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I've heard that someone was figuring out a way of solving negative square roots. Is that true?

Yes. The square-root of -1 is known as i.

This makes the square-root of -4 be 2i, the square-root of -9 be 3i, etc.

This leads to what is called imaginary numbers.

When the x-axis is real numbers and the y-axis is imaginary numbers,

the numbers in the plane are called complex numbers.

Using complex numbers, the system is complete.

Whole positive numbers and subtraction lead to negative positive numbers and 0.

Division leads to fractional numbers.

Square-roots leads to more real numbers.

Using trig formulas, powers, or many other functions leads to all real numbers.

Taking the roots of negative numbers leads to complex numbers.

Doing any sort of math on complex numbers, however, leads to nothing else -

only different complex numbers.

To calculate square-roots, cube-roots, or any other root of a complex number, convert the number to polar coordinates. That is, r is the distance to the origin and theta is the angle from the positive axis. To find a cube-root, take the cube-root of r and divide the angle by 3. Then add 120° and 240° to the result, and that will give the three cube-roots.

To find fourth-roots, compute the 4th root of r, and then divide the angle by 4.

Add 90°, 180°, and 270° to that angle to get the other 4th root.

For example, is the number is -8, then r = 8 and theta = 180°.

The cube roots would then be to find the cube-root of 8, which is 2,

and take 180°/3 = 60°. The other cube roots would be at r=2, theta = 180° and

r = 2, theta = 300°.

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