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

Physicist: In this case there’s no proof. With the exception of 0 and 1, all numbers are defined in terms of simpler numbers. “4” is Defined as “1+1+1+1”. And “2”is Defined as “1+1”.

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

1. Nathan Goedeke says:

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:

why we not multiply any trigonometry question by zero to solve it..
example- prove 2+2 =4
o*(2+2) = 4 * 0
0 = 0

3. The Physicist says:

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:

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:

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:

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:

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. 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:

Very nice explanation Saulius.

10. Bob says:

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:

@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:

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:

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:

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:

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:

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:

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:

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:

10days+4days = 2weeks

How prove 2=4

21. Ruvian says:

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. 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