91Ó°ÊÓ

Are a review of complex number arithmetic. Recall that (1) to add two complex numbers you simply add the real parts and the imaginary parts: e.g., \((2+3 i)+(5+2 i)=\) \(7+5 i ;\) (2) to multiply two complex numbers you multiply them as if they were polynomials and use the fact that \(i^{2}=-1:\) e.g., \((2+3 i)(5+2 i)=10+4 i+15 i+6 i^{2}=4+19 i .\) Finally, if you know how to multiply two complex numbers then you also know how to square them, since \((a+b i)^{2}=(a+b i)(a+b i)\). (a) Plot the points corresponding to the complex numbers \((1+i), i(1+i), i^{2}(1+i),\) and \(i^{3}(1+i)\) (b) Plot the points corresponding to the complex numbers \((3-2 i), i(3-2 i), i^{2}(3-2 i),\) and \(i^{3}(3-2 i)\) (c) What geometric effect does multiplication by \(i\) have on a complex number?

Step by step solution

01

Calculate the Complex Numbers for (a) and Plot

For this step, we calculate and plot the points corresponding to the complex numbers given: \((1+i)\), \(i(1+i)\), \(i^{2}(1+i)\), and \(i^{3}(1+i)\). This calculation involves reusing the property that \(i^{2}=-1\) and \(i^{3}=-i\).

02

Calculate the Complex Numbers for (b) and Plot

In a similar fashion, we plot the points corresponding to the complex numbers \((3-2i)\), \(i(3-2i)\), \(i^{2}(3-2i)\), and \(i^{3}(3-2i)\). Keep in mind, \(i^{2}=-1\) and \(i^{3}=-i\).

03

Analyze the Geometric Effect of Multiplication by \(i\)

Time to observe our plots and analyze: what transformation does multiplication by \(i\) effect on the corresponding complex number? We'll look for common trends on the transformations.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Number Arithmetic

Complex number arithmetic involves performing operations such as addition, subtraction, and multiplication on complex numbers. A complex number is expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. Understanding how to handle these operations is crucial.

• Addition: When adding two complex numbers, simply combine their real parts and imaginary parts separately. For example, to add \((2 + 3i)\) and \((5 + 2i)\), you add the real parts \(2 + 5\) to get 7 and the imaginary parts \(3i + 2i\) to get \(5i\). So, \((2 + 3i) + (5 + 2i) = 7 + 5i\).
• Multiplication: Multiplying complex numbers is a bit more involved. You treat them like polynomials and remember that \(i^2 = -1\). For instance, multiplying \((2 + 3i)\) by \((5 + 2i)\) involves distributing the terms: \[(2 + 3i)(5 + 2i) = 10 + 4i + 15i + 6i^2 = 10 + 19i - 6 = 4 + 19i\]
• Squaring: Squaring a complex number follows the same principle. For \((a + bi)^2\), it simplifies to \((a + bi)(a + bi)\).

Complex number arithmetic is foundational for solving advanced problems in both mathematics and engineering.

Geometric Transformation

Geometric transformation refers to the changes a shape undergoes through various operations, such as rotation, reflection, or dilation. In the context of complex numbers, multiplying by the imaginary unit \(i\) represents a specific type of geometric transformation, notably a rotation.

• Plotting Points: For complex numbers \((1+i)\), \(i(1+i)\), \(i^2(1+i)\), and \(i^3(1+i)\), the transformation can be visualized by plotting these points on the complex plane. Similarly, plotting \((3-2i)\), \(i(3-2i)\), \(i^2(3-2i)\), and \(i^3(3-2i)\) shows the effect of sequential rotations.
• Effect of Multiplying by \(i\): Every multiplication by \(i\) corresponds to a counter-clockwise rotation by 90 degrees around the origin. For example, \((1+i)\) transforms to \(-(1+i)\) when multiplied by \(i^2\), which indicates a 180-degree rotation.

This geometric perspective is not only visually intuitive but also aids in understanding the behavior of complex numbers during mathematical operations.

Imaginary Unit

The imaginary unit \(i\) is defined as the square root of \(-1\), which is a concept introduced to address quadratic equations that have no real solutions. The existence of \(i\) allows for complex numbers, which expand and enrich the number system with imaginary numbers.

• Properties: The crucial properties of \(i\) include: - \(i^2 = -1\) - \(i^3 = -i\) - \(i^4 = 1\), which forms a repeating pattern every four powers.
• Role in Mathematics: The imaginary unit is central to complex number algebra, facilitating operations that involve imaginary components. It also plays a significant role in various applications, including engineering, physics, and signal processing.

The imaginary unit essentially provides the framework for solving mathematical problems that extend beyond the limitations of real numbers alone.