Problem 75 - Algebra 2

Question

Critical Thinking Graph \(1+3 i\) and \(6+2 i .\) Also graph their sum. Draw the quadrilateral that has three points and the origin as vertices. Repeat with other complex numbers. What do you notice?

Step-by-Step Solution

Answer

The complex numbers \(1+3i\) and \(6+2i\) graph as points (1, 3) and (6, 2). Their sum, \(7+5i\), graphs as the point (7, 5). The quadrilateral with vertices at the origin and these three points is a parallelogram. This pattern holds when repeating with other complex numbers.

Step 1: Graph the complex numbers

To graph the complex numbers, represent them as points in the complex plane where the x-axis represents the real part and the y-axis represents the imaginary part. Plot the point (1, 3) for the complex number \(1+3i\) and the point (6, 2) for the complex number \(6+2i\).

Step 2: Calculate and graph the sum

To find the sum of the two complex numbers, add their real parts and their imaginary parts separately. The sum is \((1+6) + (3+2)i = 7+5i\). Plot the point (7, 5) for the sum on the complex plane.

Step 3: Draw the quadrilateral

Draw a quadrilateral by connecting the points representing the complex numbers and the origin. Connect the point for \(1+3i\) to the origin, then to the point for \(6+2i\), then to the point for \(7+5i\), and back to the first point.

Step 4: Repeat with other complex numbers

Repeat the process with different pairs of complex numbers to see if there is a pattern. Notice that the quadrilaterals formed always seem to be parallelograms. This is because opposite sides are parallel and equal in length due to the properties of complex number addition.

Key Concepts

Complex Plane Graphing

The complex plane is a fascinating and powerful tool for visualizing and working with complex numbers. It essentially consists of a two-dimensional graph where each point represents a complex number. On this graph, the horizontal axis (usually labeled as the real axis) corresponds to the real part of the complex numbers, while the vertical axis (imaginary axis) corresponds to the imaginary part.

To graph a complex number like \(1+3i\), you simply locate the point where the real part, 1, intersects with the imaginary part, 3, on their respective axes. In the given exercise, when we plot \(1+3i\) and \(6+2i\), we get two distinct points on the complex plane. Graphing serves not only as a visual aid but also helps in understanding operations with complex numbers, like addition or multiplication.

The practice of graphing complex numbers enhances comprehension, making abstract numerical concepts more concrete and easier to intuit. This very act of drawing a point on the graph turns a seemingly enigmatic number into something quite approachable and manageable.

Complex Number Addition

Complex number addition is similar to vector addition, and it is straightforward: you add the real parts and imaginary parts separately. If you have two complex numbers, \(a+bi\) and \(c+di\), their sum is given by \((a+c) + (b+d)i\).

In our exercise, to find the sum of \(1+3i\) and \(6+2i\), we add the real parts (1 and 6) to get 7, and the imaginary parts (3i and 2i) to get 5i, resulting in the complex number \(7+5i\). When graphing this sum on the complex plane, we plot the point (7, 5), which signifies the vector sum of the original complex numbers.

Understanding this operation is crucial as it touches upon the essence of complex numbers as extensions of the real numbers, where the 'i' unit adds an extra dimension of calculation. Much of the power of complex numbers is realized through their addition, which reflects the combined effects of separate real and imaginary parts.

Parallelogram in Complex Plane

A fascinating geometric representation emerges when we add complex numbers graphically. In our problem, after plotting the individual complex numbers and their sum, we can connect these points to the origin to form a quadrilateral. Interestingly, this shape is a parallelogram.

Why a parallelogram? Well, when you add two complex numbers, the resulting sum can be considered as the diagonal of a parallelogram whose sides are the vectors representing the original complex numbers. In other words, when we graphically perform complex number addition, the points representing the individual complex numbers and the sum always lie at the vertices of a parallelogram with the origin.

This happens because the parallel and equal sides of the parallelogram correspond to the equal magnitudes and directions of the real and imaginary components of the complex numbers. The exploration of this geometric property not only aids in visual learning but also reinforces the concept of complex numbers as two-dimensional entities that can be manipulated similarly to vectors.