Why is the square root of a number not plus or minus?

Question

For example, $\sqrt{4}$. I've asked a bunch of people and I get mixed answers all the time, as to whether it is $-2$ and $+2$ or just $+2$.

How about if there's a negative in front of the square root sign, for example, $-\sqrt{4}$? Would that still be plus or minus or just minus?

Answer 1

This is a common source of confusion, because people don't clearly separate in their minds a few related but different situations.

• Does $\sqrt4$ mean $\pm2$ or just $2$? The answer: it means $2$. By definition, the $\sqrt{\cdot}$ function always evaluates to a nonnegative number (as long as it's being applied to a nonnegative number; otherwise it's not defined at all). The definition of $\sqrt x$ is: the nonnegative number $y$ such that $y^2=x$.
• Does $(\sqrt x)^2$ always equal $x$? (We'd better assume that $x$ itself is nonnegative for the notation to make sense.) The answer: yes. This is exactly the definition of the $\sqrt\cdot$ function, as described above.
• Does $\sqrt{x^2}$ always equal $x$? The answer: no, since "the nonnegative number whose square is $x^2$" is not always $x$. But $|x|$ is a nonnegative number and its square is $x^2$; therefore $\sqrt{x^2}=|x|$.
• Suppose that $x^2=4$; does that mean that $x=\pm2$ or just $x=2$? The answer: it means $x=\pm2$. If we apply the square root function to both sides of the equation $x^2=4$, we get $|x|=2$, which is equivalent (by the definition of absolute value) to $x=\pm2$.