3 ms·

When 1+1+1 Equals 1

41 points by surprisetalk 1y ago

pyuser583 1y ago
Interesting. I always associate "1 + 1 = 1" with idempotency. Here, "1 + 1 + 1 = 1", but "1 + 1 = 0". I'm not a math whiz, so I'm just stuck with "1 + 1 = 2."
- aatd86 1y ago
  you're wrong. 1 + 1 = 10
  - lisper 1y ago
    Which of course is 0 mod 2.
    - aatd86 1y ago
      And 0^0 x 2
  - aleatorianator 1y ago
    [dead]
  - not_a_bot_4sho 1y ago
    My son once asked me what 1+1 was equal to. I said, "two!" He said, "no, it's eleven." And that's when I realized he's going to be a JavaScript coder.
- deleted 1y ago
  [deleted]
- harperlee 1y ago
  XOR is a simple operation that shows that behavior.
khannn 1y ago
Terrence Howard has entered the chat
- kevin_thibedeau 1y ago
  Shhh! Don't give them any ideas.
susam 1y ago
A simple example where 1 + 1 + 1 = 1 is ℤ₂, the group of integers modulo 2 under addition. In fact, in any group with binary operation, say +, the identity element 0, and a non-identity element a, we have a + a + a = a if and only if a + a = 0 (i.e. a has order 2). There are plenty of groups with elements a satisfying a + a = 0. ℤ₂ as mentioned above has its unique non-zero element of order 2. The Klein group V₄ has three non-identity elements, each of order 2. Dihedral groups D₂ₙ (the symmetry groups of regular n-gons) contain reflections, all of which have order 2. Symmetric groups Sₙ (n ≥ 2) contain transpositions, each of which has order 2. For example, in the dihedral group D₈, if we let a be a reflection of the square, then a + a = 0 and a + a + a = a. But this is conventionally written in multiplicative notation as a² = the identity element, so a³ = a. Similarly, in the symmetric group S₃ under the binary operation of composition, if a denotes the transposition (12), then (12)(12) is the identity element and (12)(12)(12) = (12). In other words, applying a transposition three times is the same as applying it once. In the last two examples, it is conventional to use product notation instead of +, although whether we use + or · for the binary operation does not matter mathematically. It is conventional to use + in some subjects (coding theory, additive groups of integers modulo n, etc.) and · in others (permutation groups, dihedral groups, etc.). Often + is used for the binary operation in abelian groups and · in non-abelian ones. I'm sure none of this is particularly insightful to someone who has studied group theory, but still I wanted to share a few concrete examples here.
- HWR_14 1y ago
  [My post below is wrong] > In fact, in any group with binary operation +, identity element 0, and a non-identity element a, we have a + a + a = a if and only if a + a = 0 (i.e. a has order 2). The "if" is correct. The "only if" is not. (I assume that '+' and '0' are used as shorthand for "any binary operation" and "the identity of that binary operation", as I don't recall cases where "+" and "*" are used for specific types of binary operations).
  - susam 1y ago
    > The "if" is correct. The "only if" is not. Both "if" and "only if" are correct. Let a + a + a = a. Adding the inverse of a to both sides, we get a + a = 0. Let a + a = 0. Adding a to both sides, we get a + a + a = a. > I assume that '+' and '0' are used as shorthand for "any binary operation" and "the identity of that binary operation" Yes. As I mentioned in my previous comment, "In the last two examples, it is conventional to use product notation instead of +, although whether we use + or · for the binary operation does not matter mathematically." In multiplicative notation, the statement becomes: a·a·a = a holds if and only if a·a = e, where e denotes the identity element.
kazinator 1y ago
1 + 1 + 1 ≡ 1 (modulo 2) In the modulo 2 congruence, 1 + 1 + 1 is the same element as 1.
- discoinverno 1y ago
  The piece is about something else, cases when 1 + 1 + 1 ≡ 1 but 1 + 1 != 0
voxleone 1y ago
Beautiful math tricks. For things like these I think math education should start with sets and groups instead of numbers. https://d1gesto.blogspot.com/2025/11/math-education-what-if-we-started-with.html https://d1gesto.blogspot.com/2025/11/math-education-what-if-...
- deleted 1y ago
  [deleted]
- cbm-vic-20 1y ago
  I like Chapter 1 of Evan Chen's An Infinitely Large Napkin for some more theory. https://web.evanchen.cc/napkin.html https://web.evanchen.cc/napkin.html
CyberDildonics 1y ago
I'm not a mathematician but I don't think that's right.
anthk 1y ago
1 as a boolean? true
moxons-master 1y ago
In tropical geometry, tropical multiplication (⊙) is replaced by standard addition (+), and tropical addition (⊕) is replaced by the minimum (min) function, so 1⊕1⊕1=min{1,1,1}=1.
SilasX 1y ago
>there are other operations that aren’t quite involutions – “near-involutions”, one might call them2 – that nonetheless have the property that thrice is the same as once, four times is the same as twice, etc. ... Unlike mod-two arithmetic, which is about counting “zero, one, zero, one, zero, one, …,” the kind of counting that governs these operations goes “zero, one, two, one two, one two, …” Interesting! Earlier I had a shower thought about "what would be an variant of idempotence?" That's where an operation has the same effect whether done one or many times. One variant would be "has the same effect whether one two or many times". Another would be "can be in any one of two possible states after done one or many times" (as opposed to one possible state for idempotence). This looks like the latter!
ttz 1y ago
modulo 2
solomonb 1y ago
The intuitionistic negation example is so mind blowing. I can only wrap my head around it when I think about it in terms of functions and types: not : Type -> Type not P = P -> ⊥ modus-ponens : P -> ((f : P -> Q) -> Q) modus-ponens p = λf. f p -- p implies not-not-p not-not : P -> ((f : P -> ⊥) -> ⊥) not-not p = λf. f p -- not-p implies not-not-not-p not-not-not : (P -> ⊥) -> (((P -> ⊥) -> ⊥) -> ⊥) not-not-not np = λf. f np -- not-not-not-p implies not-p not-p : (((P -> ⊥) -> ⊥) -> ⊥) -> (P -> ⊥) not-p nnnp = λp. nnnp (not-not p) Now it all just turns into function application :)

41 points by surprisetalk 1y ago

Interesting. I always associate "1 + 1 = 1" with idempotency. Here, "1 + 1 + 1 = 1", but "1 + 1 = 0". I'm not a math whiz, so I'm just stuck with "1 + 1 = 2."

aatd86 1y ago
you're wrong. 1 + 1 = 10
- lisper 1y ago
  Which of course is 0 mod 2.
  - aatd86 1y ago
    And 0^0 x 2
- aleatorianator 1y ago
  [dead]
- not_a_bot_4sho 1y ago
  My son once asked me what 1+1 was equal to. I said, "two!" He said, "no, it's eleven." And that's when I realized he's going to be a JavaScript coder.
deleted 1y ago
[deleted]
harperlee 1y ago
XOR is a simple operation that shows that behavior.

khannn 1y ago

Terrence Howard has entered the chat

kevin_thibedeau 1y ago
Shhh! Don't give them any ideas.

susam 1y ago

A simple example where 1 + 1 + 1 = 1 is ℤ₂, the group of integers modulo 2 under addition. In fact, in any group with binary operation, say +, the identity element 0, and a non-identity element a, we have a + a + a = a if and only if a + a = 0 (i.e. a has order 2). There are plenty of groups with elements a satisfying a + a = 0. ℤ₂ as mentioned above has its unique non-zero element of order 2. The Klein group V₄ has three non-identity elements, each of order 2. Dihedral groups D₂ₙ (the symmetry groups of regular n-gons) contain reflections, all of which have order 2. Symmetric groups Sₙ (n ≥ 2) contain transpositions, each of which has order 2. For example, in the dihedral group D₈, if we let a be a reflection of the square, then a + a = 0 and a + a + a = a. But this is conventionally written in multiplicative notation as a² = the identity element, so a³ = a. Similarly, in the symmetric group S₃ under the binary operation of composition, if a denotes the transposition (12), then (12)(12) is the identity element and (12)(12)(12) = (12). In other words, applying a transposition three times is the same as applying it once. In the last two examples, it is conventional to use product notation instead of +, although whether we use + or · for the binary operation does not matter mathematically. It is conventional to use + in some subjects (coding theory, additive groups of integers modulo n, etc.) and · in others (permutation groups, dihedral groups, etc.). Often + is used for the binary operation in abelian groups and · in non-abelian ones. I'm sure none of this is particularly insightful to someone who has studied group theory, but still I wanted to share a few concrete examples here.

HWR_14 1y ago
[My post below is wrong] > In fact, in any group with binary operation +, identity element 0, and a non-identity element a, we have a + a + a = a if and only if a + a = 0 (i.e. a has order 2). The "if" is correct. The "only if" is not. (I assume that '+' and '0' are used as shorthand for "any binary operation" and "the identity of that binary operation", as I don't recall cases where "+" and "*" are used for specific types of binary operations).
- susam 1y ago
  > The "if" is correct. The "only if" is not. Both "if" and "only if" are correct. Let a + a + a = a. Adding the inverse of a to both sides, we get a + a = 0. Let a + a = 0. Adding a to both sides, we get a + a + a = a. > I assume that '+' and '0' are used as shorthand for "any binary operation" and "the identity of that binary operation" Yes. As I mentioned in my previous comment, "In the last two examples, it is conventional to use product notation instead of +, although whether we use + or · for the binary operation does not matter mathematically." In multiplicative notation, the statement becomes: a·a·a = a holds if and only if a·a = e, where e denotes the identity element.

kazinator 1y ago

1 + 1 + 1 ≡ 1 (modulo 2) In the modulo 2 congruence, 1 + 1 + 1 is the same element as 1.

discoinverno 1y ago
The piece is about something else, cases when 1 + 1 + 1 ≡ 1 but 1 + 1 != 0

voxleone 1y ago

Beautiful math tricks. For things like these I think math education should start with sets and groups instead of numbers. https://d1gesto.blogspot.com/2025/11/math-education-what-if-we-started-with.html https://d1gesto.blogspot.com/2025/11/math-education-what-if-...

deleted 1y ago
[deleted]
cbm-vic-20 1y ago
I like Chapter 1 of Evan Chen's An Infinitely Large Napkin for some more theory. https://web.evanchen.cc/napkin.html https://web.evanchen.cc/napkin.html

CyberDildonics 1y ago

I'm not a mathematician but I don't think that's right.

anthk 1y ago

1 as a boolean? true

moxons-master 1y ago

In tropical geometry, tropical multiplication (⊙) is replaced by standard addition (+), and tropical addition (⊕) is replaced by the minimum (min) function, so 1⊕1⊕1=min{1,1,1}=1.

SilasX 1y ago

>there are other operations that aren’t quite involutions – “near-involutions”, one might call them2 – that nonetheless have the property that thrice is the same as once, four times is the same as twice, etc. ... Unlike mod-two arithmetic, which is about counting “zero, one, zero, one, zero, one, …,” the kind of counting that governs these operations goes “zero, one, two, one two, one two, …” Interesting! Earlier I had a shower thought about "what would be an variant of idempotence?" That's where an operation has the same effect whether done one or many times. One variant would be "has the same effect whether one two or many times". Another would be "can be in any one of two possible states after done one or many times" (as opposed to one possible state for idempotence). This looks like the latter!

ttz 1y ago

modulo 2

solomonb 1y ago

The intuitionistic negation example is so mind blowing. I can only wrap my head around it when I think about it in terms of functions and types: not : Type -> Type not P = P -> ⊥ modus-ponens : P -> ((f : P -> Q) -> Q) modus-ponens p = λf. f p -- p implies not-not-p not-not : P -> ((f : P -> ⊥) -> ⊥) not-not p = λf. f p -- not-p implies not-not-not-p not-not-not : (P -> ⊥) -> (((P -> ⊥) -> ⊥) -> ⊥) not-not-not np = λf. f np -- not-not-not-p implies not-p not-p : (((P -> ⊥) -> ⊥) -> ⊥) -> (P -> ⊥) not-p nnnp = λp. nnnp (not-not p) Now it all just turns into function application :)