Explain the difference between these square roots

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Discussion Overview

The discussion revolves around the interpretation of square roots in mathematics, specifically the distinction between the notation for square roots and the implications of using the positive and negative roots. Participants explore the definitions and conventions surrounding the square root symbol and its implications in mathematical expressions.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant notes that the square root symbol (V¯) is conventionally understood to represent the positive square root, leading to the assertion that V¯(4) = 2 and not -2.
  • Another participant clarifies that while both 4 and -4 squared yield 16, the correct notation for the square root of 16 should be ±√16 = ±4, emphasizing the distinction between the square root function and the solutions to the equation x² = a.
  • A third participant argues that stating V¯(4) = ±2 is incorrect, reinforcing that V¯(4) = 2 due to the definition of the square root as the positive number whose square equals the input.
  • Additionally, a participant discusses the necessity for the square root to be treated as a function, which requires it to be defined in a way that assigns a single output for each input, thus supporting the positive root convention.

Areas of Agreement / Disagreement

Participants express disagreement regarding the interpretation of the square root notation, with some asserting that the square root should only yield the positive value while others question the implications of using ± in certain contexts. The discussion remains unresolved on the nuances of these interpretations.

Contextual Notes

There are limitations in the discussion regarding the assumptions underlying the definitions of square roots and the implications of function mapping, which are not fully explored. The scope of the discussion is primarily focused on the notation and definitions rather than broader mathematical principles.

greenneub
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Hey guys, I was just wondering what the difference between these two statements are:

V¯(x) = ± 4

V¯(x) = - 4 ---> does not exist.

This is the quote from my text, "...we remind you of a very important agreement in mathematics. The square root sign V¯ always means take the positive square root of whatever is under it. For instance, V¯(4) = 2, it is not equal to -2, only 2. Keep this in mind in this section, and always. "

Maybe I've been staring at the pages too long, but how is (4) different from (-2)²? And why can we right ± 2, but not -2? I know this is basic but I'm embarrassingly confused about this.
 
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\sqrt{x} means the positive square root of x (this way you can refer to \sqrt{x} and only be talking about a single number, not two numbers). If the author says \sqrt{16}=\pm 4, he is just making the point that both 4^2 and (-4)^2 equal 16, however the correct notation is\pm \sqrt{16}=\pm 4.
 
As qntty said, the first, \sqrt{4}= \pm 2[/itex] is simply <b>wrong</b>. \sqrt{4}= 2 because \sqrt{x} is <b>defined</b> as the <b>positive</b> number y such that y^2= x. That is why we must write the solution to x^2= a as \pm\sqrt{a}- because \sqrt{a} does not include &quot;\pm&quot;.
 
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And also because that we wish that square-root should be a "function", and for being a function it has to be defined like that only. By definition, a function takes a value from a set A and maps it into B, and no two numbers in A can map to the same number in B.
 

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