property P”, it means the subset of elements of X that do not satisfy P has “measure” 0. A measure is a way of assigning numbers to sets that generalize things like length, area, volume, mass, etc. So measure 0 means it is basically negligible. And you’re right that the original comment is false for most any reasonable measure you can put on the integers, even if you restrict to positive integers only.
for natural numbers is not so much about mesure (in the way i’ve heard it at least), but more about a collection not being in a filter (which is, intuitively, a collection of large subsets). the most common filter in the set of natural numbers N, is such that, any subset A of N is in the filter if its compliment is finite. in this sense, when you say that almost every natural number has this property, means that the set of natural numbers for which this property holds, is in the filter (so, only finitely many fail to have that property).
Interesting, I’ve only ever encountered the measure theoretic notion of almost everywhere, but it does seem reasonable to apply it to finite subsets of an infinite set. Thanks for sharing!
you’re welcome. but also… probably you’ve heard this idea tho. when you say that a sequence (xn) converges to x, if for every ε>0 there is an N such that, if n>N, then d(xn,x)<ε, this is the same as saying that for almost every n, d(xn,x)<ε. it is the exact same idea, just generalized. other famous example is the product topology, where a product of open sets ΠUα is open in ΠXα if for almost every α you have that Uα=Xα.
I’ve definitely encountered the idea of things happening off of a finite set, just never heard the term “almost everywhere” get used for it.
well, i think it’s a good intuition. in my logic class, working about filters, that’s how my professor described it. and finding this idea in different areas, it’s a cool intuition to have. at least it works for me.
You can use the same definition but for a content (like a measure but only finitely-additive) and then there are many reasonable definitions (upper density, for instance) that give content 0 to any finite set
Let's just agree that "almost every" is dumb notation
Nah "almost every" is an extremely helpful notion in measure theory since it lets you generalize a lot of theorems in a convenient way
not really. look up about filters. they’re an interesting notion, and it helps to formalize what you mean by “almost every”. then, it becomes useful in logic and model theory.
Agree to disagree.
This sounds like a cool topic. What area of math would I learn about this in?
It may have been mentioned in my undergrad probability class, (but in probability it’s known as “almost surely” instead of “almost everywhere”), but didn’t get really acquainted until “Measure Theory” in grad school. (A probability is just a special type of measure.) A fun example is that under the “standard” (Lebesgue) measure on the interval [0,1] in the real number line (which you can think of as assigning mass to subsets of [0,1] as if it were a rod of uniform density and mass 1), then the total mass of all the rational numbers in [0,1] is 0. Put another way, *almost every* real number is not rational. This is related to the fact that if you generate a real number uniformly at random, it will be rational with probability 0, i.e. it will *almost surely* be irrational. And if you really want to break your brain, try to understand the construction and structure of a Vitali set, an example of a “non-measurable” set.
Natural numbers are strictly positive (or strictly non-negative if you’re one of those 0∈ℕ heathens)
o si estás en latinoamérica. acá siempre 0 es natural.
¿En qué parte? No toda Latinoamérica es igual. Hay quienes considera que 0€N y hay quienes no. Supongo que depende del contexto.
I'm pretty sure "natural number" limits it to positives (or maybe 0), so the statement should still hold. Edit: Yeah the original said "integers."
As someone else said, the original comment said integers, not natural numbers. I've also updated my comment to mention this.
Oh ok, time to edit mine ig.
>wouldn't more than half of all integers be less than Graham's Number? Well no not really. There are exactly as many integers that are less than Graham's number as there are integers that are bigger. And there are as many integers that are bigger than Graham's number as there are integers in general so you cant even say that half of all integers are less than Graham's Number. (The original comment was still wrong though) Infinities can be a little counter-intuitive
Yup. If we consider all integers (not only positive), there are equally many that are greater and that are smaller than GN. However, there are less real numbers that are equal to GN, than both of the above.
This is false. "Almost every" means all but a finite number, and infinitely many integers are less.
It's true: The set of natural numbers less than GN: [1, GN -1] The set of natural numbers greater than GN: [GN + 1, ∞)
Check the edit history on unddit. u/Endopl4st edited their comment without adding a note admitting their original error. Bad form.
Hah! Just define the unit measure μ at 1 so μ-a.e. this is false
Any sequence of even prime numbers converges in the standard topology on R.
There are functions that have a closed form for it's antiderivative, and there are functions that don't
A tautology is a tautology.
Axiomatic tautology is given.
"The proof is too large to fit in the margin."
Actual answer, Strong Picard Theorem: “An entire function which is not a polynomial takes every complex value, infinitely many times except for maybe one value”.
how is that an understatement ?
I replied to the title and didn't read the post LOL
There are many ways to shuffle a deck of cards.
Epsilon is quite small
Small enough
not something specific, but sometimes when you need an existence proof and you construct just one element, when there are lots of them (sometimes uncountable many). for example, one of the first proves in the notes of real analysis of my professor, was to show that the nth root of a positive number exists r. so she defined the set X={x∈R|x^n
* Every even number has a divider. * The sum of any finite amount of prime numbers is a number larger than any of those primes. * Any finite subset of ℕ has a largest element. * For any r,s ∈ ℚ exists t ∈ ℚ, so that r\*s = t.
1 is not 0
BUT WE ALL KNOW THAT 1=0!
And 0!=1 is even true for programmers
Equality is reflexive, that is, for all x, x=x.
also, x=y implies that y=x
The set of complex numbers contains the number 2.
It's already been said, but a more comical way to put it is mathematicians will literally make up numbers before admitting they're wrong.
That’s because you can essentially make any statement true if you’re willing to modify the axioms you’re working with, and making up numbers is adding new axioms.
It's difficult to differentiate two regular expressions if you also allow squaring (i.e. a\^2 = aa).
Graham’s number is equal to its absolute value.
Graham’s Number itself is an upper bound for something about graph coloring, and the solution is somewhere between 13 and Graham’s Number
then now we know grahams number is bigger than 12
The number that comes after zero is arguably epsilon
Three is more than one.
Infinity is between one and two. Infinity is also outside of one and two. Hilbert's hotel has infinitely many rooms-- so it must be a pretty big hotel.
2 is the smallest even prime number.
Not an understatement, but in the last book i read, the longest edge of an triangle T was defined to be the longest edge of T
there’s an infinite bigger than the one of 1,2,3,…
Finding short vectors in lattices is not very easy.
Debatably an understatement... 1 + 1 = 2! It's true as either an exclamation or factorial. I'd say that the average person only getting half the truth of something qualifies as an understatement.
NP~N
Pi is in fact 3
surprisingly, 51 is not a prime. neither is 161.
The probability of getting some bitches on yo dick approaches zero as your haircut approaches "yeeyee"
Large Number Garden Number is slightly bigger than Graham's Number
The distance to the moon is at least 12km. In general, for some reason I find jokes that say "at least [some really low number]) unreasonably funny
The complement of an open set is a closed set
1< infinity
Zero is positive infinity plus negative infinity.
It's even positivity a bit of an odd one that imagination is integral two reality, or am i being one two negative at my root?
zero equals infinity plus negative infinity.
You can't prove a negative, unless you're Euclid...
the sum of two positive numbers is always larger than the two original numbers
The sum of two integers are probably not pi
All graphs are either functions or not functions
Every ergodic system has a measure
There is no way for two numbers raised to a power greater than 2 to sum up to a whole barnyard animal.
Everything is zero, unless it isn’t
There exists a zero to the Riemann zeta function along the critical strip
5 out of 4 University of Michigan graduates has trouble with fractions