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Chapter 7: Problem 9

Graph each complex number, and find its absolute value. $$2-6 i$$

Short Answer

Expert verified
Plot the point (2, -6). Absolute value is \( 2\sqrt{10} \).

Step by step solution

01

Identify the components of the complex number

The given complex number is presented in the form of \(a + bi\). Here, \(a\) is the real part and \(bi\) is the imaginary part. For the complex number \(2 - 6i\), \(a = 2\) and \(b = -6\).
02

Plot the complex number on the complex plane

To plot the complex number \(2 - 6i\) on the complex plane, treat the real part (\(a = 2\)) as the x-coordinate and the imaginary part (\(b = -6\)) as the y-coordinate. Plot the point \( 2, -6 \) on the complex plane.
03

Find the absolute value of the complex number

The absolute value of a complex number \(a + bi\) is given by the formula \[ \left| a + bi \right| = \sqrt{a^2 + b^2} \]. For \(2 - 6i\), substitute \(a = 2\) and \(b = -6\) into the formula: \[ \left| 2 - 6i \right| = \sqrt{2^2 + (-6)^2} = \sqrt{4 + 36} = \sqrt{40} = 2\sqrt{10} \].

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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 visually represent complex numbers. Think of it as a graph where the x-axis represents the real part and the y-axis represents the imaginary part of the complex number.
Each point on this plane corresponds to a unique complex number. For example, the complex number 2 - 6i can be represented as a point (2, -6) on the complex plane.
This helps make the concept of complex numbers easier to understand.
By plotting them in this way, we can see how these numbers relate to each other geometrically.
Absolute Value
The absolute value of a complex number can be thought of as its distance from the origin (0,0) on the complex plane. It is always a non-negative number.
This distance can be found using the Pythagorean theorem.
The formula for the absolute value of a complex number a + bi is \[ \left| a + bi \right| = \sqrt{a^2 + b^2} \].
In our example, for the complex number 2 - 6i, we substitute a = 2 and b = -6, giving us: \[ \left| 2 - 6i \right| = \sqrt{2^2 + (-6)^2} = \sqrt{4 + 36} = \sqrt{40} = 2\sqrt{10} \].
This helps measure the magnitude of the number.
Plotting Complex Numbers
When plotting complex numbers, the real part determines the horizontal position (x-coordinate), and the imaginary part determines the vertical position (y-coordinate).
To plot a complex number like 2 - 6i on the complex plane, simply start at the origin (0,0).
Move 2 units to the right because the real part is 2, and then move 6 units down because the imaginary part is -6.
This point represents the complex number 2 - 6i on the complex plane.
Graphing multiple complex numbers this way allows us to see patterns and relationships between them.
Real and Imaginary Parts
Complex numbers consist of two parts: the real part and the imaginary part. A complex number is expressed in the form \((a + bi)\), where:
  • a is the real part
  • bi is the imaginary part, and i is the imaginary unit (\(i^2 = -1\))
For the complex number 2 - 6i, the real part is 2 and the imaginary part is -6i.
These two parts can be manipulated separately, making complex number arithmetic manageable.
Understanding these components is crucial for working with complex numbers in various mathematical, engineering, and physics applications.

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