What does it mean to be equal?

Question

Essentially, what I'm asking is what it means for two numbers to be equivalent.

Why I'm asking this:

• If $a=b$ and $b=c$, then $a=c$. But if $a=5$, $b=\sqrt{25}$ and $c=-5$, obviously $a$ is inequal to $c$.

• Also, is it correct to say that the square root of $25$ is equal to the fourth root of $625$?

Thanks!

Answer 1

Hint:

By definition, in the real numbers, $\sqrt{25}=+5$ (the positive number such that its square is $25$).

And the same for any root of even index.

Note that if we define:

$b=\{$a real number such that $b^2=25\}$ than $b$ can have the two different values $b=\pm5$ and we cannot write an identity as $a=b$ or $c=b$.

Answer 2

We say that $y$ is a square root of $x$ if $y^2 = x$.

We define a function $\sqrt{\cdot} : \mathbb{R}^+ \to \mathbb{R}$ ("the square root function") by $$\sqrt{x} := \text{the nonnegative number $y$ such that $y^2 = x$}$$

So you can see that $\sqrt{x}$ is a square root of $x$.

Not every square root of $4$ is equal to $\sqrt{4} = 2$. It turns out to be the case that $-\sqrt{4} = -2$ is also a square root of $4$.

When we refer to the square root of $x$, we mean $\sqrt{x}$; that is, the unique nonnegative number which squares to give $x$. When we refer to a square root of $x$, we mean any of the numbers which square to give $x$. It is a fact that there are usually two of these, and that one is the negative of the other; so in practice, we may refer to $\pm \sqrt{x}$ if we wish to identify all the square roots of a number. Only the positive one - that is, $\sqrt{x}$ - is the "principal" square root (or "the square root", or if it's really confusing from context, "the positive square root"); but both are square roots.

Answer 3

If you have an operator that yields more than one solution, then obviously the solutions are going to be different (otherwise there would only be one). But this does not mean they are equal.

Your example involves the function $f(x)=\sqrt x$, with $f(25)=\pm5$. Although both $-5$ and $5$ satisfy $\sqrt{25}$, $-5$ is clearly not the same as $5$. In other words, saying that the negative square root is equal to that of the positive makes no sense.

As suggested in a comment, to avoid this confusion, we use the principal square root: $f(x)=|\sqrt x|$

This may be extended to equations outputting two or more solutions. An example of this is solving the cubic $$x^3-2x^2-x+2=(x+1)(x-1)(x-2)=0$$ Here, $-1$, $1$ and $2$ are solutions, but $-1\neq1\neq2$.