Question

Why does the square root of a negative number not exist?

Answer 1

Square root of a negative does not exist beacause...because square of positive no. Or negative no. Is always positive...so square root of imaginative no. Is imaginative

Answer 2

a square root is a number that when multiplied by itself is equal to a given number. For example, 4 is the square root of 16, since 4 x 4 = 16.
Note, however, that -4 x -4 = 16, too. We call 4 the positive square root of 16, and -4 the negative square root of 16.
 

Now, you want to know if we can find the square root of a negative number. Let's take -16. We need to find a number, call it x, such that: x times x (x^2) = -16


Now, we know that any number times itself must be positive, not negative. Therefore, there is no such number x in the set of real numbers.
 

A number x is defined, however, in the set of complex numbers. The complex numbers are a superset of the real numbers. That is, the complex numbers form a bigger set. The reals are a subset of the complex.
 

A complex number has the form a + bi, where a and b are real numbers and the i is a special number. The "a" is called the real part; the "bi" is called the imaginary part. If we let a equal 0, then we have an imaginary number. The set of imaginary numbers is also a subset of the complex numbers. If we let b equal 0, then we have a regular real number. This is why the reals are a subset of the complex: the reals are just complex numbers that all have b=0, that is, no imaginary part.
 

Now, the number i is defined to be equal to the square root of -1. This means that i^2 (i squared) is equal to -1. So now we can find the square root of -16.

Since -16 = (-1) 16, we can write:
      sqr(-16) = sqr(-1) times sqr(16) (property of square roots)
     sqr(-16) = i times 4
 
This is usually written as 4i. We can check by squaring 4i. We get 4 x 4 = 16 times i x i = sqr(-1) times sqr(-1) = -1, giving 16 times -1 or -16.