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I'll repeat myself: it's very obvious in context that they meant the positive solution of xˆ2 = 2. My definition of the square root is as follows: the square r

I'll repeat myself: it's very obvious in context that they meant the positive solution of xˆ2 = 2. My definition of the square root is as follows: the square root of a positive real number x is the positive number, noted √x, such that (√x) ^ 2 = x. To make this a useable definition, we need to prove that equation has a solution (using the fact the function t -> t^2 is zero for t=0, diverges to +inf when t -> +inf, and is continuous between the two) and that solution is unique (using the fact the same function is strictly increasing). Do you have any other definition of √x?

thehappypm 4y ago
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.
- mannerheim 4y ago
  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 4y ago
    Sure, but doesn’t that weaken the proof? It is an arbitrary restriction on the function that is required
    - mannerheim 4y ago
      Barely? It proves it for the principal root, and you can trivially prove it for the negative root using the principal root's irrationality.
- simiones 4y ago
  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 4y ago
    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 4y ago
      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 4y ago
        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 https://news.ycombinator.com/item?id=34441863

thehappypm 4y ago

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.

mannerheim 4y ago
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 4y ago
  Sure, but doesn’t that weaken the proof? It is an arbitrary restriction on the function that is required
  - mannerheim 4y ago
    Barely? It proves it for the principal root, and you can trivially prove it for the negative root using the principal root's irrationality.
simiones 4y ago
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 4y ago
  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 4y ago
    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 4y ago
      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 https://news.ycombinator.com/item?id=34441863