I have being told all my life that $\sqrt{9}$ equals to $\pm3$. That all changed when I saw a video talks about it. It said that the square root does not spit out a negative number. I wanted to see if it's true, and if it is true, then why? Logically, $(-3)(-3)$ equals to $+9$ too. Thanks in advance.
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$x^2=9$ has two solutions, but $\sqrt{9}$ only has one. They are not the same thing. – Biggs Oct 08 '16 at 19:18
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The square root function is much too polite to spit. – MJD Oct 08 '16 at 19:23
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@Biggs: when you say "$\sqrt{9}$ only has one [solution]" you are making a type error. $\sqrt{9}$ is an expression, and expressions aren't the kinds of objects that have solutions. (equations are inequalities are.) note, for example, that you would not say "9+1" has one solution. it is correct to say that $x^2=9$ has two solutions, but $\sqrt{9}$ is defined to be the nonnegative solution of the equation $x^2=9$. – symplectomorphic Oct 08 '16 at 19:49
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@symplectomorphic you are correct. Expressions don't have solutions. I worded that comment badly. – Biggs Oct 08 '16 at 20:13
4 Answers
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It is a choice. The most convenient approach seems to be to consider $f(x)=\sqrt x$ as a function, which implies a choice. The canonical choice is that of the positive square root.
Martin Argerami
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A function must be defined so that each input is mapped to only one output. In this case the function $f(x) = \sqrt{x}$ must be chosen so that the result is only one number. While the square root of a number has two possible results, it seems more natural to go with the positive solution for the square root function. Thus, we ignore the negative solution.
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No. This is a common misconception about square roots that I carried for most of my life. The problem occurs when people think about the meaning square roots; normally if we see something like $\sqrt{9}$ we think "what is the number that squared gives us 9?" and following this logic we say $3$ and $-3$. However this is not the rigorous definition of square root, instead what we are doing is solving the equation $x^2=9$ which is NOT equivalent to $\sqrt{9}$. The square root of a positive number is always a positive number.
If anybody were to graph the function $y=\sqrt{x}$ on a $x$ an $y$ plot the result will be a line that is not defined for $x<0$.
Since we can express the square root of a number as the number elevated to $\frac12$ it follows that the result cannot be positive because a number elevated to any real power is always positive (note that $-x^{\frac12}$ is different from $(-x)^{\frac12})$
PunkZebra
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The radical sign √ refers to the principal (positive) square root only. source: Square Root Calculator
Mize
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That is why there is a ± in front of the √ in the quadratic formula....so that both the positive and negative square root answers are used...and not just the positive square root answer. – Mize Oct 09 '16 at 01:45
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When I put $-1$ into the second box of that "square root calculator", it says "$\sqrt{1} = -1$, thus $-1 \times -1 = 1$." So much for that... – Caleb Stanford Oct 09 '16 at 02:05
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Now that is just incorrect. "Square root of 1" refers to the same thing as "1". – Caleb Stanford Oct 09 '16 at 18:44
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For me, Square root of 1 = +1 , Square root of 1 = -1, √1 only = +1 due to √ refers to the principal (positive) square root only. – Mize Oct 09 '16 at 20:58
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Normally, $1$ is the square root of $1$, but both $1$ and $-1$ are each a square root of $1$. Maybe that agrees with your opinion. – Caleb Stanford Oct 09 '16 at 20:59
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If I say, "the square root of 2 is irrational", it means the positive one and not the negative one. So any time someone says the square root of it indicates the symbol $\sqrt{}$, i.e. the principal square root. Indeed, conversely, the symbol $\sqrt{}$ is always read off as "the square root of". – Caleb Stanford Oct 09 '16 at 21:00