The discussion revolves around the nature of square roots, particularly why there are two answers when taking the square root of a positive number, and the implications for zero. Participants explore definitions, properties of square roots, and the distinction between principal square roots and their negatives.
Participants generally agree on the definition of square roots and the distinction between principal and negative roots, but there is some contention regarding the interpretation of square roots in the context of zero and the implications of complex numbers, indicating multiple competing views.
Limitations include the lack of consensus on the interpretation of square roots in various contexts, particularly regarding zero and complex numbers, as well as the dependence on definitions that may vary among participants.
By definition. Wikipedia says that a square root of a number $a$ is a number $y$ such that $y^2$ = $a$. If $y$ is a square root of $a$ according to this definition, then so is $-y$ since $(-y)^2=((-1)y)^2=(-1)^2\cdot y^2=y^2$. It is probably slightly more difficult to explain why there are at most two square roots. The notation $\sqrt{a}$ denotes the principal square root, which by definition is a nonnegative square root.RTCNTC said:Why do we get two answers when taking the square root?
We can, but $0$ and $-0$ is the same number.RTCNTC said:What about 0?
Can we say sqrt{0} = - 0 and 0?
Evgeny.Makarov said:By definition. Wikipedia says that a square root of a number $a$ is a number $y$ such that $y^2$ = $a$. If $y$ is a square root of $a$ according to this definition, then so is $-y$ since $(-y)^2=((-1)y)^2=(-1)^2\cdot y^2=y^2$. It is probably slightly more difficult to explain why there are at most two square roots. The notation $\sqrt{a}$ denotes the principal square root, which by definition is a nonnegative square root.
We can, but $0$ and $-0$ is the same number.
No. A number can have two square roots but the two square roots of a are not "a" and "-a", they are "\sqrt{a}" and -\sqrt{a}" where \sqrt{a} is the squae root I referred to before.RTCNTC said:Why do we get two answers when taking the square root?
For example, let a = any positive number
sqrt{a} = - a and a.
Because (\sqrt{a})^2= a by the definition of "square root" and (-\sqrt{x})^2= (-1)^2(\sqrt{a})^2= (1)(a)= a.Why is this the case?
What about 0?
Can we say sqrt{0} = - 0 and 0?
HallsofIvy said:And, if we allow complex numbers, then the equation x^3= a. where a can be any complex number, has three solutions, x^4= x has four solutions, and, in general, x^n= a has n solutions.
If we restrict ourselves to real numbers, then the equation x^n= a, for a any real number and n odd, has one solution, while x^n= a, for n even, has 0 solutions if a<0, 1 solution if a= 0, and 2 solutions if a> 0.