How are binary operations used in the real world?

Question

Not necessarily a mathematical question, but how could binary operations be used in the real world? What applies to it?

Answer 1

A binary operation is any operation acting with two arguments, irrespective of the base, or indeed, whether they are numbers or not.

Adding two decimal numbers is a binary operation: $15 + 13 = 28$. Here, the addition operation takes two numbers, $15$ and $13$, and acts on them. In some notation systems, we can make this more clear by writing it as $(+\ 15\ 13) = 28$.

Multiplication is also a binary operation: $3\cdot 4 = 12$.

A unary operation takes a single argument. For example, differentiation is a unary operation: $\frac{d}{dx}\left(x^2\right) = 2x$.

The extension of these concepts can be brought beyond mathematics. For example, making a sandwich could be thought of as an order-5 operation:

$$S = \text{MakeSandwich}(\text{bread},\text{meat},\text{cheese},\text{veggies},\text{condiments}).$$

Indeed, if you so desired, you could define algebras on sandwiches!