Problem 6 - Algebra and Trigonometry

Question

Graph the complex number and find its modulus. $$ -3 i $$

Step-by-Step Solution

Answer

Point (0, -3); Modulus is 3.

Step 1: Identify the Complex Number

The given complex number is -3i. In rectangular form, a complex number is expressed as \( a + bi \). Here, \( a = 0 \) and \( b = -3 \), so the complex number is \( 0 + (-3)i \).

Step 2: Plot the Complex Number

The complex number \( 0 - 3i \) is plotted on the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part. Since \( a=0 \), the point is on the imaginary axis. Move 3 units down from the origin to \( (0, -3) \).

Step 3: Calculate the Modulus

The modulus of a complex number \( a + bi \) is given by \( \sqrt{a^2 + b^2} \). For \( 0 - 3i \), substitute \( a = 0 \) and \( b = -3 \):\[|z| = \sqrt{0^2 + (-3)^2} = \sqrt{9} = 3\]

Step 4: Finalize the Solution

The complex number \( -3i \) is represented by the point \( (0, -3) \) on the complex plane, and its modulus is 3.

Key Concepts

Modulus of a Complex Number

The modulus of a complex number is essentially its size or magnitude and is similar to the notion of absolute value in real numbers. If you have a complex number in the form of \(a + bi\), the modulus is symbolized as \(|z|\). The formula for computing the modulus is \[ |z| = \sqrt{a^2 + b^2} \]. This expression helps to find out the distance of the complex number from the origin on the complex plane.

To elaborate, when we have the complex number \(0 - 3i\), it means that \(a = 0\) and \(b = -3\). By plugging these values into our modulus formula, we find that: \[ |z| = \sqrt{0^2 + (-3)^2} = \sqrt{9} = 3 \].

The result of 3 reflects the distance of the point \(0 - 3i\) from the origin (0,0) on the complex plane. Understanding this helps us visualize and quantify the effect of the imaginary part of the complex number.

Rectangular Form of Complex Numbers

The rectangular form is a way to represent complex numbers in terms of a real part and an imaginary part. This form is usually given as \(a + bi\), where \(a\) is the real component and \(b\) is the imaginary component. The rectangular form allows you to easily visualize and plot complex numbers on the complex plane.

For example, consider the complex number \-3i\ from our exercise. It can be expressed in rectangular form as \(0 - 3i\). Here:

• \(a = 0\) - since there is no real part in this expression.
• \(b = -3\) - represents the vertical placement on the complex plane along the imaginary axis.

Writing complex numbers in rectangular form makes it simple to undertake operations like addition, subtraction, and finding the modulus.

Complex Plane

The complex plane is a helpful concept to visualize complex numbers as points. In this plane, each complex number corresponds to a point where:

• The horizontal axis (x-axis) represents the real part of the number.
• The vertical axis (y-axis) represents the imaginary part of the number.

In our example, we plotted the number \(0 - 3i\). This point is at the coordinates \( (0, -3) \) because:

• the real part \(a = 0\) has no effect on the horizontal location, resulting in no movement along the x-axis.
• the imaginary part \(b = -3\) moves the point 3 units downward along the y-axis.

Thus, the complex plane allows us to treat complex numbers geometrically, with the real part determining the left-right position, and the imaginary part indicating the up-down shift. This makes complex arithmetic and visualization more intuitive.