Why there is no answer when negative value is being square root?

[wenxian]

why there is no answer when negative value is being square root?
e.g: square root of -9
when i try to find answer from calculator, ''math error '' appears..
so is there an explanation for this question??
this question may looks so weird..but i m juz asking out of curiousity..

 

[MIB]

to our intuition we see that there is no number that if we multiplied by itself will be negative , since if it is negative then squaring it will result in a positive number , and the same for positive numbers , precisely we see that from the properties of the order field of real numbers that if x > 0 and y > 0 , the xy > 0 , and if x<0 and y<0 the xy<0 and we can see why x^2 > 0 or = 0 for any real number x , and hence there is no real number such that its square is negative . This led to what is Called by complex numbers , where the square root of -9 is 3i , where i is called imaginary unit and defined informally by square root of -1 .

 

[JHamm]

Well take a number $a \geq 0$ and square it, obviously two positive numbers multiplied gives a positive number. Take the second case of $a < 0$ and write it as $-a$ for $a \geq 0$ and square this $(-1)a \times (-1)a = (-1)(-1) \times a^2 \Rightarrow a^2$ which is again positive.
Now there is a special number called $i$ which is defined by $i^2 = -1$ so you can in fact find the square root of a negative number.

EDIT: sniped :)

 

[HallsofIvy]

The product of two positive numbers is positive and the product of two negative numbers is also positive. And, of course, the product of 0 with itself is 0. That is, for x positive, negative, or 0, $x^2$ is never negative.

 

[mathman]

why there is no answer when negative value is being square root?
e.g: square root of -9
when i try to find answer from calculator, ''math error '' appears..
so is there an explanation for this question??
this question may looks so weird..but i m juz asking out of curiousity..

Your calculator is designed for real numbers only. If you had a calculator which works with complex numbers, you would get an answer.

 

[wenxian]

so is it the same reason for the statement '' square root of x is always positive'' ??

 

[Curious3141]

so is it the same reason for the statement '' square root of x is always positive'' ??

That's a convention in that $\sqrt{x}$ is defined to be the positive solution y of the equation $y^2 = x$, where $x \in \mathbb{R}^+$.

So when you just use the √ symbol, people assume you're referring to the positive value. To refer to the negative value, you need to put the minus sign in front.