Question

What's 1+1?

Answer 1

Usually `2`, but it depends on the context.

Explanation:

In normal arithmetic of natural numbers, integers, rational numbers, etc. `1+1 = 2`.

If we write numbers in base `2` (binary) notation, then we would write: `1+1=10` instead.

In the context of arithmetic modulo `2` or in a field of characteristic `2` we have `1+1=0`.

Sometimes `+` is used to represent logic "or" (disjunction) with `0` representing false and `1` representing true. so in that context you would have `1+1 = 1`.

Historical note

At the beginning of the 20th century there was a little bit of a crisis in the understanding of the foundations of mathematics. A couple of guys called Alfred North Whitehead and Bertrand Russell tried to put things right by developing basic mathematics from scratch using a formal system they published in three volumes under the title "Principia Mathematica".

In their system, it took until page 379 to get to proving `1+1=2`, a result they described as "occasionally useful".

Sadly for them, Kurt Gödel publishing his second incompleteness theorem in 1931 showed that all such formal systems are doomed to be incomplete.