Why does 1+1=2?

Question

I don't care how good people are in math, no one can seem to answer this.

Answer 1

OKAY, i'll take a different approach in answering this question.

You have to trust that the counting is like: 1, 2, 3, 4, 5...
Now this is simply because if you see one object (say a bird), you say one bird... and if you see the next one alongside the first, you say 2 birds... simply because of the counting.
You don't have to call it 1,2,3... any nomenclature would work, any language too.

Now, when the question is asked if 1+1 = 2, i would say... just imagine a jar and throw a marble into it... how many marbles you got. The answer would be 1.
Now when another marble is thrown into the jar or if i may use the word added to the jar, you would call it 2 marbles...
Now WHY? The reason is simple counting...

Maths does not tell you to take 1+1=2 on the face value or just because...
the reason is if you can separate each entity you are trying to add (here numbers 1), just reduce to the smallest unit and then ADDITION is simply counting them together. Hence the result is 2.

I hope this is a good explanation...

Answer 2

I am not sure how high into math you have gotten, but this type of proof comes from abstract algebra. First we define binary operations and identity elemets. Consider the results in terms of groups, ant the number systems.

The proof starts from the Peano Postulates, which define the natural numbers N. Note: N = { 1, 2, 3, 4, 5, 6, ...}

N is the smallest set satisfying the following postulates:

P1. 1 is in N.
P2. If x is in N, then its "successor" x' is in N.
P3. There is no x such that x' = 1.
P4. If x isn't 1, then there is a y in N such that y' = x.
P5. If S is a subset of N, 1 is in S, and the implication
(x in S => x' in S) holds, then S = N.

Then you have to define addition recursively:

Def: Let a and b be in N. If b = 1, then define a + b = a'
(using P1 and P2). If b isn't 1, then let c' = b, with c in N
(using P4), and define a + b = (a + c)'.

Then you have to define 2:
Def: 2 = 1'

2 is in N by P1, P2, and the definition of 2.

Theorem: 1 + 1 = 2

Proof: Use the first part of the definition of + with a = b = 1.
Then 1 + 1 = 1' = 2 Q.E.D.


Note: Since some books also include 0 as a natural number, there is an alternate formulation of the Peano Postulates which replaces 1 with 0 in P1, P3, P4, and P5. Then you have to change the definition of addition to this:

Def: Let a and b be in N. If b = 0, then define a + b = a.
If b isn't 0, then let c' = b, with c in N, and define
a + b = (a + c)'.

You also have to define 1 = 0', and 2 = 1'. Then the proof of the
Theorem above is a little different:

Proof: Use the second part of the definition of + first:
1 + 1 = (1 + 0)'
Now use the first part of the definition of + on the sum in
parentheses: 1 + 1 = (1)' = 1' = 2 Q.E.D.

Angel, univesity mathematics instructor

Answer 3

I'd have to check the local university library stacks for an actual reference, but I think someone actually did their PhD thesis on this exact same question, and oddly enough, the proof required a lot more letters than numbers (Greek ones, in particular). The reason no one without a really advanced degree in mathematics can explain WHY 1+1=2 is because (at least on our level of comprehension) it is based on assumptions. Any system of mathematics must at its very beginning levels make certain assumptions about the world it is trying to describe. These assumptions are collectively known as the Philosophy of Mathematics, a branch of philosophy that studies the assumptions made in mathematics. Within this philosophy are those who believe that 1+1 always equals 2 because that's the way the universe works, while others who believe that 1+1 =2 under certain circumstances but not necessarily under all circumstances -- the only way to know for sure is empirically, through observation of separate objects coming together to make pairs.

So to answer your question, the reason 1+1 equals 2 always and forever in the practical world is because a) we say it does, and b) everything we've seen in the world so far exhibits the behavior of becoming a pair when two things are joined into a group, sort of a combination of the logicist and emipiricist views of mathematical philosophy (there are others, but it would only confuse the issue) Clear as mud?

Answer 4

Because 2-1=1

Answer 5

because 1 + 1 = 2

Answer 6

First off, there is one person here who said "mathematics suck". Let me say htat life as we know it owe a great deal to the study of math. Plus, just working your brain to understand the logic of any type of math is refreshing, feels good, people just don't give it a chance.
Well back to the question:
In the past our ancestors developed a very anticuated notion of number that few species "have" (studies have shown that some animals like birds react to changes in the number of their eggs for ex.,crows can distinguish the number of people hat enter a building -up to 5- etc.). These old civilizations started to use their fingers to count, each finger represented one. Now, nobody can tell what cmae first- ordinal numbers or cardina numbers- but people then gave their fingers a dif value(ORDINAL) to represent an order of the numbers. Guess: Whan they looked at the number 2 (2 fingers needed) they saw that needed to 2 fingers to represent it, and though very primitive, these "tribes" knew 1 finger represented 1, they remberes the value they gave to each individual finger! And they had 2, so they could conlude 1+1=2.
These are the first accomplishments for human logic. But then rules that for some reason made sense to the brain, where developed by civilzations like the ancient Greek, like for ex: a+b=b+a, a+a=2a, etc. These rules don't need to be represented in real life (take for example comparison of infinte sets developed by Georges Cantor) our brain simply computes them, analyzes them and for some reason make sense to us.
Now (sorry for the long answer), the sum of 2 numbers is not always based on the rules and axioms we know. It may be defined differently depending on our uses. When I studied linear algebra, I was shown examples of the type: (a,b) +(a,b) = (a+2, b+2) for example, where the sum is not defined as we know it, THIS is because for its dif. applications and uses. These examples are rarely given, and are not as "strong" as the properties of numbers we all know because the latter ones were developed by our brain and logic, whereas the other wer made for practical uses and to explain some phenomena.

Answer 7

By definition, in one sense. We generally agree that objects can be viewed, in most instances, as separate entities. Each entity can be expressed, numerically, as "1."

When we consider a set of objects that consist of individual members, in order to group them, it is convenient to express discreet common qualities in terms of multiple integers. Put one object with a like object and you'll have two like objects. It's how people agree to describe the world so that we can communicate basic facts to each other.

It also turns out that mathematics has stunning and unpredictable characteristics that don't depend upon definition alone. There seems to be an underlying structure that makes certain aspects of mathematics universal, as long as we accept certain fundamental axioms.

The most basic axiom of mathematics is identity, expressed by the equation "x=x"; that symmetry accounts for a great deal. Any quantity on one side of an equation has to equal the quantity on the other side, in order for that equation to make sense.

There is also the circumstantial, inductive evidence, which isn't always reliable, but can't be discounted out-of-hand. 1+1 has always equaled 2. 2-1=1, too.

Answer 8

because you are dealing with math- and math isn't based on emotions or opinions. It is based on pure fact. You don't even have to use "one" to make this point. Call it "a car"
a car plus a car equals two cars. yeah I know its weak, but its all I got.

however, I remember sitting in 7th grade math, and one of the days I was awake, my math teacher blew our minds. He showed us how 1 + 1 +3 . You would have to use the rule of rounding off.

1.3 + 1.3 = 2.6, right? well if you round off the 1.3 to the closest number, then you get 1. Round off 2.6 and you get 3. Hence, 1+ 1 = 3. Again, I know its weak, but its all I got.

Answer 9

Who says it does? There's an old joke of a company that was looking to hire an accountant. They interviewed all candidates, and asked them about cash flow, asset liquidation, best depreciation methods to use, and profit margin. All gave great answers, but, when asked the final question of what does 1+1 equal, all but the final candidate said, "2". The final candidate made sure the office was not bugged, closed the door, motioned for the interviewer to lean forward, and then whispered, "Whatever you want me to make it." He got the job.

Answer 10

Numbers are symbols that humans invented to stand for quantities. The reason 1+1=2 is because the quantity we recognize as 1 when added to itself equals the quantity we recognize as 2.