Square Root Calculator
Calculate square roots for positive and negative numbers. For negative values, the result is shown as an imaginary number with i.
How To Calculate: Square Root Calculator
Square Root Calculator is calculated by using the required inputs and applying the formula below. Use consistent units for reliable results.
Formula:
For n >= 0: sqrt(n) = x where x^2 = n. For n < 0: sqrt(n) = i x sqrt(|n|)
Worked Example:
Example: sqrt(81) => 9 and sqrt(-4) => 2i.
How to calculate square roots of negative numbers:
Negative numbers do not have real square roots. In complex numbers, i is defined by i^2 = -1.
sqrt(-a) = sqrt(a) x i (for a > 0). This follows from writing -a as (-1) x a, so sqrt(-a) = sqrt(-1) x sqrt(a), and sqrt(-1) is defined as i.
sqrt(-1) = i. The symbol i is the imaginary unit defined by i^2 = -1. Since no real number squares to -1, the result is expressed in complex numbers.
For sqrt(-4), split -4 as (-1) x 4: sqrt(-4) = sqrt(-1) x sqrt(4) = i x 2 = 2i. Check: (2i)^2 = 4i^2 = -4. The displayed 2i is the principal square root.
Common search phrases:
Examples: "square root of -1", "sqrt(-1)", "square root of -4", "sqrt of negative number", and "imaginary square root". This section answers all of those.