Calculator Negative Square Root

The negative square root of a number is one of the two square roots of a negative number. Unlike real numbers, which have only one square root, complex numbers have two square roots that are negatives of each other.

What is a Negative Square Root?

The negative square root of a number \( x \) is denoted as \( -\sqrt{x} \). For real numbers, the square root is only defined for non-negative numbers. However, in the complex number system, every non-zero number has two square roots that are negatives of each other.

For example, the square roots of 4 are 2 and -2. Similarly, the square roots of -4 are \( 2i \) and \( -2i \), where \( i \) is the imaginary unit (\( i^2 = -1 \)).

Note: The negative square root is distinct from the principal square root, which is always non-negative.

Formula

The negative square root of a number \( x \) can be calculated using the following formula:

\( -\sqrt{x} = - \sqrt{x} \) for real \( x \geq 0 \)

\( -\sqrt{x} = -i \sqrt{|x|} \) for real \( x < 0 \)

Where:

\( \sqrt{x} \) is the principal (non-negative) square root of \( x \)
\( i \) is the imaginary unit
\( |x| \) is the absolute value of \( x \)

Examples

Let's look at some examples to understand how to calculate the negative square root:

Number	Negative Square Root	Explanation
9	-3	The principal square root of 9 is 3, so the negative square root is -3.
-9	-3i	The principal square root of -9 is \( 3i \), so the negative square root is \( -3i \).
0	0	The square root of 0 is 0, so the negative square root is also 0.

Properties

The negative square root has several important properties:

For any non-negative real number \( x \), \( (-\sqrt{x})^2 = x \)
For any negative real number \( x \), \( (-\sqrt{x})^2 = |x| \)
The negative square root is the additive inverse of the principal square root
It is not defined for complex numbers unless considering the complex plane

FAQ

Is the negative square root the same as the principal square root?

No, the negative square root is the additive inverse of the principal square root. The principal square root is always non-negative, while the negative square root is negative for positive real numbers.

Can the negative square root be calculated for complex numbers?

Yes, but it requires working in the complex plane. The negative square root of a complex number \( z \) is \( -i \sqrt{|z|} \) when \( z \) is purely imaginary.

What is the difference between \( -\sqrt{x} \) and \( \sqrt{-x} \)?

\( -\sqrt{x} \) is the negative of the principal square root of \( x \), while \( \sqrt{-x} \) is the principal square root of \( -x \). For positive \( x \), \( -\sqrt{x} \) is negative, while \( \sqrt{-x} \) is imaginary.