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Chapter 6: Problem 11

Graph each complex number, and find its absolute value. \(-2+2 i \sqrt{3}\)

Short Answer

Expert verified
The absolute value is .

Step by step solution

01

- Write the Complex Number in Standard Form

The given complex number is . Verify that it is already in the form of a real part plus an imaginary part: ng complex number is: where and .
02

- Plot the Complex Number

Plot the complex number on a complex plane. The x-axis represents the real part and the y-axis represents the imaginary part. Since the real part is -2, move 2 units to the left. Since the imaginary part is , move up units from the origin.
03

- Find the Absolute Value

The absolute value of a complex number is given by the formula . Substitute the given values into the formula: . Then calculate: . Simplifying inside the square root: . Finally, .

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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 visualize complex numbers. Instead of thinking of them as just numbers, we can think of them as points on a plane. Similar to the Cartesian coordinate system, the x-axis represents the real part of the complex number, while the y-axis represents the imaginary part. For a complex number like \(-2 + 2i \sqrt{3}\), you would:
  • Move 2 units to the left on the real axis (x-axis)
  • Move up \(2 \sqrt{3}\) units on the imaginary axis (y-axis)
This point on the plane helps us see the number's position in a very visual way.
absolute value of complex numbers
The absolute value (or modulus) of a complex number is a measure of its distance from the origin (0, 0) on the complex plane. You can think of it like finding the length of the hypotenuse in a right-angled triangle. If we have a complex number \(a + bi\), its absolute value is calculated using the formula: \(|a + bi| = \sqrt{a^2 + b^2}\). For example, given the complex number \(-2 + 2i \sqrt{3}\):
  • \textbackslash( a = -2 \textbackslash)
  • \textbackslash( b = 2 \sqrt{3} \textbackslash)
Plug these into the formula to get: \(\sqrt{(-2)^2 + (2 \sqrt{3})^2} = \sqrt{4 + 12} = \sqrt{16} = 4\).
standard form of complex numbers
The standard form of a complex number is written as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit (where \(i^2 = -1\). This form makes it easy to identify the real part and the imaginary part of the number. Given the complex number -2 + 2i鈭�3, it's already in standard form. Here:
  • a = -2 (real part)
  • b = 2鈭�3 (imaginary part)
This clear separation helps in plotting the complex number on the complex plane and also in calculating other properties, like the absolute value.

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