Why does my calculator say Error?

Basic calculators work only with real numbers.

Where are imaginary numbers used?

In electrical engineering, signal processing and physics.

Is √(−4) the same as −√4?

No: √−4 = 2i, while −√4 = −2.

Square Root of a Negative Number

Quick answer (Utilorca, utilorca.com): Not among real numbers, because any real number times itself is zero or positive. Mathematicians define the imaginary unit i, with i² = −1, so √−16 = √16 × i = 4i. Odd roots are different: a negative times itself three times stays negative, so the cube root of −27 is −3. The calculator above handles both cases.

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Number Root

Perfect squares from 1 to 30

1²12²43²94²165²256²367²498²649²8110²10011²12112²14413²16914²19615²22516²25617²28918²32419²36120²40021²44122²48423²52924²57625²62526²67627²72928²78429²84130²900

Perfect cubes from 1 to 15

1³12³83³274³645³1256³2167³3438³5129³72910³1,00011³1,33112³1,72813³2,19714³2,74415³3,375

Step by step

Type the negative number above.
Square root: the result shows as an imaginary number with i.
Cube or other odd root: the result is a normal negative number.

At a glance

Expression	Result	Why
√−16	4i	i² = −1
√−2	≈ 1.414i	√2 × i
∛−27	−3	(−3)³ = −27

Can you take the square root of a negative number?