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Chapter 7: Problem 15

Graph each complex number, and find its absolute value. $$3+3 i$$

Short Answer

Expert verified
\|3 + 3i\| = 3\sqrt{2}

Step by step solution

01

Plot the complex number

In the complex plane, plot the complex number by identifying the real part and the imaginary part. For the complex number \(3 + 3i\), the real part is 3 and the imaginary part is 3. Plot this point at \(3, 3\).
02

Understand the absolute value

The absolute value (or modulus) of a complex number \(a + bi\) is found using the formula \(|a + bi| = \sqrt{a^2+b^2}\). This represents the distance of the complex number from the origin in the complex plane.
03

Calculate the absolute value

For the complex number \(3 + 3i\), apply the formula: \(|3 + 3i| = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}\).

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Key Concepts

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

Complex Numbers
A complex number is a combination of a real number and an imaginary number. It is generally written in the form of \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. Here, \(i\) represents the imaginary unit, which is defined by the property \(i^2 = -1\).
  • Real Part: This is the 'a' in the complex number \(a + bi\).
  • Imaginary Part: This is the 'bi' in the complex number \(a + bi\), where 'b' is a real number and 'i' is the imaginary unit.
To grasp this better, consider the example provided in the exercise: the complex number 3+3i. Here, \(3\) is the real part and \(3i\) is the imaginary part.
Modulus
The modulus, also known as the absolute value, of a complex number determines its distance from the origin in the complex plane. It is found using the formula \[|a + bi| = \sqrt{a^2 + b^2}\].
The modulus gives a non-negative value regardless of the signs of 'a' and 'b'.
  • For the complex number 3 + 3i, the modulus is calculated as follows:
  • First, square both the real part and the imaginary part: \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}.
  • This represents the distance of the complex number 3 + 3i from the origin.
Understanding the modulus helps in visualizing the magnitude of complex numbers and their locations on the complex plane.
Complex Plane
The complex plane, also known as the Argand plane, is a two-dimensional plane used to visualize complex numbers. The horizontal axis is known as the real axis, and the vertical axis is known as the imaginary axis.
When plotting a complex number \(a + bi\), 'a' (the real part) is represented along the horizontal axis, and 'bi' (the imaginary part) is represented along the vertical axis.
  • To plot 3 + 3i from the exercise: start at the origin (0,0).
  • Move 3 units along the horizontal axis (real part).
  • Then, move 3 units upwards along the vertical axis (imaginary part).
  • The point where these movements intersect is the location of the complex number 3 + 3i on the complex plane (3,3).
Visualizing complex numbers this way helps in understanding their behavior and how they interact with each other geometrically.

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