Whenever you make a decision you have to consider a universe of facts. Suppose you’re trying to decide where to go on vacation.
First you might make a list of all the cities you would consider as a vacation spot and all the facts you know about them. These are the integers, and you can get pretty far with a few simple operations. You know that Paris has great museums and London has bad food and Berlin has great nightlife.
But there are also huge gaps in the facts you know. If I teach you division, you can turn the facts you know into a much larger set of facts you don’t know — the rational numbers. What are the museums like in London? How’s the food in Berlin and the nightlife in Paris? You can spend your whole life wandering around in the rational numbers, making pretty good decisions and really honing your long division skills.
But there’s a much larger infinity of things you’ve never thought of. Can you take a staycation, or take a cruise? Is that hand gesture you always make considered rude in any of these cities? These unknown unknowns are like irrational numbers. You need new, exotic operations to find them but once you learn how to look for them they’re everywhere you look — and there are so so many of them.
We spend so much time practicing our long division for 3, 4 and 5 digit numbers — staying in the known unknowns of the rational numbers — and not nearly enough time developing and using the new tools that will bring the unknown unknowns to our attention. This is one of the most important ideas you can have in your cognitive toolkit.