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Chapter 2: Problem 98

HOW DOYOU SEE IT? The coordinate system shown below is called the complex plane. In the complex plane, the point that corresponds to the complex number \(a+b i\) is \((a, b) .\) Match each complex number with its corresponding point. $$\begin{array}{lll}{\text { (i) } 3} & {\text { (ii) } 3 i} & {\text { (iii) } 4+2 i} \\ {\text { (iv) } 2-2 i} & {(v)-3+3 i} & {(\text { vi) }-1-4 i}\end{array}$$

Short Answer

Expert verified
The matching points for the complex numbers are: (i) 3 -> (3,0), (ii) \(3i\) -> (0,3), (iii) \(4+2i\) -> (4,2), (iv) \(2-2i\) -> (2,-2), (v) \(-3+3i\) -> (-3,3), (vi) \(-1-4i\) -> (-1,-4).

Step by step solution

01

Understand Each Complex Number and Its Corresponding Point

The key is knowing that a complex number \(a+bi\) corresponds to the point (a, b) in the complex plane. Analyze each of the given complex numbers:
02

Match Complex Number to Corresponding Point (i)

Number (i) is 3. This is a real number which corresponds to the point (3, 0) on the complex plane. Meaning, it鈥檚 on the real (horizontal) axis.
03

Match Complex Number to Corresponding Point (ii)

Number (ii) is \(3i\). This corresponds to the point (0, 3) on the complex plane. It鈥檚 a pure imaginary number, so it鈥檚 on the imaginary (vertical) axis.
04

Match Complex Number to Corresponding Point (iii)

Number (iii) is \(4+2i\). This corresponds to the point (4, 2) on the complex plane.
05

Match Complex Number to Corresponding Point (iv)

Number (iv) is \(2-2i\). This corresponds to the point (2, -2) on the complex plane.
06

Match Complex Number to Corresponding Point (v)

Number (v) is \(-3+3i\). This corresponds to the point (-3, 3) on the complex plane.
07

Match Complex Number to Corresponding Point (vi)

Number (vi) is \(-1-4i\). This corresponds to the point (-1, -4) on the complex plane.

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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, also known as the Argand plane, is a two-dimensional coordinate system used to visualize complex numbers. This plane is similar to the Cartesian coordinate system, but it replaces the usual variables with real and imaginary components.
In the complex plane, each number is a point with a pair of coordinates:
  • The real part, denoted as \(a\), along the horizontal axis.
  • The imaginary part, denoted as \(b\), along the vertical axis.
This arrangement enables us to graphically represent complex numbers, facilitating operations like addition, subtraction, and finding magnitudes. The transformation of each complex number into a corresponding point helps us understand their relationships visually and geometrically.
Real and Imaginary Components
Every complex number can be expressed in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. These components give complex numbers their unique identity, allowing for comprehensive representation on the complex plane.
  • **Real Component (\(a\))**: This is the part without the imaginary unit \(i\). For example, in the complex number \(4 + 2i\), 4 is the real part.
  • **Imaginary Component (\(bi\))**: This includes the imaginary unit \(i\). In \(4 + 2i\), **2i** is the imaginary part.
Both components are crucial because:
  • They define the position of the number on the complex plane.
  • Understanding them aids in performing algebraic operations on complex numbers.
Graphing Complex Numbers
Graphing complex numbers on the complex plane is a straightforward process once you comprehend their real and imaginary components.
  • Locate the real component on the horizontal axis.
  • Locate the imaginary component on the vertical axis.
  • Identify where these two components intersect, marking the point that represents the complex number.
For example, consider the complex number \(2 - 2i\). Here, you would:
  • Move 2 units along the real axis.
  • Move 2 units down the imaginary axis (because it's negative).
  • Mark the point (2, -2) on the plane.
This graphical representation helps in visualizing not just individual numbers, but also operations like addition and multiplication, as well as features such as magnitude and direction. Each complex number has a unique spot, making the complex plane a versatile tool for analysis.

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