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When people take the square root of a number during working with real numbers, they almost always mean the non-negative root. It's a convention. Only when they

When people take the square root of a number during working with real numbers, they almost always mean the non-negative root. It's a convention. Only when they start considering complex numbers, this convention breaks, and the square root ceases to be a function - even according to the modern definition -, because "non-negative" still doesn't narrow it down to a single number in the general case.

nybble41 7y ago
Isn't that basically what I said? The convention is that sqrt(x) is generally read as abs(sqrt(x)). The definition is that sqrt(x²) = x, which has positive and negative solutions in x for any x² > 0. You can choose to ignore the negative solutions (or the positive solutions) to make it a function, but I wouldn't consider that any simpler or closer to a single formula than the piecewise-defined version of abs(x). It's an arbitrary restriction—much like the mistaken idea that a function must be defined by exactly one formula. Why not just say that abs(x) = ±x, ignoring the negative solutions? If you'll accept "the non-negative square root of x²" then I see no reason to reject "the non-negative component of ±x". Both are single formulas with positive and negative solutions combined with a qualifier rejecting the negative solutions.
- abjKT26nO8 7y ago
  > The definition is that sqrt(x²) = x, which has positive and negative solutions in x for any x² > 0. When people write "sqrt", they mean the function, or, "principal square root". "A square root" is a different thing. Saying "the definition" makes sense only within the context where the definition is established, otherwise we have to fallback to the common usage. Wittgenstein would laugh at this conversation. Also, I'm not arguing for this convention ("it needs to have a single formula to be a function"). And when you start substituting symbols, almost nothing withstands this test. Because then you can always always twist the formula into several cases. And generally thinking about functions as something that is defined by formulas is a very limiting view. Almost all functions cannot be defined by formulas.

nybble41 7y ago

Isn't that basically what I said? The convention is that sqrt(x) is generally read as abs(sqrt(x)). The definition is that sqrt(x²) = x, which has positive and negative solutions in x for any x² > 0. You can choose to ignore the negative solutions (or the positive solutions) to make it a function, but I wouldn't consider that any simpler or closer to a single formula than the piecewise-defined version of abs(x). It's an arbitrary restriction—much like the mistaken idea that a function must be defined by exactly one formula. Why not just say that abs(x) = ±x, ignoring the negative solutions? If you'll accept "the non-negative square root of x²" then I see no reason to reject "the non-negative component of ±x". Both are single formulas with positive and negative solutions combined with a qualifier rejecting the negative solutions.

abjKT26nO8 7y ago
> The definition is that sqrt(x²) = x, which has positive and negative solutions in x for any x² > 0. When people write "sqrt", they mean the function, or, "principal square root". "A square root" is a different thing. Saying "the definition" makes sense only within the context where the definition is established, otherwise we have to fallback to the common usage. Wittgenstein would laugh at this conversation. Also, I'm not arguing for this convention ("it needs to have a single formula to be a function"). And when you start substituting symbols, almost nothing withstands this test. Because then you can always always twist the formula into several cases. And generally thinking about functions as something that is defined by formulas is a very limiting view. Almost all functions cannot be defined by formulas.