Cube+roots,+branch+cuts,+and+Wolfram+Alpha

Wolfram Alpha is an incredibly useful tool, but it often behaves in ways that we don't expect. One of the biggest surprises is the way that it handles cube roots of negative numbers. It will insist on returning a complex (non-real) value for the cube root of a negative real number. In this note, I will explain why Wolfram Alpha exhibits this behavior.

The first thing to understand is that Wolfram Alpha is based on [|Mathematica], which is possibly the most powerful computer algebra system in existence. Mathematica can perform calculations in the real number system, but it is equally at home in the complex numbers. To the extent possible, Mathematica wants all operations with real numbers to extend to the complex number system.

In the real number system, every number has exactly one cube root. The cube root of a positive number is positive, and the cube root of a negative number is negative. For example, the cube root of 8 is 2, and the cube root of -8 is -2. This works great in the real number system, but it does not work very well in the complex number system.

Below is an applet for exploring cube roots of complex numbers. Note that every complex number z ≠ 0 has three (complex) cube roots. It is often desirable to designate one of these three values as the **principal cube root**. This is done by drawing an infinite wedge with an angle of 120° based at the origin. Exactly one of the cube roots lies inside the wedge, and this cube root is considered to be the principal cube root. (The principal cube root will be shown in red.)

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The technical term for this process is a **branch cut**. You can explore different branch cuts by checking the box in the upper left, then dragging the slider that appears on the left side. You will see a wedge which encompasses all principal cube roots. Observe that the wedge can cover the positive real axis or the negative real axis, but it cannot cover both at the same time. This means that, in the complex numbers, we can arrange that cube roots of positive numbers are positive, or that cube roots of negative numbers are negative, but we cannot have both at the same time.