• # Question: To what extent does 1+1 = 2?

• Keywords:
Asked by on 17 Jun 2020. This question was also asked by .
• Chris Budd answered on 17 Jun 2020:

1+1 = 2 is true in all bases, apart from binary in which 1 + 1 = 10

• Katy Tant answered on 17 Jun 2020:

2=1+1 is a definition of two, it’s a belief that we then build on.

However, we can work in different systems where we can write things like 2=0 ! How can we do this? Think about how we read clock faces. The numbers go from 1 to 12, but when you get to “13 o’clock”, it actually becomes 1 o’clock again. And so we can write 13=1 (mod 12), 14=2 (mod 12) and so on. This is called modular arithmetic and it means we’re working in a circular system. So, if I had a clock face with only two numbers, (0 and 1) then 2=0 (mod 2). Clock arithmetic is great fun – look it up!

• Daniel Bearup answered on 17 Jun 2020:

There are so many contexts in which to think about that question. Here are a couple:

Firstly, consider maths as a tool to describe reality. If I have Apple A and Apple B it is highly unlikely (actually impossible) that those apples are identical. To what extent does it make sense to aggregate them as members of the same “class”?
Well from a practical perspecitive, it depends on your purpose.
Are you checking how many of your five-a-day you’ve had? If so, it doesn’t matter what types of apple they are, but it might matter whether they are the same size.
Are you taking stock in a shop? Then you probably care that one is a Granny Smith and the other is a Braeburn.

Secondly, consider maths as a framework, a set of tools that we use to manipulate numbers. Then, in standard arithmetic, 2 = 1+1 is basically the definition of the symbol 2.
We could decide that 2 is not equal to 1+1 but then we would need a different symbol for 1+1 or we would mean something different by either 1 or +. In any of these cases we step outside of standard arithmetic. This is fine … exploring different rules is one of the things mathematicians do … but it might not be useful for counting apples.
One example of a non-standard arithmetic is modular arithmetic, where we add normally until we reach a certain (whole positive) number and then reset to zero. For example: 1+1 = 0 (mod 2), 1+1+1 = 1 (mod 2), 1+1+1+1 = 0 mod(2).
This may seem a bizarre thing to do … but actually it’s embedded in the way we write numbers. We write ten as 10, “one ten” and “zero ones”. We’re actually working mod 10 in some sense (we usually call this base 10). If you think about imperial units of measurement, they often don’t use base 10 … and thus would use mod n arithmetic (for some n).
More practically, the basis for computer calculations is basically mod 2 (a transistor is either ON 1 or OFF 0 depending on input).