Why Do We Need Real Numbers Anyway?
The other day I decided to try an experiment. I decided to ask an AI tool to explain Real Numbers to me. Specifically, I gave it this prompt:
"Pretend I'm a first year calculus student. How would explain real numbers to me and why do they matter in Calculus?"
The AI tool responded with some things which are not surprising, like the definition of Real Numbers and a few examples. But what I didn't expect was the analogy it gave me. This is what it said:
"Imagine trying to build a smooth road using only LEGO bricks. You can get close to a smooth surface, but there will always be bumps and gaps. Real numbers are like having access to infinitely fine sand, allowing you to create a perfectly smooth surface for calculus to operate on."
That surprised me, first, because I didn't ask for an analogy. And, second, because I'd never heard any teacher/professor explain Real Numbers like that. As a matter of fact, I don't recall ANY of my teachers ever attempting to explain Real Numbers or why we need them.
As an engineering student, I was required to take three semesters of Calculus. Three. And, yes, I made through all three semesters but it would have been nice to understand why it all mattered. If you ask me now to explain any of it I'd have to reference my old text books (yes, I still have them).
They say that you never forget what you truly learned. And I've also heard that when truly learn something you don't just commit it to memory but when you can associate that something with another something you already understand.
This analogy of LEGO bricks is something I won't forget even though I just read it recently. I won't forget it because now I have a visual to go along with the definition of Real Numbers. I have an explanation that most people will understand because most people understand LEGO bricks and most people understand what sand is.
Conclusion
Does this mean that I think teachers will be replaced by AI? No, on the contrary. I think as humans we crave interaction with other humans. We still need teachers to inspire and nurture students. These are human qualities that AI still can't replicate.
But I do think that AI is an extremely helpful tool. AI can distill lots information and present it to you in a way that makes sense. We still need to get better at knowing how to coax that information out of AI and we still need to closely examine the information it returns. But, when used properly, AI can most definitely fill in the gaps we may have in our knowledge.