Now, holding the end of the eraser fixed, rotate the pencil counter-clockwise by ninety degrees, so that the pencil and the eraser are perpendicular. If we call the eraser p, then we will denote the action of rotating the pencil counter-clockwise ninety degrees by Ap.

Now rotate the pencil again, counter-clockwise, by another ninety degrees.

The pencil is now once again parallel with the ruler, but now the point is facing the opposite direction.

If we think of the ruler as representing the real number line, then we can think of the pencil as now pointing towards negative values.

We will denote this as \[ (-1)p. \]

Note that we got the pencil to reverse its direction by applying the rotation, that we denoted by \[A,\]twice.

Therefore we can write \[ A(Ap) = A^{2}p = (-1)p.\]

Or, \[ A^{2} = -1 .\]

So a rotation of ninety-degrees squares to a reversal, or to (-1).

Actually, there are many things in mathematics that satisfy the equation:

\[ = A^{2} = (-1),\]

but, of course, there are no real numbers that satisfy this equation. (Because (-1)(-1) = 1, all real numbers square to a positive real number.)

Numbers that satisfy \[ z^{2} = -|a|, \]where \[|a| \]is a positive real number, are called **Imaginary Numbers**. This name is historical, and somewhat unfortunate, for, as we have seen in the case of rotations, such numbers "actually exist."

We call the square root of minus one, i, (\[ i^{2} = - 1). \]Then every imaginary number can be written as \[ iy, \]where \[ y \]is a real number.

We can then form numbers that have both real and imaginary parts.

We refer to such a number as a **Complex Number**.

For example \[ z = x + iy, \]where both \[ x \]and \[ y \]are real numbers, is a complex number.

We refer to the real number \( x, \) as the **Real Part** of \( z. \)

\[ Re(z) = x. \]

and to the real number \( y, \) as the **Imaginary Part** of \( z. \)

\[ Im(z) = y. \]

Note that this terminology is a bit quirky in that the "imaginary" part of a complex number is always a real number.