Why does 1 + 1 = 2?

Why does 1 + 1 = 2?

From our childhood, it is taught that one plus one is equal to two. You may not have cross questioned this your entire life. But, what if someone asks this to you? Well you can say I don’t know. That’s easy but it’s better to answer correctly. After reading this text you will be able to give complete proof. So, stay tuned.

In mathematics, at first, we suppose inherent truth which cannot be proven. These are called axioms. These are simple statements. On the basis of these, we prove other statements. Those statements are called theorems. Just like this, axiom is needed to prove the identity above. The axiom is called Peano axioms.

The Peano axiom actually construct the set of natural numbers. I will state the Peano axiom in informal language so, it is just for understanding purpose.

i. 1 is in the set of natural number N
ii. If there is some number n in N then there is successor S(n).
iii. 1 isn’t the successor of any element in N. (0 isn’t in set of N)
iv. All the elements (except 1) are successor of other element in N.
v. If there is subset of N such that it contains 1 and n is in subset implies S(n) is also in subset. Then, the subset is N itself.

using these postulates,

Let a and b are in set of N.

a + b = S(a) if b =1

Define S (1) = 2; 2 is in N from (i) and (ii)

Then,

put a =1 in above relation;

1 + 1 = S (1)

But, S (1) is defined to be 2.

Hence, 1 + 1 = 2

Here, 2 is just symbol for the successor of 1.

For further reading

i. http://mathworld.wolfram.com/PeanosAxioms.html

ii. https://onemorebrown.com/2007/08/02/why-does-112/

iii. https://www.quora.com/Why-does-1+1-2

Photo reference

i. http://www.hulkshare.com/hayr2/a-contrari-project-sashko-infernal-dubstep-remake