Problem 10 - A Graphical Approach to Precalculus with Limits

Question

Graph each complex number as a vector in the complex plane. Do not use a calculator. $$\sqrt{2}+i \sqrt{2}$$

Step-by-Step Solution

Answer

Plot a point at \((\sqrt{2}, \sqrt{2})\) and draw a vector from the origin.

Step 1: Understanding Complex Numbers

A complex number is of the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. For \(\sqrt{2} + i\sqrt{2}\), \(a = \sqrt{2}\) and \(b = \sqrt{2}\).

Step 2: Identifying the Real and Imaginary Components

In the complex number \(\sqrt{2} + i\sqrt{2}\), the real component is \(\sqrt{2}\) and the imaginary component is \(\sqrt{2}\).

Step 3: Plotting the Complex Number

To graph the complex number \(\sqrt{2} + i\sqrt{2}\), plot a point at the coordinate \((\sqrt{2}, \sqrt{2})\) on the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part.

Step 4: Drawing the Vector

Draw a vector from the origin \((0,0)\) to the point \((\sqrt{2}, \sqrt{2})\) on the complex plane. This vector represents the complex number \(\sqrt{2} + i\sqrt{2}\).

Key Concepts

complex plane

A complex plane is a visual way to represent complex numbers. It's like a graph where each number is a point on a flat map. Imagine an axis similar to the number line. However, instead of just one, the complex plane uses two axes. The horizontal axis is for the real part of the number, like normal numbers, and the vertical axis is for the imaginary part.

Plotting a point is straightforward on the complex plane. For example, with complex number \( \sqrt{2} + i\sqrt{2} \), you find the real part \( \sqrt{2} \) in the horizontal axis. Then, you find the imaginary part \( \sqrt{2} \) on the vertical axis. The point where these meet is your complex number. You can think of it as a simple graph of points, and the position gives you the full picture. It helps in visualizing how complex numbers add, subtract, or multiply.

real part

The real part of a complex number is just like a regular number. It's the number you see before adding the imaginary part. In the expression \( a + bi \), \( a \) is the real part. It represents how far you move horizontally in the complex plane.

• In \( \sqrt{2} + i\sqrt{2} \), the real part is \( \sqrt{2} \).
• It behaves like a component in two-dimensional vectors.

The real part helps define the location of a point on the complex plane. Similar to any dot or point in geometry, it tells you how much to move in a left or right direction. So, while the complex numbers sound complicated, the real part remains quite simple and grounded.

imaginary part

The imaginary part of a complex number may sound unfamiliar at first. It’s represented by the letter \( i \), which denotes \( i = \sqrt{-1} \). In a complex number \( a + bi \), \( b \) marks the imaginary part. It tells you how far to move vertically on the complex plane.

• For \( \sqrt{2} + i\sqrt{2} \), the imaginary part is \( \sqrt{2} \).
• This component adds a whole new dimension to the graphs, unlike normal numbers.

The imaginary part allows complex numbers to involve rotation and unusual number behaviors. In practical terms, it gives structure to operations that rely on more than just simple movement in the real world. Thus, while the word "imaginary" may hint at something unreal, in math, it plays an essential and tangible role.