Problem 11 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The complex number is \( \sqrt{3}+i \) and its modulus is 2.

Step 1: Plotting the Complex Number

To graph the complex number \( \sqrt{3} + i \), we interpret it as a point in the complex plane. The real part is \( \sqrt{3} \) and the imaginary part is \( 1 \). Plot the point \( (\sqrt{3}, 1) \) on the Cartesian plane, where \( x = \sqrt{3} \) represents the real axis and \( y = 1 \) represents the imaginary axis.

Step 2: Finding the Modulus

The modulus of a complex number \( a + bi \) is given by the formula \( \sqrt{a^2 + b^2} \). For the complex number \( \sqrt{3} + i \), substitute \( a = \sqrt{3} \) and \( b = 1 \).

Step 3: Calculating the Modulus

Calculate the modulus using the formula: \[\sqrt{(\sqrt{3})^2 + 1^2} = \sqrt{3 + 1} = \sqrt{4} = 2.\]

Step 4: Conclusion and Interpretation

The complex number \( \sqrt{3} + i \) is plotted at the point \( (\sqrt{3}, 1) \) in the complex plane. Its modulus, or distance from the origin, is 2. This confirms both the plotted point and calculated modulus are consistent on the graph.

Key Concepts

Complex Plane

The complex plane is a fascinating mathematical visualization tool. It provides a graphical way to interpret complex numbers. Think of it just like a typical Cartesian plane that you might use in geometry class, but with a twist! In the complex plane, the horizontal axis, known as the real axis, represents the real part of a complex number. Meanwhile, the vertical axis, called the imaginary axis, represents the imaginary part of the complex number.
This system allows us to visualize each complex number as a unique point on the plane.

• The point's coordinates reflect the real and imaginary parts. For example, the complex number \( \sqrt{3} + i \) will have a real part of \( \sqrt{3} \) and an imaginary part of \( 1 \).
• Thus, it's plotted at \((\sqrt{3}, 1)\) on the complex plane.

By utilizing the complex plane, we can easily see the magnitude, direction, and overall behavior of complex numbers, making them much easier to work with, even if they seem, well, complex!

Modulus of a Complex Number

The modulus of a complex number is like a fancy way of saying "its length." It's similar to how we measure the length of a vector in geometry.To find the modulus of a complex number \( a + bi \), we use the formula: \[\sqrt{a^2 + b^2}\]Here's why this works:

• Imagine the complex number as a right triangle on the complex plane where the real part \( a \) and the imaginary part \( b \) are the legs of the triangle.
• The modulus is the hypotenuse, or the "longest side," of this triangle.
• It's calculated just like you would in the Pythagorean theorem.

In the case of \( \sqrt{3} + i \):

• The modulus is \( \sqrt{(\sqrt{3})^2 + 1^2} = 2 \).
• This tells us that the "distance" from the origin \((0,0)\) to our point \((\sqrt{3}, 1)\) is exactly \( 2 \).

Effectively, the modulus helps us understand how "far" the complex number is from the origin in the complex plane.

Graphing Complex Numbers

Graphing complex numbers is a rewarding endeavor that brings abstract concepts to life. To graph a complex number, we look at its location in the complex plane, pinpointing it using its real and imaginary parts.Here's a simple way to think about it:

• Every complex number can be thought of as a point with coordinates \((a, b)\), where \(a\) is the real part and \(b\) is the imaginary part.
• To graph \(\sqrt{3} + i\), we plot it at \((\sqrt{3}, 1)\).

When graphing:

• The real part tells you "how far to go left or right" from the origin on the real axis.
• The imaginary part tells you "how far to go up or down" on the imaginary axis.

By visualizing complex numbers this way, graphing helps us not only plot them but also gain insight into their properties and relationships. For example, graphing several complex numbers can reveal patterns and symmetries that aren't apparent from looking at the numbers alone.