Meaning of the square root and its contradiction

[kntsy]

Is square root a function in this way?

$$f:\mathbb C\rightarrow\mathbb R^+$$

However contradiction can be drawn:$$\sqrt{x^2}=|x|\text{ and } i^2=-1$$

$$\sqrt{-1}=\sqrt{i^2}=|i|=1$$
$$\sqrt{-1}=1$$ ??What is the problem in the definition or the deduction?

 

[HallsofIvy]

Is square root a function in this way?

$$f:\mathbb C\rightarrow\mathbb R^+$$

No, it's not- the square root function is from $\mathbb C$ to $\mathbb C$. And functions, on the complex numbers, are not in general "single valued".

However contradiction can be drawn:


$$\sqrt{x^2}=|x|\text{ and } i^2=-1$$

$$\sqrt{-1}=\sqrt{i^2}=|i|=1$$
$$\sqrt{-1}=1$$ ??


What is the problem in the definition or the deduction?

The problem is with your definition, which is wrong.

 

[Gerenuk]

No, it's not- the square root function is from $\mathbb C$ to $\mathbb C$. And functions, on the complex numbers, are not in general "single valued".

That's a quite narrow view. It might not be the most mathematically useful way, but you might as well define Sqrt to be the principal value. Just as some people use Ln() instead of ln(). This makes writing out equations much easier because you have actual functions.

In this case the answer to the question is
$$\sqrt{x^2}\neq |x|$$
for complex numbers!

 

[kntsy]

No, it's not- the square root function is from $\mathbb C$ to $\mathbb C$. And functions, on the complex numbers, are not in general "single valued".

What does it mean for function not being single-valued? Does it violate the definition of function?

In this case the answer to the question is
$$\sqrt{x^2}\neq |x|$$
for complex numbers!

Is it due to the definition of absolute value on complex number?

 

[MidBlowfish]

Is it due to the definition of absolute value on complex number?

I suppose you could say that. A very important identity for complex numbers is $$x \bar{ x } = | x |^2$$, so the correct identity relating square roots and absolute value (usually called modulus when extended to complex numbers) is $$| x | = \sqrt{ x \bar{ x } }.$$ The reason that you're familiar with the identity $$| x | = \sqrt{ x^2 }$$ for real numbers is that for a real number x, $$x=\bar{x}$$, so $$x\bar{x}=x^2$$.

 

[HallsofIvy]

The definition of "function" for real numbers requires that f(a) be a unique number. To use Gerunuk's phrase, that is too "narrow" a view for complex numbers.