Let me try to explain - or at least point in the direction of an explanation.
What's going on is metaphorical. We speak of the real line, because we can identify each point of the Euclidean line with one and only one real number. In the same way, we can identify each point of the Euclidean plane with one and only one real number: the Euclidean plane is to the complex numbers, as the Euclidean line is to the reals. Metaphor, then: if we say "the real line", can we not likewise say "the complex line", meaning the Euclidean plane (conceived in a particular way)?
It's a flawed metaphor, of course; the Euclidean line has many properties that the Euclidean plane does not. The only question is, is the metaphor useful? That depends on what you're trying to do; and there are branches of geometry in which the metaphor is sufficiently powerful to be worth accepting (though not, of course, without recognizing its potential to mislead or confuse).
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Date: 2007-12-10 10:54 pm (UTC)What's going on is metaphorical. We speak of the real line, because we can identify each point of the Euclidean line with one and only one real number. In the same way, we can identify each point of the Euclidean plane with one and only one real number: the Euclidean plane is to the complex numbers, as the Euclidean line is to the reals. Metaphor, then: if we say "the real line", can we not likewise say "the complex line", meaning the Euclidean plane (conceived in a particular way)?
It's a flawed metaphor, of course; the Euclidean line has many properties that the Euclidean plane does not. The only question is, is the metaphor useful? That depends on what you're trying to do; and there are branches of geometry in which the metaphor is sufficiently powerful to be worth accepting (though not, of course, without recognizing its potential to mislead or confuse).