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Let $n \in \mathbb N$ be a natural number and $a \in \mathbb R$ be a real number. The $n$-th root of the number $a$ is defined as follows:

Case I: $n$ is an odd number. In this case the $n^{\text{th}}$ root of $a$ is defined to be that number $b \in \mathbb R$ such that $b^n = a$.

Case II: $n$ is an even number. In this case the $n^{\text{th}}$ root of $a$ is defined to be that number $b \geq 0$ such that $b^n = a$.

Why is it that when $n$ is even, we only consider $b \geq 0$. For example, both $+2$ and $-2$ squared equal $4$, but when we say the square root of $4$ is $2$. Is there a reason for this?

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    Nothing more than convention...2011-05-28
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    @Fabian: No, there is a lot more to it then convention!2011-05-28
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    See also http://math.stackexchange.com/questions/13094/significance-of-displaystyle-sqrtnan2011-05-28
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    Actually Fabian is right. Even if you work with complex numbers and chose a particular branch, that is just a convention; you could pick another one. And while it becomes pretty clear that that particular branch is the "natural" choice, I don't see why this would be more natural than chosing the positive over the negative as the root of a positive number.2011-05-28
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    @user9176: A few things: The branch you pick can depend on what you are doing, and what you want your function to look like, so no that is not just convention. When you have functions with multiple branches, different choices can have cancellation in different ways. Most important, even when there is a convention, it is in no way helpful to say that it is "Nothing more then convention." There are always reasons to why we choose a particular convention, and understanding those reasons is very important.2011-05-28
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    Personally, I remember having this question come up in high school, and the answer being: "We just choose it that way." Why? "Because we do." This is the same answer as "Nothing more than convention." There is a lot more, or at least, there is understanding why we chose that convention in the first place.2011-05-28
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    That's precisely why I asked this question! Frankly, as I went through high school a lot of these questions were answered as "because that's the way it is..". But I know that there is always a reason (in Mathematics at least), so I decided to ask here.2011-05-28

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