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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Recent
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 The 2012 Venus transit
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 π day!
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 Q: What is the Riemann Hypothesis? Why is it so important?
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 Q: Could God have existed forever? Is it actually feasibly possibly for some ‘being’ to have just existed, infinitely?
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 Entanglement omnibus!
 Q: How are imaginary exponents defined?
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 Q: What is Bayes’ rule and how do I use it to improve my life?
 Q: Are there universal truths?
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 Q: Is there an equation that determines whether a question gets answered on ask a mathematician/physicist?
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 Q: What is the three body problem?
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 Q: CERN’s faster than light neutrino thing: WTF?
 Q: What’s the point of purely theoretical research?
 Q: Why does lightning flash, but thunder rolls?
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 Q: How much of physics can be deduced from previous equations/axioms?
 Q: If God were allseeing and allknowing, the doubleslit experiment wouldn’t work, would it? Wouldn’t God’s observation of the location of the photon collapse its probability wave function?
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 Q: Does light experience time?
 Q: Would it be possible for humans to terraform mars?
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 Q: If you stood in the beam of a particle accelerator, what would happen?
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 Q: How can we have any idea what a 4D hypercube or any nD object “looks like”? What is the process of developing a picture of a higher dimensional object?
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 Q: Why does the Earth orbit the Sun?
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 Q: If we meet aliens, will they have the same math and physics that we do?
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 Mathematical proof of the existence of God.
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 Q: Are there physical limits in the universe other than the speed of light?
 Q: Is it of any coincidence that mathematics is able to describe physical reality – given that both are inventions of the human mind?
 Q: If you were to break down an average human body into its individual atoms, and then laid the atoms out in a single straight line, how far would it stretch?
 Q: What’s it like when you travel at the speed of light?
 Q: Is there a real life example where two negatives make a positive?
 Q: Do the “laws” of physics and math exist? If so, where? Are they discovered or invented/created by humans?
 Q: Do we have free will?
 Q: How did mathematicians calculate trig functions and numbers like pi before calculators?
 Q: How can planes fly upsidedown?
 Q: A flurry of blackhole questions!
 Q: Why does going fast or being lower make time slow down?
 Q: What’s so special about the Gaussian distribution (i.e. the normal distribution / bell curve)??
 Q: Is the universe infinitely old?
 Q: Have aliens ever visited Earth?
 Q: Why is the sky blue?
 Q: Is there a formula for how much water will splash, most importantly how high, and in what direction from the toilet bowl when you *ehem* take a dump in it ?
 Q: What is the meaning of the term “random”? Can thinking affect the future?
 Q: Is it possible to choose an item from an infinite set of items such that each one has an equal chance of being selected?
 Q: Do aliens exist?
 Q: Is it true that all matter is simply condensed energy?
 Q: Which is better: Math or Physics?
 Q: Why is the number 1 not considered a prime number?
 Q: If the universe is expanding and all the galaxies are moving away from one another, how is it possible for galaxies to collide?
 Q: What happens when you fall into a blackhole?
 Q: Is the total complexity of the universe growing, shrinking or staying the same?
 Q: If two trains move towards each other at certain velocities, and a fly flies between them at a certain constant speed, how much distance will the fly cover before they crash?
 Q: Why does oxygen necessarily indicate the presence of life?
 Q: What’s the relationship between entropy in the informationtheory sense and the thermodynamics sense?
 Q: Would it be possible to kill ALL of Earth’s life with nuclear bombs?
 Q: Will black holes ever release their energy and will we be able to tell what had gone into them?
 Q: What are the Intersecting Chord and Power of a Point Theorems?
 Q: How far away is the edge of the universe?
 Q: Why do superconductors have to be cold?
 Q: Why does the leading digit 1 appear more often than other digits in all sorts of numbers? What’s the deal with Benford’s Law?
 Q: How does the Monty Hall Problem work?
 Q: How/Why are Quantum Mechanics and Relativity incompatible?
 Q: What the heck are imaginary numbers, how are they useful, and do they really exist?
 Q: What’s that third hole in electrical outlets for?
 Q: Do physicists really believe in true randomness?
 Q: Could a simple cup of coffee be heated by a hand held device designed to not only mix but heat the water through friction, and is that more efficient than heating on a stove and then mixing?
 Q: What did Einstein mean by: “Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.”
 Q: Why does saliva boil in the vacuum of space?
 Q: Can things really be in two places at the same time?
 Q: Why do weird things happen so much?
 Q: Why?
 Q: Will CERN create a black hole?
 Q: What’s the highest population growth rate that the Earth can support?
 Q: What is time?
 Q: What is “Dark Matter”?
 Q: Why do heavy objects bend space and what is it they are bending?
 Q: Why does math work so well at modeling the world around us?
 Q: Why is it that when you multiply a positive number with a negative number you get a negative number?
 Q: What is the best way to understand relativity theory? Why is it so counter intuitive?
 Q: Will we ever go faster than light?
 Q: If you were on a space station, would you be able to tell the difference between centrifugal force and normal gravity?
 Q: Is teleportation possible?
 Q: If black holes are “rips” in the fabric of our universe, does it mean they lead to other universes? If so, then did time begin in that universe at the inception of the black hole? Could we be in a black hole?
 Q: Since pi is infinite, do its digits contain all finite sequences of numbers?
 Q: What is the connection between quantum physics and consciousness?
 Q: What is the probability that in a group of 31 people, none of them have birthdays in February or August?
 Q: What is the meaning of life?
 Q: Why is e to the i pi equal to 1?
 Q: How does a refrigerator work?
 Q: How do I find the love of my life?
 Q: Why does “curved spacetime” cause gravity?
 Q: What is monotony?
 Q: How do we know if science is right?
 Q: How plausible is it that the laws of physics may actually function differently in other parts of the universe?
 Q: Are there an infinite number of prime numbers?
 Q: How can we prove that 2+2 always equals 4?
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?
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 ……….
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.
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.
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
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/2008/12/224proof.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).
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.
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 Xray detectors — when a detector is hit by such an intense Xray 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…
Very nice explanation Saulius.
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?
@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?
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 🙂
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.
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.
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.
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.
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????
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
10days+4days = 2weeks
How prove 2=4
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.
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……
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.