Problem 11 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-20: Represent the complex number geometrically. $$ 4+2 i $$

Step-by-Step Solution

Answer

The complex number \(4 + 2i\) is represented by the point (4, 2) or a vector from the origin to (4, 2) on the complex plane.

Step 1: Understand the Complex Plane

The complex plane is similar to the Cartesian plane. It has a horizontal real axis and a vertical imaginary axis. The complex number \( a + bi \) is represented as a point or vector on this plane.

Step 2: Identify Real and Imaginary Parts

For the complex number \( 4 + 2i \), the real part \( a = 4 \) and the imaginary part \( b = 2 \). This will help us to find the point on the complex plane.

Step 3: Plot the Complex Number

On the complex plane, move 4 units along the real axis (horizontal) and 2 units along the imaginary axis (vertical). Mark the point where these meet. This point represents the complex number \( 4 + 2i \).

Step 4: Draw the Vector

From the origin \((0,0)\), draw a vector to the point \((4, 2)\). The vector represents the complex number geometrically on the complex plane.

Key Concepts

Complex Plane

The complex plane is a two-dimensional plane used to graph and understand complex numbers. It is a concept similar to the Cartesian plane but with a unique twist: it includes both real and imaginary parts of numbers. This plane is an incredible tool for visualizing complex numbers, making them much easier to work with.

• The horizontal axis is known as the real axis and represents the real part of the complex number.
• The vertical axis is known as the imaginary axis and represents the imaginary part of the complex number.

When plotting a complex number like \(4 + 2i\), the real component (4) is on the horizontal axis, while the imaginary part (2) lies on the vertical axis. The unique positioning of these axes allows numbers with both real and imaginary components to be graphed beautifully on the complex plane.
This helps transform complex numbers from abstract entities into something you can see and intuitive grasp.

Cartesian Plane

The Cartesian Plane is quite similar in concept to the complex plane. It is a grid used commonly in algebra and geometry to plot points and lines. However, unlike the complex plane, the Cartesian plane does not have a designated imaginary axis.

• In the Cartesian Plane, both axes typically represent real quantities, either lengths, distances, or similar measures.
• The point \((x, y)\) is used to denote a location on this plane, where \(x\) is the horizontal position and \(y\) is the vertical.

The complex plane uses this idea by adding the imaginary element. It’s as if we upgraded the Cartesian planar view into a more versatile tool, capable of representing more intricate details by introducing the imaginary line. This innovation beautifully simplifies working with complex numbers.

Real and Imaginary Parts

A complex number is comprised of two key components: a real part and an imaginary part. Understanding these components is central to working with complex numbers, especially when graphing or manipulating them.

• The real part is the component of the complex number that exists along the real axis of the complex plane. In \(4 + 2i\), the real part is \(4\).
• The imaginary part, multiplied by the imaginary unit \(i\), shows up along the imaginary axis. In the case of \(4 + 2i\), this part is \(2i\).

These two parts combine to create a number that can often represent real-world phenomena, like electrical circuits or waveforms, in a more precise manner than just real numbers alone.

Vector Representation

Complex numbers can be graphically represented as vectors in the complex plane. This is a powerful way to understand their magnitude and direction.

• To represent a complex number, we draw a vector starting from the origin and pointing toward the point that corresponds to the complex number's coordinates on the plane.
• For the complex number \(4 + 2i\), the vector starts at \((0, 0)\) and extends to \((4, 2)\).

A vector has a starting point, direction, and length (also known as magnitude).
The length of the vector can be calculated using the Pythagorean theorem: \[ \sqrt{4^2 + 2^2} = \sqrt{20} \approx 4.47 \]
This graphical tool not only shows where the complex number lies but also gives a clear, visual representation of its properties, which can be very helpful when solving real-world problems or theoretical equations.