COMPLEX NUMBERS What happens to a complex number in the complex plane when it is (a) multiplied by \(j,(b)\) multiplied by \(j^{2} \cdot(c)\) divided by \(j\) ?

When we multiply a complex number by the imaginary unit, denoted as \(j\), an interesting transformation occurs within the complex plane. Imagine a complex number \(z = a + bi\), which sits on this plane akin to coordinates on a map. Now, when \(z\) is multiplied by \(j\), effectively \(jz = j(a + bi) = -b + ai\), the original position of \(z\) undergoes a quarter turn; that is, a 90-degree rotation counterclockwise. This rotation switches the real and imaginary parts of \(z\), with the real part now taking a position on the imaginary axis and vice versa, but with a change of sign for the real part. This is like turning a key in a lock, with \(j\) being the key that rotates our complex number. The actions we perform frequently in the physical world find a parallel in the abstract realm of complex numbers, providing a robust mental model for understanding this phenomenon.

It's essential to visualize this rotation to grasp how multiplication by \(j\) alters the position of complex numbers. Think of a point on a clock face moving from 3 o'clock to 12 o'clock; that is the effect of multiplying by \(j\) on the complex plane.

Exploring further, the concept of \(j^2\) might seem perplexing at first, but it's a straightforward extension of multiplication by \(j\). By definition, \(j\) is designed to have a property such that \(j^2 = -1\). If we revisit our complex number \(z = a + bi\) and consider its multiplication by \(j^2\), we find \(z \times j^2 = (a + bi) \times -1 = -a - bi\). This operation does not produce another rotation, but rather a reflection of the complex number about the origin, akin to flipping the number over a line that intersects both the real and imaginary axes. In lay terms, this is like viewing the number in a mirror placed at the point where the axes of the complex plane cross.

Therefore, the effect of \(j^2\) on a complex number is a 180-degree turn, flipping it to the opposite quadrant. It's like retreating; reversing direction bearing the same distance away from our starting point, but now on a completely different path.

Now, let's converse about division by \(j\). Dividing by a number typically involves multiplication by its reciprocal. For the imaginary unit \(j\), its reciprocal is unexpectedly \(-j\), considering that \(j \times -j = 1\). When we divide our complex number \(z = a + bi\) by \(j\), we get \(\frac{z}{j} = z(-j) = -a \times j - b \times j^2 = b - aj\). The division by \(j\) results in another kind of rotation, similar to the one we saw with multiplication, but this time it's in the opposite direction: a 90-degree turn clockwise.

This action is similar to the reversing of a twist or unscrewing a bolt; a counter-rotation that adjusts the position of the complex number by turning it towards the right from its original location. Division by \(j\) can be considered a corrective or counter-balancing movement for the rotations introduced by multiplication by \(j\).

The theme of rotation permeates the operations involving \(j\). Complex number rotation on the plane is the geometrical interpretation of multiplying or dividing complex numbers by \(j\) and its powers. When you rotate a complex number, you essentially move it around the origin of the complex plane while preserving its distance from that origin.

Multiplying by \(j\) rotates a complex number by 90 degrees counterclockwise, akin to taking a step to the left. Multiplying by \(j^2\) leads to a rotation by 180 degrees, which is like turning around to face the opposite direction. Conversely, dividing by \(j\) rotates the number by 90 degrees but clockwise, embodying a step to the right. These rotations are pivotal in fields such as signal processing and control theory where phase shifts and frequency modulations are the norm. With this key to rotations, complex numbers reveal their dynamic nature not as mere points, but as arrows capable of pointing in any direction, pivoting around a central axis that anchors them to the realm of mathematics.

Visualizing Rotations

A useful strategy to internalize these rotations is through visualization exercises. Try drawing the complex plane and plotting a complex number—then apply these operations and trace the movement. It's a dance of points, directed by the mathematical choreography of \(j\).