Problem 11 - Precalculus Mathematics for Calculus

Question

Graph the complex number and find its modulus. $$\sqrt{3}+i$$

Step-by-Step Solution

Answer

Graph at \((\sqrt{3}, 1)\); modulus is 2.

Step 1: Understand the Complex Number

The complex number given is \( \sqrt{3} + i \). It can be represented in the form \( a + bi \), where \( a = \sqrt{3} \) and \( b = 1 \). The real part is \( \sqrt{3} \), and the imaginary part is \( 1 \).

Step 2: Graph the Complex Number

To graph \( \sqrt{3} + i \), start by drawing a complex plane with a horizontal axis (real axis) and a vertical axis (imaginary axis). Plot the point \((\sqrt{3}, 1)\) where \( \sqrt{3} \) is along the real axis and \( 1 \) is along the imaginary axis.

Step 3: Determine the Modulus

The modulus of a complex number \( a + bi \) is given by the formula \( |z| = \sqrt{a^2 + b^2} \). Substitute \( a = \sqrt{3} \) and \( b = 1 \) into the formula: \[ |\sqrt{3} + i| = \sqrt{(\sqrt{3})^2 + (1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2 \]

Step 4: Interpret the Modulus

The modulus represents the distance from the origin \((0,0)\) to the point \((\sqrt{3}, 1)\) on the complex plane. Therefore, the modulus is 2.

Key Concepts

Modulus of a Complex Number

When dealing with complex numbers, one important concept to understand is their modulus. The modulus of a complex number, denoted as |z|, is a measure of its 'size' or 'length'. It's similar to how we calculate the length of a line segment on a graph.

For a complex number in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part, the modulus is calculated using the formula:

• \( |z| = \sqrt{a^2 + b^2} \)

This formula is derived from the Pythagorean theorem and essentially gives the distance from the complex number's position on the plane to the origin \((0,0)\).

For example, for the complex number \( \sqrt{3} + i \):

• Real part \( a = \sqrt{3} \)
• Imaginary part \( b = 1 \)

The modulus is calculated as 2, indicating that the point \((\sqrt{3}, 1)\) is 2 units away from the origin.

Graphing Complex Numbers

Graphing complex numbers involves plotting them on a two-dimensional plane known as the complex plane. In this plane:

• The horizontal axis is labeled as the real axis, representing the real component of the complex number.
• The vertical axis is labeled as the imaginary axis, representing the imaginary component.

To graph the complex number, like \( \sqrt{3} + i \), you'd plot the point \((\sqrt{3}, 1)\).

Here's how to do it step-by-step:

• Find \( \sqrt{3} \) on the real axis (horizontal).
• Find 1 on the imaginary axis (vertical).
• Plot the point where these two meet \((\sqrt{3}, 1)\).

This visual representation helps in understanding its magnitude and provides a graphical way to comprehend the modulus calculated previously. Graphing is not just about locating a point but also about understanding spatial relationships on the complex plane.

Complex Plane

The complex plane is a geometric representation of complex numbers, extending the conventional x-y Cartesian coordinate system into a system with a real and an imaginary component.

In this plane:

• The real numbers are represented along the horizontal axis, known as the real axis.
• The imaginary numbers are represented along the vertical axis, known as the imaginary axis.

A complex number \( a + bi \) is expressed as the point \((a, b)\) on this plane.

Key aspects of the complex plane include:

• Each point on the plane corresponds to a unique complex number.
• The distance of the point from the origin is its modulus.
• Graphical operations on the plane can translate into algebraic operations with complex numbers.

Understanding the complex plane aids in visualizing not just individual numbers but also their interactions, facilitating easy comprehension of complex arithmetic and other operations.