# Why can I square both sides?

There is the equation:

$$x= 2^\frac{1}{2}$$

we can square both side like this:

$$x^2= 2$$

But I don't understand why that it's okay to square both sides.

What I learned is that adding, subtracting, multiplying, or dividing both sides by the same thing is okay. For example:

$$x = 1$$
$$x-1 = 1-1$$
$$x-1 = 0$$
$$x \times 2 = 1 \times 2$$
$$2x = 2$$

like this.

But how come squaring both sides is okay too?

$$x = 2$$
$$x \times 2 = 2 \times 2$$
$$2x = 4$$
$$2x \times x = 4 \times x$$

This does not induce it.

Can you answer this silly question?

• If $x=2^\frac12$ then multiplying by $x$ on both sides gives $x^2=2^\frac12x=2^\frac12\cdot2^\frac12=2$. This is the same as squaring both sides because both sides have been multiplied by equal quantities. Jan 2, 2022 at 18:19
• If two numbers are equal then their squares (or for that matter any other function of them) are equal. The danger here is that if two numbers have the same square then they may not be equal ($1^2=(-1)^2$ for example), so the operation of "squaring both sides" is valid, but only goes in one direction.
– Ian
Jan 2, 2022 at 18:19
• note that $2 = x$ and $x = 2$ so $2x = x^2$ and $2x = 2^2$
– sato
Jan 3, 2022 at 4:03
• More generally, you can multiply equalities like: $$a=b, \;c=d \;\;\implies\;\; a \cdot c = b \cdot d$$ If you take for example $a=x, b=1, c=d=2$ then you get (as in the question): $$x=1, \;2=2 \;\;\implies\;\; 2x=2$$ Or, if you take $a=c=x, b=d=\sqrt{2}$ then you get: $$x=\sqrt{2}, \;x=\sqrt{2} \;\;\implies\;\; x \cdot x = \sqrt{2} \cdot \sqrt{2} \;\;\iff\;\; x^2 = 2$$
– dxiv
Jan 3, 2022 at 6:20
• Does this answer your question? Why can't you square both sides of an equation?
– user53259
Jan 3, 2022 at 9:08

In maths, we use the equals sign, $$=$$, to mean that two things are identical. If you take two identical objects, and do the same thing to them (i.e. adding 2, subtracting some number $$x$$, squaring them etc. or any combination of operations you can come up with), then since you've done the same thing to the same objects, it seems reasonable that they would still be equal afterwards.

Note that if you square both sides of an equation, you are multiplying the thing on the left of the equation by itself, and the thing on the right of the equation by itself. So if you have $$x=2$$ then $$x²=2²$$ is what you get when you square both sides.

• However, one can still ask why squaring is a function in the first place. Jan 2, 2022 at 18:32

In your example, remember that $$x=2$$ so that you have $$x\cdot2=2\cdot 2\\x\cdot x=2\cdot 2\\x^2=2^2$$

More generally if $$f(x)$$ is any polynomial, and $$x=2$$ then you have $$f(x)=f(2)$$

• This holds for any function $f$, by definition. (then the interesting part becomes proving $x \mapsto x^2$ is a function.)
– qwr
Jan 3, 2022 at 7:54

We start from the fact that if $$x = y$$, then $$a\cdot x = a\cdot y$$ and $$x\cdot a = y\cdot a$$ for any $$a$$.

Therefore, if $$x = y$$, letting $$a = x$$ we get $$x\cdot x = x\cdot y$$ and letting $$a = y$$ we get $$x\cdot y = y\cdot y$$, so $$x^2 = x\cdot x = x\cdot y = y\cdot y = y^2.$$

Thus, we have shown that given equality $$x = y$$ it is ok to square both sides, i.e. $$x = y$$ implies $$x^2 = y^2$$.

• A nice thing about this explanation is it uses tools the OP, per the question, is already comfortable with (i.e., multiplying both sides by the same number). Jan 3, 2022 at 13:53

The reason for this is that the operations you describe - squaring, addition, and the like - are all functions, and functions have the general property that if $$a = b$$, then $$f(a) = f(b)$$, where $$f$$ is the function in question. One can consider this the quintessential defining aspect of a function: that it relates each input unambiguously to an output, so when you give it the same input on two different occasions or in two different forms, like on the two sides of the equation, it must give you the same output in both instances.

Unfortunately, the idea of functions tends to get introduced rather late, which is more like history than it is like the logical structure of modern maths. Functions are actually one of the most basic and elementary concepts in maths, and you have been using them ever since you did addition.