[thehappypm]

The most satisfying answer I have for the nature of the square root is to consider complex numbers.

For an arbitrary complex number, a + bi, we can plot this as a vector on a two axis scale (x axis real and y axis imaginary). We can also convert any complex number to the form z = r e^(i θ). In other words, draw the complex number vector as an angle and a magnitude, in polar form.

So any number can be drawn as a vector onto the complex plane. Multiplying two vectors together causes the angles to add and the magnitude to multiply — creating a rotated and extended (or shrunken) new vector as the product.

So what’s the square root of a complex number z? It’s the vector a that, when multiplied by itself, winds up with z.

When z has no imaginary component, it lies flat on the x axis; its magnitude is z and the angle is 0. What’s its square root?

There are two. |a|e^(i 0) and |a|e^(i 180). 180 degrees multiplied together rotates the vector back to 0. And the e^(i 180), in radians, is e^(i pi), or negative one.

So putting it together: there are only two solutions, with the same magnitude, and different signs.

[simiones]

The initial definition of a square root of a number X (from where the name actually comes from) is "the length of the sides of a square whose area is X". There are some straightedge and compass constructions for this even in Euclid's Elements. That's why the square root is always a positive number; bringing in complex numbers only serves to confuse the issue, and is anachronistic.

[thehappypm]

Um, I don’t really think that’s right.

using the square root to convert area to perimeter, for example, is an application of square roots, one where the negative root is kinda useless. That’s not the definition of the operation. The definition of the operation is the solution to y = xx. And sure for many applications the negative root is useless, but you can’t argue against that both -2 -2 and 2*2 = 4.

[simiones]

You may not think I'm right, but that is the actual history. This operation was invented at a time where geometry was the main way mathematics was done. The square root is a much older concept than negative numbers (edit: at least in the Hellenistic world; Chinese and Indian mathematics may have had different histories).

So, by definition, the square root of 4 is +2. x * x = 4 has two solutions, which we dub +sqrt(2) and -sqrt(2).

Edit: to discuss just how much older, the concept of a square root appears in Euclid's Elements, c. 300 BC; on the other hand, Diophantus, in Arithmetica, c. 280 AD, was claiming that the equation 4x + 20 = 4 doesn't have any solutions/is absurd. So, the square root is more than 600 years older than negative numbers in the Hellenistic tradition.

[mannerheim]

The negative root is also a square root, but it isn't the root denoted by the radical symbol.

But anyway, when talking about THE nth root, we're almost always talking about the principal root.

https://news.ycombinator.com/item?id=34441863

[thaumasiotes]

> Diophantus, in Arithmetica, c. 280 AD, was claiming that the equation 4x + 20 = 4 doesn't have any solutions/is absurd. So, the square root is more than 600 years older than negative numbers in the Hellenistic tradition.

The gap is much wider than that; double-entry bookkeeping speaks of "credit" and "debit" (defined to be exactly the same concept as positive and negative numbers) because negative numbers were still nearly unknown to Europeans in the 15th century.

[mannerheim]

The radical symbol always refers to a single-valued function by convention.

x^(1/n) is used for multi-valued functions in complex analysis, not the 'n-radical' symbol, which would usually refer to the principal root.

[thehappypm]

Sure, but doesn’t that weaken the proof? It is an arbitrary restriction on the function that is required

[mannerheim]

Barely? It proves it for the principal root, and you can trivially prove it for the negative root using the principal root's irrationality.