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I'm wondering why when $n$ is even we always assume the positive root for $\sqrt[n]{}$.

For example, if we have $x = \sqrt{4}$, we always assume $x = 2$. But if we have $x^2 = 4$, we do $x = \pm\sqrt{4} \Longrightarrow x = 2, -2$.

The problem is that if I take the 1st equation, and square both sides, I get $$ x^2 = 4 \Longrightarrow x = \pm\sqrt{4} \Longrightarrow x = 2, -2. $$ My teacher says that if you introduce the radical sign, use $\pm$, but those two equations are the same under the rules about radicals she taught us, so in reality, (I think) "taking the positive root" rule has no meaning.

I get that people want to make the radical mean something without ambiguity, but making arbitrary rules (at least I thing they're arbitrary), like taking the positive answer, seems to confuse things and leads to inconsistency, like what is shown above.

There are other examples, such as solving for $x$, then substituting the original equation with the value of $x$, and not getting it to work because you can only take the positive root.

For things like the Pythagorean theorem, people like to say it's obvious to take the positive one only, but there are ways to represent the Pythagorean theorem without relying on human judgement to decipher the final answer. Say you have a right triangle with legs $a=3$, $b=4$, and you want to find c. So you do: $3^2 + 4^2 = c^2$, $c > 0$. and solve the systems of equations, just like any other: $c = \pm\sqrt{25} \Longrightarrow c = -5, 5$. $c = -5, 5$ intersects $c > 0$ at $c = 5$, so the answer is $c = 5$. I don't see why you would need to redefine square root: $\pm\sqrt{}$ to mean principal square root: $\sqrt{}$ to find the correct answer.

Does it have something to do with imaginary numbers, or am I missing something? Thanks for any help.

Edit: I am not asking if $\sqrt{}$ means positive, I'm asking why.

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    $\begingroup$ One way to dodge this seemingly magical "take the $\pm$ when you take square roots on both sides of an equation" is to instead introduce the $\pm$ when you solve an absolute value equation, by introducing an additional step. In this situation you're doing $x^2=4$ hence $|x|=2$ hence $x=\pm 2$. In that first step we took the principal root; we only introduced the other one through the absolute value. $\endgroup$ – Ian Nov 6 '16 at 2:15
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    $\begingroup$ Possible duplicate of Square roots -- positive and negative $\endgroup$ – naveen dankal Nov 6 '16 at 2:16
  • $\begingroup$ I don't see how you can classify $-2, 2$ as a single number under the absolute value sign, even though both absolute values would be 2. $\endgroup$ – Vityou Nov 6 '16 at 2:17
  • $\begingroup$ Not a duplicate, his what asking what, mine is asking why. $\endgroup$ – Vityou Nov 6 '16 at 2:18
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    $\begingroup$ No, really, there is no "law of mathematics" that says that the equations $x=\sqrt{4}$ and $x^2=4$ are equivalent equations (i.e. that they characterize the same solution set). The rules only tell you that the solution set of the former is contained in the solution set of the latter. You won't be able to grasp this concept until you get that idea out of your mind. $\endgroup$ – Ian Nov 6 '16 at 2:59
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Here is a short answer to the question in the title of OP:

Well, if we don't do so, what could a better alternative be?

What is the notation $\sqrt{}$?

The confusion seems to be from understanding of the notation $\sqrt{}$. When writing, for instance $\sqrt{16}$, one pronounces it as "square root of $16$". However, what one really means is "the principal square root of $16$".

Let's go back to the definitions. A square root of a real number $a$ is a number $y$ such that $y^2 = a$; in other words, a number $y$ whose square is $a$. For example, $4$ and $−4$ are square roots of $16$ because $4^2=(-4)^2=16$. Note carefully that the notation $\sqrt{}$ is not involved in this definition at all.

Now, for every given positive real number, say $16$ again, there are two "square roots" (note carefully again that we don't write $\sqrt{x}$ for "square roots of $x$" yet) of it. What if one wants specifically to refer to the positive one? Instead of explicitly saying "I'm refering to the positive square root of $16$", one uses the notation $\sqrt{}$ to define $\sqrt{16}$ as the positive square root of $16$. Here comes the notation $\sqrt{}$. Of course you are losing "information" when you write $\sqrt{16}$ to mean "the positive square root of $16$". Because it is by definition so. What does one do for the "lost information"? One naturally has $-\sqrt{16}$ as the negative square root of $16$.

One can put two definitions together to see what is really going on:

  • A "square root" of a real number $a$ is a number $y$ such that $y^2=a$;

  • Given a positive real number $x$, the notation $\sqrt{x}$ is defined as a positive real number $y$ such that $y^2=x$. And in this case, we write $y=\sqrt{x}$.

Why is $\sqrt{}$ defined in the way above?

If one does not define $\sqrt{a}$ as the positive square root of $a$ and instead as the "square roots of $a$", then one would have $\sqrt{16}=\pm 4$. Now how would you write the answer to the following question?

What is the positive real number $x$ such that $x^2=\pi$?


[Added: ]Compare the following two possible definitions for the notation $\sqrt{}$:

  • I. For any positive real number $a$, define $\sqrt{a}$ as the square roots of $a$;
  • II. For any positive real number $a$, define $\sqrt{a}$ as the positive square root of $a$;

Now, if one uses definition I, then $\sqrt{16}=\pm4$. With this definition, you have perfectly what you might want: $$ x^2=16\Rightarrow x=\pm 4;\quad\text{and }x=\sqrt{16}=\pm4. $$

If one uses definition II instead, on the other hand, one would have $\sqrt{16}=4$.

You might be happier with definition I and ask why on earth one prefers II. Here is "why". Suppose you are asked to solve the following problem.

Find the solution to the equation $x^2-\pi=0$ such that $x>0$.

If one uses definition II, then one immediately has $x=\sqrt{\pi}$.

Now if one uses definition I, $x=\sqrt{\pi}$ would be the WRONG answer.


One more lesson from Terry Tao:

It’s worth bearing in mind that notation is ultimately an artificial human invention, rather than an innate feature of the mathematics one is working on; sometimes, two writers happen to use the same symbol to denote two rather different concepts, but this does not necessarily mean that these concepts have any deeper connection to them.

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  • $\begingroup$ $\sqrt{\pi}$. I understand the notation, I'm wondering why, not what. Take for example, the first example I provided. As a programmer, I have a good understanding of what a function is. I see the square root function as an unbalanced chemical equation. Even though "9/10 times", you want the positive root, mathematics doesn't care, and you will get conundrums like my first example. I get that it's easier to say $\pm\sqrt{x}$ than $-\sqrt{x}$ (like you said, when we would like to refer to the negative). I don't get why we treat the square root function like the mathematical $\pm\sqrt{}$. $\endgroup$ – Vityou Nov 9 '16 at 1:59
  • $\begingroup$ By unbalanced chemical equation, I mean that you can deduce a balanced one, but if you try to use the unbalanced one, you will get incorrect results. $\endgroup$ – Vityou Nov 9 '16 at 2:01
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    $\begingroup$ @codersarecool, that mathematics doesn't care is irrelevant: mathematicians do. Our notations are purely based on convenience, and the accepted practice is the most convenient one. $\endgroup$ – Mariano Suárez-Álvarez Nov 9 '16 at 3:40
  • $\begingroup$ Accept when it doesn't actually work. Mathematics exists regardless of what mathematicians define as standards. $\endgroup$ – Vityou Nov 9 '16 at 4:00
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What happens is that when you squared you got an equation that seems to be equivalent to the original, but in reality it is not. The original equation implicitly has the restriction that the $x$ must be a non-negative number, but the second equation does not.

As the equation you are solving is the first and not the second one, the restriction must remain present, and if it cannot be implicitly, you must write it explicitly:

$$ x^2 = 4, x\geq0 $$

And so nothing has changed. When we solve an equation we must identify firstly what the restrictions are and keep them present, because sometimes algebraic manipulations change the set of solutions.

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Instead of looking at a specific equation, like $x^3 = 8$, you need to look at the bigger problem $x^3 = y$. What you want is, given $y$, to find what $x$ is. In other words, you want a function $f$ such that $x = f(y)$. In this example, $x = \sqrt[3] y$ and you problem is solved. This point of view works great when the exponent is an odd number.

Now consider the equation $x^2 = y$. We have a problem because, for example, $3^2 = 9$ and $(-3)^2 = 9$. A function $x = f(y)$ can only return one value for each $y$. So either $3 = f(9)$ or $-3 = f(9)$ but we can't have both if we want $f$ to be a function. So, if we want $f(y) =\sqrt y$ to be a function, then we have to choose. The choice was $f(y) = \sqrt y$ is the positive square root of $y$.

So, when you see an equation like $x^2 = 25$. Then $x = \sqrt{25}$ gives you a solution $x=5$. If you want both solutions, then you have to write $x = \pm \sqrt{25}$.

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  • $\begingroup$ $x^2=25$ gives you $x = 5$ or $x = -5$. $x = \sqrt{25}$ gives you $x = 5$. The only way you could plot something like $y^2 = x$ is to make a graph of y and graph $f(y)=y^2$. It's not impossible to represent in a visual way even. $\endgroup$ – Vityou Nov 9 '16 at 2:08
  • $\begingroup$ @codersarecool - It's not hard to plot $y^2 = x$. But it is not the graph of a function. The function $ y=\sqrt x$ is a branch of $y^2 = x$ that provides a solution. To get both solutions you need to use the other branch function $y = -\sqrt x$. $\endgroup$ – steven gregory Nov 10 '16 at 13:19
  • $\begingroup$ I guess another question I'd have is why do functions need to have only one solution. I'd assume it's by definition. So then I'd ask what purpose does having one solution have. Simplicity? Passing the vertical line test? Those all seem like arbitrary goals. You lose sign info when squaring, and the square root should be able to deal with that. What's the problem with having an x correspond to 2 y values? $\endgroup$ – Vityou Nov 12 '16 at 17:37
  • $\begingroup$ Functions do not have one solution. They have one value for each input. The value nowadays is that that value can be computed on a calculator. From that value, the other solutions can, in may cases, be calculated easily. $\endgroup$ – steven gregory Nov 13 '16 at 18:13

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