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Chapter 7: Problem 17

Represent the complex number geometrically. $$-2 i(3-2 i)$$

Short Answer

Expert verified
The complex number is \((-4, -6)\) in the complex plane.

Step by step solution

01

Expand the Expression

First, expand the expression by distributing \( -2i \) over \( (3 - 2i) \) to calculate the complex multiplication. This gives: \[ -2i(3 - 2i) = -2i \times 3 + (-2i \times -2i) \] Simplify each term separately.
02

Simplify the Terms

Calculate each of the terms obtained in the expansion:- \(-2i \times 3 \) results in \(-6i\).- \(-2i \times -2i \) results in \((4i^2)\), and since \(i^2 = -1\), it becomes \(-4\).Thus, the expanded and simplified equation results in: \[ -6i + (-4) = -4 - 6i \]
03

Interpret as a Point in the Complex Plane

The complex number \(-4 - 6i\) can be represented as a point on the complex plane. The real part is \-4\ and the imaginary part is \-6\.This means you have a point located at \((-4, -6)\), where the x-coordinate is \-4\ and the y-coordinate is \(-6\).

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Key Concepts

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

Complex Plane
The complex plane is a way to represent complex numbers graphically. It is similar to the Cartesian coordinate system used in geometry. In the complex plane, each complex number corresponds to a unique point.
  • The horizontal axis represents the real part of the complex number.
  • The vertical axis represents the imaginary part.
To plot a complex number like \(-4 - 6i\), we consider the real part, \-4\, as the x-coordinate and the imaginary part, \-6\, as the y-coordinate. So, the point \((-4, -6)\) finds its place on the complex plane:
  • Move left by 4 units from the origin along the x-axis.
  • Move down by 6 units from that point along the y-axis.
This visualization helps in understanding operations involving complex numbers, like addition or multiplication, which often result in movements to different points in the plane.
Complex Multiplication
Complex multiplication can be a bit tricky but becomes clearer with practice. It involves distributing parts of one complex number over the components of another using rules of algebra.
When multiplying two complex numbers, like \-2i \(3 - 2i\), follow these steps:
  • Distribute: \-2i \(3\) and \-2i \(-2i\).
  • Simplify separately: \-2i \[\times 3 = -6i\] and \-2i \[\times -2i = 4i^2\].
  • Recall \(i^2 = -1\), so \4i^2 = -4\.
Finally, combine these to get \-6i - 4\ or \(-4 - 6i\). The resulting number can again be interpreted in the complex plane, which provides insights into both magnitude and direction.
Imaginary Unit
The imaginary unit, symbolized as \(i\), is a fundamental concept in complex numbers. It represents \sqrt{-1}\, a number that, when squared, results in \-1\.
Some important properties of \(i\) include:
  • \(i^2 = -1\)
  • \(i^3 = -i\)
  • \(i^4 = 1\)
  • The pattern \(i, -1, -i, 1\) repeats every four powers.
Understanding the imaginary unit is important because it forms the basis for all imaginary parts of complex numbers. In the example \-2i(3 - 2i)\, \(i\) allowed us to navigate through complex multiplication and derive meaningful results like \-4 - 6i\. Mastery of \(i\)'s properties simplifies calculations and eases understanding of complex number operations.

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