I was always taught that negative numbers cannot have a square root because it has no meaning, since there cannot be two numbers with the same sign whose product is negative. Then suddenly someone comes along and says, guys, I defined a number i, which is the square root of -1. And I, little me, ask: what’s the difference between that and a circular triangle? Both are mathematically nonsensical, and for one of them someone just decided to give it a name, while the other is called a logical contradiction. What’s the difference?

Then they lied to you. File a complaint and have the kindergarten teacher fired. The truth is that a negative number has no real square root, but it does have a complex square root (or at least one can define it that way). By the same token, they could have taught you in preschool that every number is divisible by two, and then in kindergarten you would learn about odd numbers and discover that not every number is like that. Then in first grade you would learn about positive numbers, and later in second grade discover negative ones. Then in fourth grade move from natural numbers to integers, and so on. It’s good that every now and then you move up a grade; that way you learn more things.

The high-school math education system does a lot of harm in that instead of explaining things, they simply say, “that’s just how it is”, Now there are cracks/negative/complex/the commutative law, and that’s how bizarre views like this come out, that complex numbers are a contradiction.

I don’t think so. They rely on the students’ intelligence. A student is supposed to understand that things are explained to him according to his current level of knowledge. In the world of real numbers, a negative number really does not have a square root. There is nothing inaccurate about saying that. When one studies or defines additional numbers, like complex numbers, it may turn out that in that domain a negative number does have a square root. Or that either a negative number has a square root or such a square root can be defined. If it is only a new definition, then there was no mistake before at all. The definition of numbers was simply changed.
When I wrote above that they lied to him, that was sarcastic criticism of him, not of the education system.

Is it possible to define a number that is the result of division by zero, or is that a meaningless thing?

I assume it’s possible.

That’s accurate, but when they say, never ever divide by zero!!!! instead of saying that division by zero has no meaning because it won’t preserve various things (that’s an explanation from an introduction to linear algebra, nothing complicated), they create the feeling that mathematics is just something that somebody arbitrarily decided is how it is, and that’s that.

If we’ve moved on to feelings, then my arguments have been stopped in their tracks. To each his own feelings.