Complex numbers: I was right and you were wrong
Occasionally, when I link to my post complex numbers made straightforward, people tell me "but students are used to extending the number line with fractions and surds and negatives already, you should just tell them that you're extending it again in an analogous way!"
So here's an example of someone who was hopelessly confused by complex numbers (or, more precisely, someone who thought they were "bullshit"), whose confusion was instantly cured by an explanation exactly along the lines in my post. Not by my post, alas: today must have been one of those rare days when I don't check Reddit enough.
Ideally we'd want more: it would be nice to know if there exist people for whom "a complex number is an ordered pair of reals" doesn't work. Probably. But this at least shows us that there is a problem with the "by analogy with the extension from N to Z" approach, and not just among the mathematically clueless. In fact, I'd argue that their very confusion shows that they have a better handle on the crucial mathematical notion of well-definedness than their teachers.
So here's an example of someone who was hopelessly confused by complex numbers (or, more precisely, someone who thought they were "bullshit"), whose confusion was instantly cured by an explanation exactly along the lines in my post. Not by my post, alas: today must have been one of those rare days when I don't check Reddit enough.
Ideally we'd want more: it would be nice to know if there exist people for whom "a complex number is an ordered pair of reals" doesn't work. Probably. But this at least shows us that there is a problem with the "by analogy with the extension from N to Z" approach, and not just among the mathematically clueless. In fact, I'd argue that their very confusion shows that they have a better handle on the crucial mathematical notion of well-definedness than their teachers.