Q: How can we prove that 2+2 always equals 4?

35 Responses to Q: How can we prove that 2+2 always equals 4?

1. Nathan Goedeke says:

   February 9, 2011 at 10:21 pm

   What would happen if the Parallel Postulate became the Parallel Theorem? Would it only effect the scientific community in a hypothetical sense or would there be more drastic implications?
2. Vishal Agarwal says:

   May 30, 2011 at 11:31 am

   why we not multiply any trigonometry question by zero to solve it..
   example- prove 2+2 =4
   o*(2+2) = 4 * 0
   0 = 0
   please answer ……….
3. Error: Unable to create directory uploads/2024/12. Is its parent directory writable by the server? The Physicist says:

   May 30, 2011 at 1:04 pm

   Although you end up with a true statement (0=0), you don’t get any new information.
   For example, if you were stuck with a question like “2x+1=5, what is x?”, you could multiply both sides by zero and get the (true) statement “0=0”, but you still wouldn’t know what x is.
4. sharafali.a says:

   February 24, 2012 at 4:50 am

   X+X+X+X=(X+X)+(X+X)
   4X=2X+2X ………sice; iknow mango=mango=2mango
   or 4X=X(2+2)
   devide this by X => 4=2+2
   or 2+2=4

   hence the proof.
5. bounce says:

   February 27, 2012 at 3:11 pm

   sharafali.a, if you’re really intent on providing a proof, you could use the definitions directly instead of involving multiplication, division, distribution, and a variable. 4=1+1+1+1=(1+1)+(1+1)=2+2. Hardly deserves to be called a proof.

   vishal, deriving a true fact from an initial statement does not prove the initial statement. This is backwards. I can use that type of argument to “prove” almost anything:

   2+2 = 3
   o*(2+2) = 3 * 0
   0 = 0
6. Locutus says:

   April 22, 2012 at 12:24 am

   In science, theories are often not proven, but disproven. Perhaps the same concept would have to be applied here. If the numbers 0 through 4 are set quantities (so that I can’t just turn around and say, oh let’s make 2=3 (artificially, as in a linguistic change, not a mathematical one)), then by definition a physical total of two 2s would yield four. It may be impossible to prove that 2+2 always yields four, but perhaps it is impossible to disprove 2+2 equalling four ever. Because the physical evidence is overwhelmingly in 2+2=4’s favor, perhaps it is a postulate or observing type of thing, like right angles are always equal, not a proof type of thing. However, applying the Peano axioms, there is a proof. There is one listed here: http://skepticsplay.blogspot.com/2008/12/224-proof.html. Whether or not this actually proves 2+2=4 for every case is another matter.

   Regardless, unless if you can find an instance where two physical twos never make a physical four (and I don’t mean taking two sets of two black holes and combining them together to see if you end up with four black holes or something to that effect, count before you combine them), 2+2=4.

   I like pi(e).
7. Rob says:

   October 27, 2012 at 2:48 pm

   How can we prove that 2+2 always equals 4? Dah, maybe check your report card to see if you graduated from 1ts grade and thus payed attention in school.
   How can 2 people each with 2 apples share evenly if each one doesn’t end up with two apples each? Easily, make apple sauce. Thus proving anything can only be possible if we have no constraints.
8. Saulius Grazulis says:

   January 2, 2013 at 5:04 am

   My 2 cents:

   You can prove 2+2=4 if you chose a right “formal system”, that is the right definitions, axioms and the “rules of proof”. And the 2+2 need not be 4… 😉 Consider:

   a) you are doing modular arithmetics: each time you add two numbers, you take a reminder of the division by 4 as a result. Then,

   2 + 2 = 0 (since the reminder of 4 divided by 4 is 0)

   Most computers do binary arithmetics like this (just its not modulo 4, but modulo 2^32, or 2^64 nowadays…)

   b) you are doing saturation arithmetics — you are only using 4 different numbers, 0, 1, 2, and 3, and whenever a result is larger than 3, it “sticks” to 3:

   2 + 2 = 3

   This kind of addition is useful in X-ray detectors — when a detector is hit by such an intense X-ray beam that the intensity exceeds the capacity of the detector, the detector “sticks” to its highest possible reading, and indicates an overflow (which is much better than wrapping back to 0; sure the highest reading is usually not 3 but something like 30 thousand or evem more 😉

   3) if you are doing ternary logics where “0” is “false”, “1” is “unknown”, “2” is “true” and “+” is logical “or”, then

   2 + 2 = 2

   (since if two statements are true, then the statement “either of them or both of them are true” is also true).

   d) 2 liters of ethanol + 2 liters of water, when mixed, will not give you 4 liters of solution — it will be slightly less…

   e) and finally, when counting apples, sheep, you can reasonably expect that the “usual” school arithmetic

   2 + 2 = 4

   will apply well… 😉

   So basically, what “2 + 2” amounts to depends on your set of axioms regarding “2”, “+” and “=”. From there on, you can derive a formal, “strict” proof; the reference to Peano arithmetic in a post before was a good demonstration!

   Mathematitians are usually interested to derive (or prove) as many theorems as possible from the smallest amount of axioms. But what you prove dependes on what axioms you choose. Physicists are usually interested in mathematics that permits them the best description of the observed world. Which math (i.e. which set of axioms) you choose depends on what kind of phenomena you are invesigating…
9. Yoron says:

   March 19, 2013 at 5:19 pm

   Very nice explanation Saulius.
10. Bob says:

    April 2, 2013 at 7:38 pm

    An even harder question: How do you know that 4=1+1+1+1? Why doesn’t 4=1+1? Why was the symbol “4” assigned to the value of 1+1+1+1?
11. MikeDa says:

    April 15, 2013 at 8:05 am

    @Vishal: that is a fallacy of identity. let’s say I change my name to Barack Obama, and it is known that Barack Obama is the name of the President of the United States. Am I then President?
12. MikeDa says:

    April 15, 2013 at 8:09 am

    two males in prison and two females in a separate female prison will remain four. a male and a female, two pairs, given the natural order of the world don’t remain only four for very long. life isn’t a series of statics 🙂
13. Xerenarcy says:

    April 11, 2014 at 2:13 am

    for the sake of traditional arithmetic proofs, you may need to look at numbers representing sets (church numbers for instance). since the rules of sets follow the rules of arithmetic (mostly) when dealing with countables, arguably you could use sets to prove arithmetic.
14. arya tomer says:

    April 22, 2014 at 7:18 am

    according to me it is easy to prove that 2 2=4.If we draw four lines in a plane we can easily see these lines.Now cover two lines by your hand and see two lines.These are two.Now remove your hand and cover last two lines these are also two.Now remove your hand and you can see all four lines.Thus it is proved that 2 2=4.
15. Stan says:

    April 23, 2014 at 5:04 pm

    4=1+1+1+1 and 2=1+1, so 4=(1+1)+(1+1)=2+2 by definition. No need to prove it. 2+2 will not be 4 only if we change the definition for the numbers 2 or 4.
16. dwayne roth says:

    April 24, 2014 at 5:40 pm

    I was playing around with a mathematical system as follows.
    0,1 ,+ where a+b=b+a i.e. + is commutative however NOT associative
    define successor S(n)=n+1=1+n 2=S(1) 3=S(2) 4=S(3) etc.
    so that 2=1+1 3=2+1 4=3+1 (=1+3 also)
    furthermore 0 is the “additive identity” so a+0=0+a =a
    now 2+2 =(1+1)+(1+1) will be distinct from 4 =(1+1+1)+1 =1+(1+1+1)
    Remember + is non associative here.
    also 2+3=3+2 will NOT be 5=4+1=1+4
    these objects resemble “ordinary numbers but are formally distinct.
    This is why I love math. It lets you posit 2+2≠4 counterintuitively.
17. Akash Agrawal says:

    April 29, 2014 at 11:13 am

    According to the rules established by us itself, we have got 2+2=4. then how can we go on arguing whether 2+2 actually equals 4 or not????
18. Alan Christ says:

    June 4, 2015 at 1:19 am

    what ? you prove it by subistuting the word red with the understanding red + red = 4 or 1 red 3 4 5 6
    green blue and 2 are now prime colors
19. binumon says:

    June 4, 2015 at 6:31 am

    10days+4days = 2weeks
20. ahmad says:

    September 25, 2015 at 6:38 pm

    How prove 2=4
21. Ruvian says:

    October 5, 2015 at 7:21 am

    I think you can’t prove elementar things. Like “what’s mass?”, “what’s force?”, “what’s space?”. You can’t answer elementary questions like these. Because it goes so farther in the elementary state of number/things that we can’t even see “more than the amplified truth” like when we solve y=2x+1 when x=1. We just replace x by 1 and let the arithmetic (as we learnt) “amplify” the truth for us. But I think we can’t “amplify” more than this.
22. SKYFISCIENTIST@gmail.com says:

    May 22, 2016 at 7:46 am

    the question is is 2+2 always = 4.
    and you have written that 0.99….. = 1
    thus substituing 2 x 0.99… x2
    = 4 x0.99999……
    =3.999999999999…..(upto infinity)……….6
    Now you habe to prove that above no. is equal to 1.
    OR
    like that:————–
    9x 1/9 x4
    =36 X 1/9
    =4
    Now 3.9999999999999999….(upto infnity)…6 = 4
    Or theanswer is 3.9999……
23. Daniel says:

    November 30, 2016 at 2:59 am

    This can be proven.

    By Fermat’s Little Theorem, you can prove 2+2 >=4
    And using mathematical induction with the properties of Fibonacci numbers, you can prove 2+2 <= 4

    Then you have only one case, 2+2=4.
24. Vinaykumar says:

    June 3, 2018 at 7:23 pm

    2+2=5
    This can be easily prover
    20-20=25-25
    4×5-5×4=5×5-5×5
    4( 5-5)=5(5-5)
    4=5
    2+2=5 proved
25. Marvin Clarke says:

    January 22, 2019 at 3:58 pm

    My dad is teaching me about proving basic math problems, example, 5a=-35

    Answer- a= -7 -7×5=-35
26. Let's be real says:

    February 16, 2019 at 5:16 pm

    whatever. numbers equal numbers equal numbers. let us focus on the fine things in life rather than the more primitive ones- STATISTICS. get outside and play a sport. be real- because, in the grand scheme of things, WHO CARES?!?!?!?
27. Jeffy bob says:

    March 11, 2019 at 1:20 pm

    2+2=4 because it just is. There is nothing else to it. Unless it’s five. Could be but people say that it’s 4. It could be 834234452364! but it’s not NEVERMIND. I DON’T CARE. WHY DON’T YOU PEOPLE GO BACK TO PLAYING FORTNITE.
28. Simone Cinti says:

    November 18, 2020 at 10:49 am

    I think you could try with this…
    Assume we are working on integers set (Z), let’s remember the existance of “-1” who is the opposite of element “1” (assumed by the definition and the construction of the whole integers set Z) .
    If you can give a proof such that is impossibile to find a z in Z such that (z – z) != “0”, demonstrating that for every a and b from Z, always exists c = a-b and if exists also d = a-b so c and d must be the same element of Z , you can also use this proof to show that if the difference between two elements p and q of Z is equal to z then this “z” always equals to zero when p equals to q.
    Please tell me if I’m right or not.
29. Lewis Harrison says:

    February 13, 2021 at 12:12 pm

    This has been invaluable to me in understanding and explaining Godel in my work in applied game theory. Something does not need to be 100%, only good enough to achieve the desired outcome. Thanks
30. aaaaaaaaaaaaaaaaaaaaaaaaa says:

    February 24, 2021 at 11:54 am

    i think that sometimes 2+ 2 should be 22 and why? because if 2 = 2 and 22 = 22
    2+2 should be 22. Not 4. the english dictionary should have been made differently . Like Knuckle you don’t say the K so it should be “nunckle” riddle me that buster
31. Kelly Robinson says:

    May 26, 2021 at 8:46 pm

    How do you prove it? Get a small handful of pebbles. Take ’em to the playground, and sit in the sandbox with them. Run as many experiments as you like.
32. perfect says:

    December 4, 2022 at 10:09 am

    To prove something you need to believe in some statement like for example what i observe is true (a primis, an axiom) then you can move forward and prove
    Something

    If you want to prove an axiom then you have to believe in some other axiom to prove or disprove it

    Hance it’s common understanding that what we observe is real and true (a belief, axiom, primis) whatever) you can then prove or

    It’s either a circular reasoning or either infinite regress

    If you believe in some primis it xan be proven or disprove based on primis

    You can not prove anything without believing in something blindly
33. Anonymous says:

    July 24, 2023 at 2:43 pm

    2 plus 2 equals 4 as I was taught to add.
34. Alpha Light Bearer says:

    November 12, 2023 at 8:54 pm

    @MikeDa

    incorrect. a male aligned person and a female aligned person do not necessarily create more. there are factors such as sexual orientation, fertility, attraction, contraceptives, the desire to reproduce or not, intact and working sexual organs, age, etc. your statement of human heterosexual reproduction being the natural order of the world is a fallacy.

    [human] life isn’t [just] a series of sexual acts in order to multiply.
35. Probably not Paul says:

    February 29, 2024 at 3:26 am

    I know that there’s a branch of mathematics, which set about to prove mathematically, that any particular mathematical equation is actually sound.

    Specifically, there’s a Mathematical proof for 2+2=4.
    I know this because a few years ago, I read an article detailing how that branch is a dying art.
    Mostly because the people who actually know that math, are getting older without any new people learning that particularly difficult math.

    Darn it if THIS PAGE wasn’t the top my searching is currently giving me.