The discussion clarifies that the square root of a positive number 'a' yields two results: the principal square root, denoted as sqrt{a}, and its negative counterpart, -sqrt{a}. This is due to the definition of square roots, where both y and -y satisfy the equation y^2 = a. For zero, sqrt{0} is defined as 0, since -0 and 0 are equivalent. The conversation also touches on the number of solutions for equations involving even and odd powers, emphasizing that for even powers, there can be zero, one, or two solutions depending on the value of 'a'.
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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.