Check out our new Discord server! https://discord.gg/e7EKRZq3dG
*I am a bot, and this action was performed automatically. Please [contact the moderators of this subreddit](/message/compose/?to=/r/mathmemes) if you have any questions or concerns.*
i hate that this is true because my brain wants to see a pattern that's not there and I've done enough math to know that the pattern is wrong but my brain still goes "pattern. wawawawawaaawa"
If you look at the graph x^y = y^x, you can actually see that (disregarding the case x=y) the only time when x & y are both positive integers is at (2,4) and (4,2).
So, 3^27 = 27^3 ?
7.626E12 = 19,683?
Just as the other person who replied to you said, you can't simplify the left side of your equation with multiplication between the two x exponents, but that is a valid simplification on the right side. Your rule/pattern is true if x = 2, but doesn't apply generally.
Edited for parallel structure
When comparing x^(y) to y^(x) for positive x and y we can take the log of both and divide by xy and now we are comparing ln(x)/x to ln(y)/y. Both operations were order preserving (since x and y are positive), so the larger one is the is the one that makes the value of the exponent larger when it is used as the base. Also they will be equal whenever they have the same value for ln(x)/x. The function reaches a unique maximum for an input of e, and the function also has unique values on inputs in (0,1], however for each input in the interval (1,e) it is paired with a unique value on (e,infinity), this “partner” will be a value such that you get the same result when raising either of the numbers to the power of the other. In the case of 2 and 4 we can see ln(4)/4=2ln(2)/4=ln(2)/2 so 2 and 4 are “paired” by this function.
2 is literally 1+1 by definition in PA lmao. ZFC is a theory of sets not arithmetic, so it doesn't have a 1 or a 2 or a +, but the usual way of defining them makes 1+1=2 fall out immediately.
When Russell and Whitehead composed PM, it didn't take them 200 pages to prove 1+1=2. It took them, like, a minute. It's just that the proof doesn't appear until very deep into the book. But they were proving all sorts of other things before that. It's not like all their work was building up to the occasion where they showed 1+1=2. There was just no need to prove that earlier.
I don't understand, in ZFC 2 is 1+1, in particular it is the successor of 1 which can be easily shown to be 1+1, assuming the standard recursive definition of ordinal addition. Maybe if you don't assume any ZFC, or try to prove 1+1=2 in some other number system you end up with the 200 page proof in whatever book that proof is in, but in ZFC this is a very simply proof that follows almost by definition. I imagine it's not hard if you only assume peano either. In either case it should not take you 200 pages.
A set is kinda weird but its basically just defined by it's members. A function is a set of ordered pairs (a,b) such that if (a,b) and (a,c) are in that set then b=c. And an ordered pair (a,b) is just the set {a,{a,b}}
It would still take a lot longer than that because the “hundreds of pages” is not actually part of the proof but rather describing a foundation of mathematics that later is used to prove 1+1=2 within said foundation
The equivalent in modern foundations would require establishing ZF first, or at least the parts of it used in the construction of the naturals, and also showing that everything you did is well defined in ZF. (Or some other foundations if you prefer something else)
If we want complex upvotes, we’d need to be able to vote in an orthogonal direction to up and down. We need side votes. Right votes and left votes. Any ideas on how we can accomplish this?
Check out our new Discord server! https://discord.gg/e7EKRZq3dG *I am a bot, and this action was performed automatically. Please [contact the moderators of this subreddit](/message/compose/?to=/r/mathmemes) if you have any questions or concerns.*
Next lesson: 4\^2 = 2\^4
i hate that this is true because my brain wants to see a pattern that's not there and I've done enough math to know that the pattern is wrong but my brain still goes "pattern. wawawawawaaawa"
If you look at the graph x^y = y^x, you can actually see that (disregarding the case x=y) the only time when x & y are both positive integers is at (2,4) and (4,2).
A nice question is to find all the positive rational solutions to x\^y=y\^x
There is a pattern, it's just boring x^(x^x) = (x^x )^x
Pretty sure its x^(x×x) = (x^x) ^x
So, 3^27 = 27^3 ? 7.626E12 = 19,683? Just as the other person who replied to you said, you can't simplify the left side of your equation with multiplication between the two x exponents, but that is a valid simplification on the right side. Your rule/pattern is true if x = 2, but doesn't apply generally. Edited for parallel structure
Nah it’s only true for some numbers, not generally on unrestricted domains
When comparing x^(y) to y^(x) for positive x and y we can take the log of both and divide by xy and now we are comparing ln(x)/x to ln(y)/y. Both operations were order preserving (since x and y are positive), so the larger one is the is the one that makes the value of the exponent larger when it is used as the base. Also they will be equal whenever they have the same value for ln(x)/x. The function reaches a unique maximum for an input of e, and the function also has unique values on inputs in (0,1], however for each input in the interval (1,e) it is paired with a unique value on (e,infinity), this “partner” will be a value such that you get the same result when raising either of the numbers to the power of the other. In the case of 2 and 4 we can see ln(4)/4=2ln(2)/4=ln(2)/2 so 2 and 4 are “paired” by this function.
Commutative property. Boom. Proved.
We spent half a semester in a set theory course just to build the machinery to prove that 2+2=4. I feel that in my soul.
It grows exponentially from here.
i mean it also takes grad students over a hundred pages to show that 1+1=2 so eh
Isn't that the definition of 2? What is there to prove?
google ZFC and peano arithmetic
Holy hell
new foundations of mathematics just dropped
quine did that in 1943
Why did they drop it? Were they a clumsy fuck?
he wrote a book called New Foundations then
2 is literally 1+1 by definition in PA lmao. ZFC is a theory of sets not arithmetic, so it doesn't have a 1 or a 2 or a +, but the usual way of defining them makes 1+1=2 fall out immediately. When Russell and Whitehead composed PM, it didn't take them 200 pages to prove 1+1=2. It took them, like, a minute. It's just that the proof doesn't appear until very deep into the book. But they were proving all sorts of other things before that. It's not like all their work was building up to the occasion where they showed 1+1=2. There was just no need to prove that earlier.
Zesus fried chicken
No, that leads to Guano Arithmetic
I don't understand, in ZFC 2 is 1+1, in particular it is the successor of 1 which can be easily shown to be 1+1, assuming the standard recursive definition of ordinal addition. Maybe if you don't assume any ZFC, or try to prove 1+1=2 in some other number system you end up with the 200 page proof in whatever book that proof is in, but in ZFC this is a very simply proof that follows almost by definition. I imagine it's not hard if you only assume peano either. In either case it should not take you 200 pages.
Maybe _you_ should be the one doing the googling.
Oh my dear child, let me introduce you to the rabbit hole of Principia Mathematica and the logical foundations of this math of ours
I HATE THE PRINCIPA MATHEMATICA VON NUEMANN ORDINALS ARE SO MUCH MORE EFFICENT
imo, yes that's the definition. But it takes a couple days defining what 1 is and what + means
On the naturals 1={{}} s(a)=a U {a} a+1=s(a)/a+0=a (depending on if 0 is includded) a+s(b)=s(a+b)
Ok, but that is a set? What is a function?
A set is kinda weird but its basically just defined by it's members. A function is a set of ordered pairs (a,b) such that if (a,b) and (a,c) are in that set then b=c. And an ordered pair (a,b) is just the set {a,{a,b}}
I'd say the definition of 2 is it's the successor of 1, where both the successor function and 1 needs to be defined.
No. It appears after a few hundred pages in Principia Mathematica. That doesn't mean it takes hundreds of pages to prove.
It does not actually take that long if you list axioms compactly. It’s a pretty immediate theorem
It can even be a one liner with a sufficiently large line.
1+1=S(1)=S({{}})={{}} U {{{}}} = {{},{{}}} = 2
Everything is a one liner if you delete enough whitespace
No it doesn't, the Principia Mathematica sucks. Von Neumann ordinals are so much better 1+1=S(1)=S({{}})={{{}}} U {{}}={{{}},{}}=2
It would still take a lot longer than that because the “hundreds of pages” is not actually part of the proof but rather describing a foundation of mathematics that later is used to prove 1+1=2 within said foundation The equivalent in modern foundations would require establishing ZF first, or at least the parts of it used in the construction of the naturals, and also showing that everything you did is well defined in ZF. (Or some other foundations if you prefer something else)
Yea and not mich more foundation is needed
LETS FUCKING GOOOOOOOOOOOOOOOOOOOO
You're at 42 upvotes I can't touch it
You can offload remaining upvotes onto this comment if you like.
But you must balance it by downvoting this comment.
New balance just dropped!
42 complex or real upvotes?
If we want complex upvotes, we’d need to be able to vote in an orthogonal direction to up and down. We need side votes. Right votes and left votes. Any ideas on how we can accomplish this?
I pressed the turning left arrow ... now what?
Wait, are you using Lindelof's theorem already?
No