How do you solve sqrtz=-1?

2 Answers
Aug 4, 2017

z=1

Explanation:

Square both sides, sqrt(z)=-1, [(sqrt(z))^2=(-1)^2]-=[z=1]

Aug 4, 2017

There is no solution.

Explanation:

By definition, for x and y real numbers, sqrt denotes the principle square root. That is:

For x >=0,

y = sqrtx if and only if (1) y^2 = x AND (2) y >= 0.

It is true that every positive number has two square roots.
That means that for every positive number n, the equation y^2 = n has two solutions.

The square root symbol (the square root function) denotes the non-negative solution.

Although it is true that 1^2 = 1 and (-1)^2 = 1, the principle root of 1 is 1 (not -1).
There is no solution to sqrtz = -1

If we are working in the complex numbers , the square root function is unchanged for positive real numbers.

For negative real number, z, the notation sqrtz denotes the principle square root which is a complex number with positive imaginary part.

sqrt(-n) = bi if and only if

(1) (bi)^2 = -n , and
(2) b > 0

sqrt1 = 1 " " (Not -1)
sqrt4 = 2 " " (Not -2)

sqrt(-1) = i " " (Not -i which has imaginary part -1)
sqrt(-9) = 3i " " (Not -3i which has imaginary part -3)

For complex numbers more generally, there is no consensus on a "principle" square root.

So there is no principle square root of 3+4i for instance. There are two square roots, but neither is "principle".