How to logically represent a smaller number being divided by a large number on a number line?

Question

Recently I asked this question on this platform:

When we divide a number by another number ($x \div y$), we can interpret it in two ways:

1. $x$ is divided in equal groups, where each group consists of $y$
2. $x$ is divided in $y$ equal groups

Suppose we divide $500 \div 5$. We can interpret this in two ways:

1. $500$ is divided in equal groups of $5$
2. $500$ is divided in $5$ equal groups

Now, if we divide $5 \div 500$, how can we interpret it in the same way?

I understood the $2nd$ interpretation for $5 \div 500$ from the answers. But I am not able to understand the $1st$ interpretation. I want to visualize the first interpretation on a number line.

In mathematics, if something is true, then it must be true for every scenario. Likewise, if we can interpret $500 \div 5$ in both ways on a number line, then we should be able to interpret $5 \div 500$ in both the ways on a number line too, right? How can $5$ be divided into equal groups, where each group contains $500$? If both the interpretations are applicable to $500 \div 5$ then it must be applicable to $5 \div 500$ too.

I would be grateful if someone can explain me this by visualizing it on a number line. Thank you for your time and efforts

Answer 1

From what I interpret from your question, you seem to be stuck at how to divide 5 into equal groups of 500.

Firstly, this is a bad way to look at it. In math, proofs are always true for every scenario. What you have here are interpretations, or intuitions. These basically relate the concept of division into a simpler, more understandable form. But they won't always remain valid.

In your case, you want to find the number of groups of 500 that make up 5. You'll know that the answer is 0.01, but 0.01 groups makes no sense. The mistake lies in your assumption that your intuition will always stay true. Another example could be complex division. Say (3 + 4i)/(5 + 6i). This has no meaning in your two interpretations. The same goes for something like root(2)/root(5). There is simply no good intuition to generalize the concept of division.

While I do believe that intuition can help you get a feel of things in math, it is usually better to stick to the core concepts. You won't always be able to visualize everything.