How is division comparable to subtraction?

Question

I was watching a lecture where the professor was describing the mathematics of the ancient Greeks and said they had division because "that's just fancy subtraction." That line got me thinking because it doesn't seem to be quite true. Multiplication is certainly "fancy addition" because you can turn the equation 3 x 5 = 15 into an addition problem where you have "=15" at the other side. But division can't be turned into a subtraction problem. You can't take 15/3=5 and make a subtraction problem with "=5" at one side.

I have read in many places that division is subtraction in the sense of 15/3=5 turned into 15-3-3-3-3-3=0 and then you count the number of times you subtracted 3. But I take a lot of issue with that explanation of division because a) it does not parallel multiplication b) it does not have "=5" on one side of the new equation and c) it's simply a "count" of something. It's like viewing the equation from the outside and seeing how many times you did something, like a counter in programming. But it's not fundamental to the equation.

Essentially, I'm hoping you can give me some guidance on the essential theory of division in terms of how it relates to subtraction and how it parallels multiplication. I have been turning this question over in my head for many weeks now and I've come up with a theory (I'm not close to being even an amateur mathematician but I love solving problems) but it has to do with redefining multiplication and I think it would obfuscate the point.

Answer 1

That is the issue with an inverse operators, like plus is an inverse operation to minus, division is an inverse of multiplication.

The multiplication as a direct operator, calculate result of repetitive addition, like in 3x5=3+3+3+3+3=15

An inverse operation does something else, it is answer the question about the number which would make the result of direct operator as we wish. That is a/b asks about number c such that b*c=a, that is 15/3 looking for c such that 3c=15, so you have to repetitively add 3's until you get 15 and count it. Equivalently you can subtract 3's from 15 until you get 0.

Therefore, the division is similar to multiplication, but different.

Answer 2

I think you may be taking the "fancy subtraction" remark a little too seriously. Nevertheless, division is related to subtraction, as shown in the answers to How to divide using addition or subtraction.

Taking your objections one at a time:

a) it does not parallel multiplication

Why should division "parallel" multiplication? Does subtraction "parallel" addition? Subtraction and addition do not work the same: for one thing, addition is commutative but subtraction is not.

In fact subtraction undoes addition; $8 - 5$ is the unique solution for $x$ in the equation $5 + x = 8.$ Similarly, $15/3$ is the unique solution for $y$ in the equation $3 \times y = 15$.

Division is to multiplication as subtraction is to addition. Subtraction is related to but not exactly like addition; division can be calculated by repeated subtraction in a way that is related to (but not exactly like) the way multiplication can be calculated by repeated addition.

b) it does not have "=5" on one side of the new equation

This is a meaningless formality. Addition, subtraction, multiplication, and division are not defined by the symbols with which we express them. The ancient Greeks would not even understand what you meant by that objection, since they had never seen an equality sign.

c) it's simply a "count" of something

The $5$ in $3 \times 5 = 3 + 3 + 3 + 3 + 3 = 15$ is a "count" of something, too: it counts the number of times $3$ appears in the sum. For that matter, all integers can be regarded as a "count".