I found some inconsistencies of notations in my calculus textbook: for example, whenever it uses $\sqrt4$, it assumes that $\sqrt4 = 2$ instead of $\pm2$. But whenever it solves an equation such as $x^2 = 4$, it gives me the answer that $x = \pm2$. I think that square root and square are inverse operations. What I mean is that: if $x^2 = y$ then $\sqrt{y} = x$. Here one of the appropriate solutions would be either $x = \pm2$ and $y = 4$ or $x = 2$ and $y = 4$.

  • $\begingroup$ This must be a duplicate ... $\endgroup$ Commented Apr 15, 2018 at 19:56
  • 1
    $\begingroup$ You must distinguish between the solutions of an equation and a function's output. The square root function outputs the principal root by convention (to keep it a function) while a solution is just a number that satisfies an equation. $\endgroup$
    – Andrew Li
    Commented Apr 15, 2018 at 20:07
  • $\begingroup$ Square root of a square is also relevant. $\endgroup$ Commented Apr 15, 2018 at 20:43

3 Answers 3


There is no inconsistency. The square root is a function, hence has a single value. Unless otherwise specified, the positive root is taken.

The solutions of

$$x^2=4$$ are $$x=\pm\sqrt4.$$


From here,

Every non-negative real number $a$ has a unique non-negative square root, called the principal square root, which is denoted by $\sqrt a$, where $\sqrt{\,\,}$ is called the radical sign or radix. For example, the principal square root of $9$ is $3$, which is denoted by $\sqrt9 = 3$, because $3^2 = 3\cdot3 = 9$ and $3$ is nonnegative.


Every positive number $a$ has two square roots: $\sqrt a$, which is positive, and $−\sqrt a$, which is negative. Together, these two roots are denoted as $\pm\sqrt a$. Although the principal square root of a positive number is only one of its two square roots, the designation "the square root" is often used to refer to the principal square root.


Your book is correct.

The equation $$x^2=4$$ has two solutions and they are $$x =\pm 2$$

One of them is positive and the other one is negative.

The positive one is denoted by $\sqrt 4 $ and the negative one is denoted by $- \sqrt 4$ That is $$\sqrt 4 =2\\ -\sqrt 4 =-2 $$

We like $$\sqrt x : [0, \infty ) \to [0, \infty ) $$ to be a function and functions do not have more than one value at a single point.

Similarly $\sqrt 9 =3$ and $- \sqrt 9 =-3$


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