Addition vs multiplication as binary operators

Question

It is known that arithmetic operations of addition and multiplication are binary operations that take in two inputs and give out a single output. However, consider the following scenario:

Suppose I have a jar containing some number of balls, and I give you some balls to add to the jar. If I asked you to add three balls into the jar, then that's easy, you'd just toss em in there. However, let's say I asked you to double the number of balls in the jar, you wouldn't be able to do that unless I showed you many balls were in the jar.

This is sort of strange: We require to know both inputs for multiplication but we don't require to know both to do addition. Why is this so when they are both known to be binary operators?

Answer 1

You still won't know how many balls are inside the jar until you look inside the jar. You can't add two numbers when you only know one of the numbers...

Answer 2

It has nothing to do with the operators being binary, but rather how those operations deals with the operands.

Let's assume the jar has $n$ balls, and let that be the left operand. Then, adding $2$ balls to the jar would correspond to the operation $n+2$, while doubling the balls would correspond to the operation $n \times 2$

Now, addition can be seen as a repeated process of adding 1 ball to the jar some number of times. How many times? Well, that's what the right operand says. OK, but how many each time? Exactly $1$. So, we know everything to do what needs to be done from the right operand alone.

For example, adding 2 balls means adding 1 ball exactly 2 times. No need to know the left operand (i.e. what's already in the jar).

On the other hand, multiplication can be seen as a repeated process of adding n balls to the jar some number of times. How many times? Again, that's what the right operand says (well, you need to subtract $1$ if you start out with the $n$ balls already in the jar). OK, but how many each time? Well, that's the number $n$, i.e. th number of balls already in the jar i.e. the left operand.

For example, multiplying the number of balls by $2$ (i.e. doubling the number of balls) means adding n balls to the already existing $n$ balls. So this time, yes, we need to know the left operand.