Is addition really a binary function??

Question

My class 12 maths textbook says addition is a binary function. But I had this experiment which caused a doubt.

Addition is explained as a binary function because if you add more than two numbers you will have to do it in pairs of two. Now say you have 3 blue balls 7 black balls and 4 green balls. You're asked the total no. of balls. Instead of doing binary addition what if you mix them all up in a bag and randomly count each ball out. It will then no more be binary. How do we resolve this conflict??

Answer 1

Addition can be defined as an $n$-ary operation in a recursive manner

1. $f_2(x,y) = x+y$

2. $f_{n+1}(x_1,\ldots,x_{n+1}) = f_n(x_1,\ldots x_n) + x_{n+1}$, for $n\geq 2$.

Then $f_n(x_1,\ldots,x_n)= x_1+\ldots +x_n$ for each $n\geq 2$.

Answer 2

Elaborating a bit on Wuestenfux’ answer, we can certainly regard addition as an $n$-ary operation for all $n\ge2$. The only reason we don’t talk about it, is because any addition can be broken down into a sequence of binary additions, basically by associativity.

From this viewpoint, associativity should be strengthened to $$ a+(b+c)=(a+b)+c=a+b+c, $$ and something similar for $n>3$, to guarantee that the binary and $n$-ary definitions agree. For example Wuestenfux’ recursive definition does the job.

In general, any reasonable associative binary operation will be the same situation.

Answer 3

The issue is you're comparing a physical encoding of the abstract concept of addition to the mathematical description of it.

Here's an example which will really mess your intuition of binary operation. Suppose I gave you a basket of balls, and I myself have a basket of some number of balls.

If I ask add three balls in my basket, then without looking inside my basket you can simply throw balls into my basket.

But what if I ask multiply the number of balls in my basket by three? Then you can not do this unless you have some way of knowing how many balls there are in my basket.

Now does this mean multiplication is some way beyond a binary operation ( in math sense)? Well... no. Its just that this physical realization of the addition comes with more properties than the mathematical description of it.

The thing is mathematical objects and descriptions of them don't neccesarily need to conform to our intuition or a particular model in real life.

We don't go from the physical world to mathematics but rather try to explain mathematics as a thing itself using logic. That's the whole idea of axiomatic systems and proofs. One more point is that this is also a hint that the teaching mathematics by extrapolating physical intuition of a small set of scenario to a mathematical description which works anywhere is problematic.