Problem 10 - Algebra and Trigonometry

Question

Graph the complex number and find its modulus. $$ 7-3 i $$

Step-by-Step Solution

Answer

The complex number is graphed at (7, -3) on the complex plane, and its modulus is \(\sqrt{58}\).

Step 1: Understand the complex number

The complex number given is \(7 - 3i\), where \(7\) is the real part and \(-3\) is the imaginary part. It can be represented on the complex plane as \((7, -3)\).

Step 2: Graph the complex number

On the complex plane, the horizontal axis represents the real part, and the vertical axis represents the imaginary part. Plot the point \((7, -3)\), moving 7 units to the right on the real axis and 3 units down on the imaginary axis.

Step 3: Calculate the modulus

The modulus of a complex number \(a + bi\) is calculated using the formula \(\sqrt{a^2 + b^2}\). Substitute \(a = 7\) and \(b = -3\) into the formula: \[\sqrt{7^2 + (-3)^2} = \sqrt{49 + 9} = \sqrt{58}.\] So, the modulus is \(\sqrt{58}\).

Key Concepts

Modulus of a Complex Number

The modulus of a complex number is a measure of its size or magnitude. You can think of the modulus as the distance from the origin on the complex plane to the point representing the complex number. For a complex number in the form of \(a + bi\), the modulus is given by the formula:\[\text{Modulus} = \sqrt{a^2 + b^2}\]This formula comes from the Pythagorean theorem, since the complex plane is essentially a coordinate plane where the horizontal axis is the real part and the vertical axis is the imaginary part.
When you calculate the modulus of \(7 - 3i\), you substitute the real part \(a = 7\) and the imaginary part \(b = -3\) into the formula:

• Square the real part: \(7^2 = 49\)
• Square the imaginary part: \((-3)^2 = 9\)
• Add them: \(49 + 9 = 58\)
• Take the square root: \(\sqrt{58}\)

Thus, the modulus of the complex number \(7 - 3i\) is \(\sqrt{58}\). This tells you how far this point is from the origin without considering any directional aspects.

Complex Plane

The complex plane, also known as the Argand plane, is a two-dimensional plane used to visually represent complex numbers. It has two perpendicular axes, similar to the Cartesian coordinate system, but here they serve different purposes:

• The horizontal axis represents the real part of the complex number.
• The vertical axis represents the imaginary part.

A complex number, say \(a + bi\), is represented as a point \((a, b)\) on this plane. In the complex plane:

• Moving horizontally involves changing the real part.
• Moving vertically involves changing the imaginary part.

For instance, the number \(7 - 3i\) can be placed on the complex plane by moving 7 units along the horizontal axis, and 3 units downward on the vertical axis, due to the negative imaginary part. This graphical representation allows for visualizing complex operations such as addition, subtraction, and finding the modulus.

Graphing Complex Numbers

Graphing complex numbers means placing them correctly on the complex plane.
To graph a specific complex number like \(7 - 3i\), follow these steps:

• Identify its real part, which is \(7\).
• Identify its imaginary part, which is \(-3\).

Start at the origin, which is \((0, 0)\):

• Move 7 units to the right along the real axis because the real part is positive.
• Move 3 units down along the imaginary axis because the imaginary part is negative.
• Mark the point \((7, -3)\) on the plane.

This point is where the complex number \(7 - 3i\) lies in the plane. Visualizing complex numbers this way helps in understanding their properties and how operations like addition or multiplication affect them geometrically. It turns an abstract concept into something you can see and work with more intuitively.